A Short Course on Duality, Adjoint Operators, Green's Functions, and ...

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A Short Course on Duality, Adjoint Operators, Green’s Functions, and A Posteriori Error Analysis Donald J. Estep [email protected]

Department of Mathematics Colorado State University Fort Collins, CO 80523

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Outline of this course This course has four parts: • Basic background and theory of the concepts of duality,

adjoint operators, and Green’s functions • Development of an approach to a posteriori error analysis

and adaptive error control based on a generalization of the Green’s function • An application of these ideas to using the effective domain

of influence in elliptic problems to form a decomposition of a solution • Extension of the ideas to nonlinear problems.

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Duality, Adjoint Operators, and Green’s Functions

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The Green’s Function In the classic problem, u solves ( −∆u = f, x ∈ Ω, u = 0, x ∈ ∂Ω, where Ω is a domain in Rd with boundary ∂Ω. The Green’s function φ satisfies ( −∆φ(y; x) = δy (x), x ∈ Ω, φ(y; x) = 0, x ∈ ∂Ω, δy is the delta function at a point y ∈ Ω.

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The Green’s Function Function representation formula: u(y) =

Z

Z

δy (x)u(x) dx = −∆φ(y; x)u(x) dx Ω Ω Z Z = φ(y; x) · −∆u(x) dx = φ(y; x)f (x) dx. Ω

Ω

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Background in linear algebra X is a vector space with norm k k over the real numbers

An important property of the spaces depends on the notion of a Cauchy sequence: Definition A sequence {xn } in X is a Cauchy sequence if we can make the distance between elements in the sequence arbitrarily small by restricting the indices to be large. More precisely, for every > 0 there is an N such that kxn − xm k < for all n, m > N . This is a computable criterion for checking convergence.

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Background in linear algebra Example Consider the sequence {1/n}∞ n=1 in [0, 1]. It is a Cauchy sequence since 1 2 − 1 = m − n ≤ 2 max{m, n} = n m mn mn min{m, n}

can be made arbitrarily small by taking m and n large. It converges to 0 which is in [0, 1].

Sequences that converge are Cauchy sequences. Cauchy sequences do not necessarily converge. Example The sequence {1/n}∞ n=1 is a Cauchy sequence in (0, 1) but does not converge in (0, 1). Duality, Adjoints, Green’s Functions – p. 7/304

Background in linear algebra Spaces in which Cauchy sequences converge are greatly preferred. Definition A Banach space is a vector space with a norm such that every Cauchy sequence converges to a limit in the space. We also say the space is complete.

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Background in linear algebra Example Rn with the norms defined for x = (x1 , · · · , xn )> , kxk1 = |x1 | + · · · + |xn | 2 2 1/2 kxk2 = |x1 | + · · · + |xn | kxk∞ = max |xi |.

are all Banach spaces. We use k k = k k2 unless noted otherwise.

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Background in linear algebra Spaces of functions: Definition For an interval [a, b], the space of continuous functions is denoted C([a, b]), where we take the maximum norm kf k = max |f (x)|. a≤x≤b

C 1 ([a, b]) denotes the space of functions that have continuous first derivatives on [a, b], where we use the norm kf k = max |f (x)| + max |f 0 (x)|. a≤x≤b

a≤x≤b

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Background in linear algebra Spaces of functions: Definition For 1 ≤ p ≤ ∞, Lp (Ω) = {f : f is measurable on Ω and kf kp < ∞}

where for 1 ≤ p < ∞, kf kp =

Z

Ω

kf kp dx

1/p

, 1 ≤ p < ∞, and

kf k∞ = ess supΩ kf k.

L2 is particularly important.

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Background in linear algebra Theorem The Lp spaces and C([a, b]) are Banach spaces.

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Background in linear algebra Consider two vector spaces X and Y with norms k kX and k kY Definition A map or operator L from X to Y is a rule or association that assigns to each x in X a unique element y in Y . Definition A map L : X → Y is linear if L(αx1 + βx2 ) = αL(x1 ) + βL(x2 ) for all numbers α, β and x1 , x2 in X .

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Background in linear algebra Example Every linear map from Rm to Rn is obtained by multiplying vectors in Rm by a n × m matrix, i.e., they have the form Ax, where A is a n × m matrix. Example Differentiation is a linear map from C 1 ([a, b]) to C([a, b]).

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Background in linear algebra The maps we consider also have to behave continuously. Definition A map L : X → Y is continuous if for every sequence {xn } in X that converges to a limit x in X , i.e., xn → x, we have L(xn ) → L(x). Example Linear maps from Rm to Rn are continuous.

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Background in linear algebra There is an equivalent property for linear maps. Definition A linear map L : X → Y is bounded if there is a constant C > 0 such that kLxkY ≤ CkxkX for all x in X . Theorem A linear map between normed vector spaces is continuous if and only if it is bounded.

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Background in linear algebra The set of all linear transformations between two vector spaces X and Y is a vector space. Definition If X and Y are normed vector spaces, we use L(X, Y ) to denote the vector space of all bounded linear maps from X to Y . L(X, Y ) is a normed vector space under the operator norm kLxkY kLk = sup kLxkY = sup . x6=0 kxkX kxkX =1

(1)

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Background in linear algebra Example If the linear transformation L is given by the n × n matrix A, then kLk1 = kAk1 = max

1≤j≤n

kLk2 = kAk2 =

q

n X

|aij |

i=1

σ(AT A)

kLk∞ = kAk∞ = max

1≤i≤n

n X

|aij |

j=1

where σ(A> A) is the spectral radius of A> A.

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Background in linear algebra Theorem If X and Y are normed vector spaces and Y is complete, then L(X, Y ) is complete.

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Linear functionals and dual spaces The concept of duality starts with linear functionals. Definition A linear functional on a vector space X is a linear map from X to R. Example Let v in Rn be fixed. The map F (x) = v · x = (x, v) is a linear functional on Rn . Example Rb Consider C([a, b]). Both I(f ) = a f (x) dx and F (f ) = f (y) for a ≤ y ≤ b are linear functionals.

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Linear functionals and dual spaces A linear functional provides a “low dimensional snapshot” of a vector. Example In the example above with F (x) = (x, v), consider v = ei , the ith standard basis function. Then F (x) = xi where x = (x1 , · · · , xn ). We can take v = (1, 1, · · · , 1)/n and compute the average of the components of a given input vector.

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Linear functionals and dual spaces Example δy gives a linear functional on sufficiently smooth, real valued functions via Z F (u) = u(y) = δy (x)u(x) dx. Ω

Another linear functional is the average value of an integrable function, Z 1 u(x) dx. F (u) = vol. of Ω Ω We can view the formulas defining the Fourier coefficients of a function as a set of linear functionals. Duality, Adjoints, Green’s Functions – p. 22/304

Linear functionals and dual spaces We are interested in the continuous linear functionals. Definition If X is a normed vector space, the space L(X, R) of bounded linear functionals on X is called the dual space of or on or to X , and is denoted by X ∗ . The dual space is a normed vector space under the dual norm defined for y ∈ X ∗ as kykX ∗

|y(x)| . = sup |y(x)| = sup x∈X x∈X kxk kxkX =1

x6=0

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Linear functionals and dual spaces Example Consider X = Rn . Every vector v in Rn is associated with a linear functional Fv (·) = (·, v). This functional is clear bounded since |(x, v)| ≤ kvk kxk (The “C ” in the definition is kvk). A classic result in linear algebra is that all linear functionals on Rn have this form, i.e., we can make the identification (Rn )∗ ' Rn .

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Linear functionals and dual spaces Example Rb For C([a, b]), consider I(f ) = a f (x) dx. It is easy to compute kIkC([a,b])∗ =

by looking at a picture.

sup f ∈C([a,b]) max |f |=1

Z b f (x) dx a

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Linear functionals and dual spaces 1

possible functions

b

a

-1

Computing the dual norm of the integration functional. The maximum value for I(f ) is clearly given by f = 1 or f = −1, and we get kIkC([a,b])∗ = b − a. Duality, Adjoints, Green’s Functions – p. 26/304

Linear functionals and dual spaces Example Recall Hölder’s inequality for f ∈ Lp (Ω) and g ∈ Lq (Ω) with 1 1 + = 1 for 1 ≤ p, q ≤ ∞ is p q kf gkL1 (Ω) ≤ kf kLp (Ω) kgkLq (Ω) .

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Linear functionals and dual spaces Each g in Lq (Ω) is associated with a bounded linear functional 1 1 p on L (Ω) when + = 1 and 1 ≤ p, q ≤ ∞ by p q Z F (f ) = g(x)f (x) dx. Ω

We can “identify” (Lp )∗ with Lq when 1 < p, q < ∞, The cases p = 1, q = ∞ and p = ∞, q = 1 are trickier. The case L2 is special in that we can identify (L2 )∗ with L2 .

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Linear functionals and dual spaces The dual space is the collection of “reasonable” possible samples. An important characteristic of a dual space is how much we can reveal about a vector by considering samples in the dual space. Example By considering the set of n functionals corresponding to taking the inner product with {e1 , · · · , en }, we can “reconstruct” any given vector in Rn by looking at the functional values. The question of whether or not we can “recover” a vector u completely by computing sufficiently many linear functionals depends heavily on properties of the underlying space. In practice, we will often be content with one or just a few “snapshots”. Duality, Adjoints, Green’s Functions – p. 29/304

Linear functionals and dual spaces We might wonder about the number of bounded linear functionals that exist on an arbitrary normed vector space. The celebrated Hahn-Banach theorem says there is a great abundance of them. Let x0 6= 0 be a fixed element of Banach space X . The set X0 = {αx0 : α ∈ R} forms a vector subspace of X . The linear functional F (αx0 ) = α is defined on X0 and is bounded since |F (αx0 )| = |α| = kαx0 k/kxk. So, we have found a bounded linear functional on a subspace of X. Can we extend this to be defined on all of X ? Duality, Adjoints, Green’s Functions – p. 30/304

Linear functionals and dual spaces Issues: • What does it mean to extend? • Does the norm increase upon extension? • What if there is an infinite number of extensions involved?

Theorem Hahn-Banach Let X be a Banach space and X0 a subspace of X . Suppose that F0 (x) is a bounded linear functional defined on X0 . There is a linear functional F defined on X such that F (x) = F0 (x) for x in X0 and kF k = kF0 k.

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Linear functionals and dual spaces Dual spaces are connected to distribution theory and weak convergence. The dual space can be better behaved than the original normed vector space. Theorem If X is a normed vector space over R, then X ∗ is a Banach space (whether or not X is a Banach space).

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Linear functionals and dual spaces Useful notation: Definition If x is in X and y is in X ∗ , we denote the value y(x) =< x, y > .

This is called the bracket notation.

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Linear functionals and dual spaces The norms on X and its dual X ∗ are closely related. Recall that if y ∈ X ∗ , then kyk

X∗

|y(x)| = sup |y(x)| = sup . kxk x∈X x∈X kxkX =1

x6=0

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Linear functionals and dual spaces Definition The generalized Cauchy inequality is |< x, y >| ≤ kxkX kykX ∗ ,

x ∈ X, y ∈ X ∗ .

Theorem If X is a Banach space, then kxkX

|y(x)| = sup = ∗ kykX ∗ y∈X y6=0

sup |y(x)| y∈X ∗ kykX ∗ =1

for all x in X .

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Hilbert spaces and duality Example Rn with the standard Euclidean norm k k = k k2 can be identified with its dual space. L2 can be identified with its dual space.

Both of these spaces are Hilbert spaces.

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Hilbert spaces and duality One way to get a normed vector space is to place an inner product (i.e., dot product) on the space. Theorem If X has an inner product (x, y), then it is a normed vector space with norm kxk = (x, x)1/2 for x in X . Definition A vector space with an inner product that is a Banach space with respect to the associated norm is called a Hilbert space. Example Rn with k k = k k2 and L2 are both Hilbert spaces.

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Hilbert spaces and duality If X is a Hilbert space with inner product (x, y), then each y ∈ X determines a linear functional Fy (x) =< x, y >= (x, y) for x in X . This functional is bounded by Cauchy’s inequality, |(x, y)| ≤ kxk kyk. The Riesz Representation theorem says this is the only kind of linear functional on a Hilbert space. Theorem Riesz Representation For every bounded linear functional F on a Hilbert space X , there is a unique element y in X such that F (x) = (x, y), all x ∈ X and kykX ∗

|F (x)| = sup . x∈X kxk x6=0

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Hilbert spaces and duality A Hilbert space is isometric to its dual space. Definition Two normed vector spaces X and Y are isometric if there is a linear 1-1 and onto map L : X → Y such that kL(x)kY = kxkX for all x in X . We often replace the bracket notation and the generalized Cauchy inequality by the inner product and the “real” Cauchy inequality without comment.

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Hilbert spaces and duality The Sobolev spaces are Hilbert spaces based on L2 . Example For k = 1, 2, 3, · · · , we define H k (Ω) to be the distribution functions in L2 (Ω) whose partial derivatives of order k and less are also distributions in L2 (Ω).

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Hilbert spaces and duality Definition index notation: For α = (α1 , · · · , αn ) with integer coefficients, we define ∂ ∂ ∂ |α| α D= ,··· , , D = ∂x1 ∂xn ∂xα1 1 · · · ∂xαnn with |α| = α1 + · · · + αn . H k (Ω) = {u, Dα u ∈ L2 (Ω), |α| ≤ k}. The inner products and norms are X 1/2 (u, v)k = (Dα u, Dα v), kukk = (u, u)k . |α|≤k

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Hilbert spaces and duality The Riesz Representation theorem says that every linear functional on H k has the form (u, v)k for some fixed v in H k . Defining the dual space to H k runs into subtle difficulties due to a collision with the requirements for using distribution theory. We define the dual spaces to a subspace. k−1 ∂ u ∂u = ··· = = 0 on ∂Ω , H0k (Ω) = u ∈ H k (Ω) : u = k−1 ∂n ∂n

where ∂/∂n denotes the normal derivative on the boundary ∂Ω. We let H −k (Ω) is the set of all linear functionals on H0k (Ω).

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Adjoint operators - definition We now explain how a linear transformation between two normed vector spaces X and Y is naturally associated with another linear transformation between Y ∗ and X ∗ . This is the infamous adjoint operator.

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Adjoint operators - definition Suppose L ∈ L(X, Y ) is a bounded linear transformation. For each y ∗ ∈ Y ∗ , y ∗ ◦ L(x) = y ∗ (L(x)) =< Lx, y ∗ >

assigns a number to each x ∈ X , hence defines a functional F (x). F (x) is clearly linear. It is also bounded since |F (x)| = |y ∗ (L(x))| ≤ ky ∗ kY ∗ kL(x)kY ≤ ky ∗ kY ∗ kLk kxkX .

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Adjoint operators - definition There is an x∗ ∈ X ∗ such that y ∗ (L(x)) = x∗ (x) for all x ∈ X . x∗ is unique.

Thus, to each y ∗ ∈ Y ∗ , we have assigned a unique x∗ ∈ X ∗ and thus have defined a linear transformation L∗ : Y ∗ → X ∗ . We can write these relations as y ∗ (L(x)) = L∗ y ∗ (x)

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Adjoint operators - definition Using the bracket notation, < L(x), y ∗ >=< x, L∗ (y ∗ ) >

x ∈ X, y ∗ ∈ Y ∗ .

Definition This is called the bilinear identity. It defines the adjoint operator L∗ : Y ∗ → X ∗ associated to a bounded linear transformation L : X → Y . Note that we have defined the adjoint transformation via sampling by elements in the dual space.

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Adjoint operators - definition Example Let X = Rm and Y = Rn , where we take the standard inner product and norm. By the Riesz Representation theorem, the bilinear identity for L ∈ L(Rm , Rn ) reads (Lx, y) = (x, L∗ y),

x ∈ Rm , y ∈ Rn .

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Adjoint operators - definition There is a unique n × m matrix A so that if y = L(x) = Ax where 

a11  . A =  ..

···

an1 · · ·



a1m ..  , . 

anm





y1 . y =  ..  , yn





x1  .  x =  ..  xm

and yi =

m X

aij xj ,

1 ≤ i ≤ n.

j=1

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Adjoint operators - definition For a linear functional y ∗ = (y1∗ , · · · , yn∗ )> ∈ Y ∗

∗ ∗

P



∗

m j=1 a1j xj

  ∗ ∗   L y (x) = y (L(x)) = (y1 , · · · , yn ),  =

m X

y1∗ a1j xj + · · ·

j=1

=

m X n X j=1 i=1



 ..  .  Pm j=1 anj xj m X

yn∗ anj xj

j=1

yi∗ aij

xj

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Adjoint operators - definition L∗ (y ∗ ) is given by the inner product with y˜ = (˜ y1 , · · · , y˜m )> where n X y˜j = yi∗ aij . i=1

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Adjoint operators - definition The matrix A∗ of L∗ is  ∗   ∗ a11 · · · a1n a11 a21 · · ·  ..   .. . ∗ . A = . . = . a∗m1 · · · a∗mn a1m a2m · · ·



an1 ..  = A> . . 

anm

The bilinear identity is y > Ax = x> A> y

using the fact that (x, y) = (y, x).

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Adjoint operators - definition The following theorem is crucial. Theorem L∗ ∈ L(Y ∗ , X ∗ ) and kL∗ k = kLk. proof |L∗ y ∗ (x)| ≤ ky ∗ kY ∗ kLkkxkX .

Therefore, kL∗ y ∗ kX ∗

|L∗ y ∗ (x)| ≤ ky ∗ kY ∗ kLk, = sup kxk x6=0

which implies that L∗ ∈ L(Y ∗ , X ∗ ) and kL∗ k ≤ kLk. Duality, Adjoints, Green’s Functions – p. 52/304

Adjoint operators - definition To show the reverse inequality, we prove that kLxkY ≤ kL∗ k kxkX ,

x ∈ X.

The bilinear identity implies that |y ∗ (Lx)| ≤ kL∗ y ∗ kX ∗ kxkX ≤ kL∗ kky ∗ kY ∗ kxkX

and

|y ∗ (Lx)| ∗ sup ≤ kL k kxkX , ∗ y ∗ 6=0 ky kY ∗

x ∈ X.

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Adjoint operators - definition Properties of the adjoint operator: Theorem Let X , Y , and Z be normed linear spaces. Then, 0∗ = 0 (L1 + L2 )∗ = L∗1 + L∗2 , (αL)∗ = αL∗ ,

all L1 , L2 ∈ L(X, Y )

all α ∈ R, L ∈ L(X, Y )

If L2 ∈ L(X, Y ) and L1 ∈ L(Y, Z), then L1 L2 ∈ L(X, Z), (L1 L2 )∗ ∈ L(Z ∗ , X ∗ ), and (L1 L2 )∗ = L∗2 L∗1 .

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Adjoint operators - definition We are often deal with linear operators that are not defined on the entire space. Example Consider D = d/dx on X = C([0, 1]). This linear map is only defined on the subspace C 1 ([0, 1]).

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Adjoint operators - definition Definition Let X and Y be normed vector spaces. A map L that assigns to each x in a subset D(L) of X a unique element y in Y is called a map or operator with domain D(L). L is linear if (1) D(L) is a vector subspace of X and (2) L(αx1 + βx2 ) = αL(x1 ) + βL(x2 ) for all α, β ∈ R and x1 , x2 ∈ D(L).

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Adjoint operators - definition We define the dual of a linear operator by examining its behavior on its domain. We want L∗ y ∗ (x) = y ∗ (Lx)

all x ∈ D(L).

We say that y ∗ ∈ D(L∗ ) if there is an x∗ ∈ X ∗ such that x∗ (x) = y ∗ (Lx),

all x ∈ D(L).

The existence of x∗ is no longer automatic. When x∗ exists, we define L∗ y ∗ = x∗ . Duality, Adjoints, Green’s Functions – p. 57/304

Adjoint operators - definition For this to work, x∗ must be unique. In other words, x∗ (x) = 0 for all x ∈ D(L) should imply x∗ = 0. This depends on the “size” of D(L). Definition A subspace A of a normed linear space X is dense if every point in X is either in A or the limit of a sequence of points in A. Example The rational numbers are dense in the real numbers and the polynomials are dense in C([a, b]).

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Adjoint operators - definition We can define L∗ for a linear operator L : X → Y provided D(L) is dense in X . We define D(L∗ ) to be those y ∗ ∈ Y ∗ for which there is an x∗ ∈ X ∗ with x∗ (x) = y ∗ (Lx) for all x ∈ X . This x∗ is unique and L∗ y ∗ = x∗ . The Hahn-Banach theorem implies that if there is a C such that |y ∗ (Lx)| ≤ Ckxk for all x ∈ D(L), then y ∗ ∈ D(L∗ ).

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Adjoint operators - motivation Common problem: Given normed vector spaces X and Y , an operator L(X, Y ), and b ∈ Y , find x ∈ X such that Lx = b.

We explain the role of the adjoint in solving this kind of problems.

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Adjoint operators - motivation Definition The set of b for which there is a solution is called the range, R(L), of L. The set of x for which L(x) = 0 is called the null space, N (L), of L. 0 is always in N (L).

If it is the only element in N (L), then Lx = b can have at most one solution.

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Adjoint operators - motivation Theorem N (L) is a subspace of X and R(L) is a subspace of Y .

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Adjoint operators - motivation If y ∈ R(L), there is an x with Lx = y . For y ∗ ∈ Y ∗ , y ∗ (Lx) = y ∗ (y).

By the definition of the adjoint, L∗ y ∗ (x) = y ∗ (y).

If y ∗ ∈ N (L∗ ), then y ∗ (y) = 0. A necessary condition that y ∈ R(L) is that y ∗ (y) = 0 for all y ∗ ∈ N (L∗ ).

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Adjoint operators - motivation Is this sufficient? We require just one condition. Definition A subset A of a normed vector space X is closed if every sequence {xn } in A that has a limit in X has its limit in A. Theorem Let X and Y be normed linear spaces and L ∈ L(X, Y ). A necessary condition that y ∈ R(L) is y ∗ (y) = 0 for all y ∗ ∈ N (L∗ ). This is a sufficient condition if R is closed in Y .

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Adjoint operators - motivation Example Suppose that L ∈ L(X, Y ) is associated with the n × m matrix A, i.e., L(x) = Ax. The necessary and sufficient condition for the solvability of Ax = b is that b is orthogonal to all linearly independent solutions of AT y = 0.

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Adjoint operators - motivation Example In the case X is a Hilbert space and L ∈ L(X, Y ), then necessarily R(L∗ ) ⊂ N (L)⊥ , where S ⊥ is the subspace of vectors that are orthogonal to a subspace S . If R(L∗ ) is “large”, then N (L)⊥ must be “large” and N (L) must be “small”. The existence of sufficiently many solutions of the homogeneous adjoint equation implies there is at most one solution of Lx = b for a given b.

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Adjoint operators - motivation Example The setting of general partial differential equations: Theorem Holmgren Uniqueness The generalized initial value problem consisting of the equation L(u) =

X

Aα (x)Dα u = f (x),

x ∈ Rn ,

|α|≤m

where {Aα } and f are analytic functions, together with the data Dβ u(x) = gβ (x),

|β| ≤ m − 1, x ∈ S

given on an analytic noncharacteristic surface S , has at most one solution in a neighborhood of S . Duality, Adjoints, Green’s Functions – p. 67/304

Adjoint operators - motivation The adjoint has an important role in the solution of a general n × m system Ax = b, where A is a n × m matrix, x ∈ Rm , and b ∈ Rn . The reason to dwell on such problems is that differential operators do not tend to be “square”. Example y 00 = d2 y/dx2 requires two boundary conditions to define a problem for the associated differential equation that has a unique solution. We may want to study the differential operator without any boundary conditions, or more or less than two conditions.

Duality, Adjoints, Green’s Functions – p. 68/304

Adjoint operators - motivation Example Divergence, div u

=

∂un ∂u1 + ··· + , ∂x1 ∂xn

associates a scalar with a given vector function, and the associated differential equation is very “under-determined”. The gradient, grad u

=

∂u1 ∂un ,··· , ∂x1 ∂xn

,

associates a vector field with a given scalar function, and the associated equations are very “over-determined”.

Duality, Adjoints, Green’s Functions – p. 69/304

Adjoint operators - motivation Consider the n × m system Ax = b, where A is a n × m matrix, x ∈ Rm , and b ∈ Rn . We enlarge the system by adding the adjoint m × n system A> y = c, where y and c are independent of x and b. The new problem is an (n + m) × (n + m) symmetric problem Sz = d,

with z=

!

y , x

d=

!

b , c

S=

0 A>

! A . 0

Duality, Adjoints, Green’s Functions – p. 70/304

Adjoint operators - motivation Since S is symmetric, it is diagonalizable with eigenvalues satisfying Av = λu A∗ u = λv

or

AA> u = λ2 u A> Av = λ2 v

The eigenvalues of S are the singular values of A and last two equations give the left and right singular vectors of A.

Duality, Adjoints, Green’s Functions – p. 71/304

Adjoint operators - motivation Consequences: • The theorem on the adjoint condition for solvability falls out

right away. • It yields a “natural” definition of a solution or gives

conditions for a solution to exist in the over-determined and under-determined cases. • It also gives a way of determining the condition of the

solution process.

Duality, Adjoints, Green’s Functions – p. 72/304

Adjoint operators - motivation One interesting observation is that there is a reciprocal relationship between the degree of over/under-determination of the original system and the adjoint system. The more over-determined the original system, the more under-determined the adjoint system, and so forth.

Duality, Adjoints, Green’s Functions – p. 73/304

Adjoint operators - motivation Example Consider 2x!1 + x2 = 4, where L : R2 → R. L∗ : R → R2 is given

by L∗ =

2 . The extended system is 1

     4 0 2 1 y1      2 0 0 x1  = c1  , c2 x2 1 0 0

from which we see that 2c1 = c2 is required in order to have a solution.

Duality, Adjoints, Green’s Functions – p. 74/304

Adjoint operators - motivation Example On the other hand, if the problem is 2x1 + x2 = 4 x2 = 3,

with L : R2 → R2 , then there is a unique solution. The extended system is      c1 y1 0 0 2 1 0 0 0 1  y  c    2   2     =  ,  2 0 0 0 x1   4  3 x2 1 1 0 0 where we can specify any values for c1 , c2 .

Duality, Adjoints, Green’s Functions – p. 75/304

Adjoint operators - motivation In the under-determined case, we can eliminate the deficiency by posing the method of solution AA> y = b x = A> y.

or

L(L∗ (y)) = b x = L∗ (y).

Duality, Adjoints, Green’s Functions – p. 76/304

Adjoint operators - motivation This works for differential operators Example Consider the under-determined problem div F

= ρ.

The adjoint to div is -grad modulo boundary conditions. If F = grad u, where u is subject to the boundary condition u(00 ∞00 ) = 0, then we obtain the “square”, well-determined problem div grad u = ∆u = −ρ, which has a unique solution because of the boundary condition.

Duality, Adjoints, Green’s Functions – p. 77/304

Adjoint operators - motivation Example Consider Ax = b A> φ = ei

where A is a n × n invertible matrix. There are no constraints on the data for the adjoint problem. We have specified the ith standard basis vector of Rn . xi = (x, ei ) = (x, A> φ) = (Ax, φ) = (b, φ). y is the discrete Green’s vector associated to Ax = b. Duality, Adjoints, Green’s Functions – p. 78/304

Adjoint operators - computation The computation of the adjoint to a general differential operator is not easy. A tedious, formal approach: 1. Take the problem for the linear operator, including any boundary and/or initial conditions imposed with the operator, and discretize it. 2. Extract the matrix from the resulting discrete system, and compute its transpose. 3. Let the discretization parameter tend to its limit and determine the differential operator that is approached by the transposed matrix. Determination of the adjoint of a differential operator is heavily influenced by the boundary and/or initial conditions, as well as the underlying spaces. Duality, Adjoints, Green’s Functions – p. 79/304

Adjoint operators - computation Example A standard difference approximation of ( −u00 (x) = f (x), 0 < x < 1, u(0) = 0, u(1) = 0, yields the matrix 



2 −1 0 −1 2 −1      1  .. .. .. , . . .   2 h    −1 2 −1 0 −1 2

The differential operator is self-adjoint. Duality, Adjoints, Green’s Functions – p. 80/304

Adjoint operators - computation Example If we change the boundary conditions to ( −u00 (x) = f (x), 0 < x < 1, u(0) = 0, u0 (0) = 0, we get a triangular matrix after discretization. The adjoint corresponds to a problem ( −v 00 (x) = g(x), 0 < x < 1, v(1) = 0, v 0 (1) = 0,

Duality, Adjoints, Green’s Functions – p. 81/304

Adjoint operators - computation Example ( u0 = f (x), u(0) = 0

0 < x < 1,

⇒

( −v 0 = g(x), v(1) = 0,

1 > x > 0,

Example ( −u00 = f (x), 0 < x < 1, no boundary conditions ( −v 00 = g(x), ⇒ v(0) = v 0 (0) = v(1) = v 0 (1) = 0.

0 < x < 1,

Duality, Adjoints, Green’s Functions – p. 82/304

Adjoint operators - computation In the case of L2 , we can use the bilinear identity < Lu, v ∗ >=< u, L∗ v ∗ >

all u ∈ X, v ∗ ∈ Y ∗ ,

which is (Lu, v) = (u, L∗ v)

all suitable u, v ∈ L2 (Ω).

Definition We say that we are evaluating the bilinear identity when we compute < Lu, v ∗ > − < u, L∗ v ∗ >= (Lu, v) − (u, L∗ v)

for some suitable functions u and v .

Duality, Adjoints, Green’s Functions – p. 83/304

Adjoint operators - computation • Since we are considering differential operators, these will

not be defined on all of L2 (Ω), but only a subset of sufficiently smooth functions. • The L2 inner product involves integration over the domain,

and we can interpret the process producing the bilinear identity as a succession of integration by parts. This yields terms that involve the integrals over the boundary of Ω of the functions involved as well as certain derivatives.

Duality, Adjoints, Green’s Functions – p. 84/304

Adjoint operators - computation Computing the adjoint using the bilinear identity proceeds in two stages. First, we compute a formal adjoint neglecting all boundary terms by assuming that the functions involved have compact support inside Ω. Definition A function on a domain Ω has compact support in Ω if it vanishes identically outside a bounded set contained inside Ω.

Duality, Adjoints, Green’s Functions – p. 85/304

Adjoint operators - computation 1. Take the differential operator applied to a smooth function with compact support and multiply by a smooth test function with compact support 2. Integrate over the domain 3. Integrate by parts to move all derivatives onto the test function while ignoring boundary terms. Functions that have compact support are identically zero anywhere near the boundary and any boundary terms arising from integration by parts will vanish.

Duality, Adjoints, Green’s Functions – p. 86/304

Adjoint operators - computation Definition Let L be a differential operator. The formal adjoint L∗ is the differential operator that satisfies Z Z (Lu, v) = (u, L∗ v) Lu · v dx = u · L∗ v dx Ω

Ω

for all sufficiently smooth u and v with compact support in Ω.

Duality, Adjoints, Green’s Functions – p. 87/304

Adjoint operators - computation Example Consider d Lu(x) = − dx

d d a(x) u(x) + (b(x)u(x)) dx dx

on [0, 1]. Integration by parts neglecting boundary terms gives −

Z

0

1

d dx

d a(x) u(x) v(x) dx dx 1 Z 1 d d d a(x) u(x) v(x) dx − a(x) u(x)v(x) = dx dx dx 0 0 1 Z 1 d d d u(x) a(x) v(x) dx + u(x)a(x) v(x) , =− dx dx dx 0 0 Duality, Adjoints, Green’s Functions – p. 88/304

Adjoint operators - computation

Z

0

1

d (b(x)u(x))v(x) dx = − dx

Z

0

1

1 d u(x)b(x) v(x) dx+b(x)u(x)v(x) , dx 0

where all of the boundary terms vanish. Therefore, L∗ v = −

d dx

a(x)

d d v(x) − b(x) (v(x)). dx dx

Duality, Adjoints, Green’s Functions – p. 89/304

Adjoint operators - computation In higher space dimensions, we use the divergence theorem. Example A general linear second order differential operator L in Rn can be written L(u) =

n X n X

aij

i=1 j=1

∂2u ∂xi ∂xj

+

n X i=1

∂u + cu, bi ∂xi

where {aij }, {bi }, and c are functions of x1 , x2 , · · · , xn . Then, L∗ (u) =

n X n X ∂ 2 (aij v) i=1 j=1

∂xi ∂xj

−

n X ∂(bi v) i=1

∂xi

+ cv.

Duality, Adjoints, Green’s Functions – p. 90/304

Adjoint operators - computation It can be verified directly that n X ∂pi , vL(u) − uL∗ (v) = ∂xi i=1

where n X ∂(aij v) ∂u aij v pi = −u + bi uv. ∂xj ∂xj j=1

The divergence theorem yields Z Z (vL(u) − uL∗ (v)) dx = Ω

p · n ds = 0,

∂Ω

where p = (p1 , · · · , pn ) and n is the outward normal in ∂Ω. Duality, Adjoints, Green’s Functions – p. 91/304

Adjoint operators - computation A typical term, va11

∂2u

∂ ∂u ∂u ∂ ∂(a11 v) ∂u = va − 11 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x21 ∂u ∂ ∂ ∂(a11 v) ∂ 2 (a11 v) = va11 − u +u ∂x1 ∂x1 ∂x1 ∂x1 ∂x21 = va11

yielding va11

∂2u ∂x21

−

∂ 2 (a11 v) u ∂x21

∂u ∂(a11 v) ∂ a11 v −u . = ∂x1 ∂x1 ∂x1

Duality, Adjoints, Green’s Functions – p. 92/304

Adjoint operators - computation Example Let L be a differential operator of order 2p of the form Lu =

X

|α|

α

|α|

α

(−1) D

β

β

aαβ (x)D u ,

|α|,|β|≤p

then ∗

L v=

X

(−1) D

aβα (x)D v ,

|α|,|β|≤p

and L is elliptic if and only if L∗ is elliptic.

Duality, Adjoints, Green’s Functions – p. 93/304

Adjoint operators - computation Some special cases. grad div

∗

∗

curl

Lu =

= −div

= −grad ∗

X

= curl aα (x)Dα u

|α|≤p

then ∗

L v=

X

(−1)|α| Dα (aα (x)v(x)).

|α|≤p

Duality, Adjoints, Green’s Functions – p. 94/304

Adjoint operators - computation Example If Lu = ρutt − ∇ · (a∇u) + bu

then L∗ = L. If Lu = ut − ∇ · (a∇u) + bu

then L∗ v = −vt − ∇ · (a∇v) + bv

where time runs “backwards”.

Duality, Adjoints, Green’s Functions – p. 95/304

Adjoint operators - computation This procedure also works for systems Example Let L(~u) = A~ux + B~uy + C~u,

where A, B , and C are n × n matrices, then L∗ (~v ) = (−A>~v )x − (B >~v )y + C > v,

so that ~v > L~u − ~u> L∗~v = ∇ · (~v > A~u, ~v > B~u).

Duality, Adjoints, Green’s Functions – p. 96/304

Adjoint operators - computation In the second stage of computing the adjoint, we deal with boundary conditions by removing the assumption that the functions involved have compact support. Integration by parts yields additional terms involving integrals over the boundary of the functions involved and their derivatives. We want to determine boundary conditions such that the bilinear identity still holds, e.g., such that any boundary terms that arise vanish.

Duality, Adjoints, Green’s Functions – p. 97/304

Adjoint operators - computation It turns out that the form of the boundary conditions imposed in the problem for L are important, but the values given for these conditions are not. If the boundary conditions are not homogeneous, we make them so for the purpose of determining the adjoint. Definition The adjoint boundary conditions posed on the formal adjoint operator are the minimal conditions required to make the boundary terms that appear when evaluating the bilinear identity for general smooth functions vanish.

Duality, Adjoints, Green’s Functions – p. 98/304

Adjoint operators - computation Some of the boundary terms that appear when evaluating the bilinear identity will vanish because of the boundary conditions imposed in the original problem. The point of this assumption is to make the formal adjoint serve as the true adjoint by pairing it with the correct boundary conditions.

Duality, Adjoints, Green’s Functions – p. 99/304

Adjoint operators - computation This definition can be made completely precise. Issues involved: • Placing conditions on the differential operator L so that

evaluating the bilinear identity for general smooth functions results in expressions involving only values on the boundary. • Making precise the meaning of “minimal conditions” needed

for the adjoint problem, and proving these always exist.

Duality, Adjoints, Green’s Functions – p. 100/304

Adjoint operators - computation Example Consider Newton’s equation of motion s00 (x) = f (x) with x = “time”, normalized with mass 1. First, suppose we assume s(0) = s0 (0) = 0, and 0 < x < 1. We have d (vs0 − sv 0 ) s v − sv = dx 00

00

and Z

0

1

1 (s v − sv ) dx = (vs − sv ) 0 . 00

00

0

0

Duality, Adjoints, Green’s Functions – p. 101/304

Adjoint operators - computation The boundary conditions imply the contributions at x = 0 vanish, while at x = 1 we have v(1)s0 (1) − v 0 (1)s(1).

To insure this vanishes, we must have v(1) = v 0 (1) = 0. These are the adjoint boundary conditions.

Duality, Adjoints, Green’s Functions – p. 102/304

Adjoint operators - computation Example Suppose instead we impose conditions such that at the mass returns to the origin with zero speed at x = 1. This gives four boundary conditions s(0) = s0 (0) = s(1) = s0 (1) = 0 on the original problem. In this situation, all of the boundary terms are zero and no boundary conditions will be imposed on the adjoint.

Duality, Adjoints, Green’s Functions – p. 103/304

Adjoint operators - computation Solving the over-determined problem. Example Based on the discussion above, we require the data f to be orthogonal to the solution of the adjoint problem v 00 = 0, which is v = a + bx. Hence, f must be orthogonal in L2 (0, 1) to 1 and x. Assume for example that f (x) = a + bx + cx2 . It is easy to see that (f, 1) = 0 and (f, x) = 0 forces a = c/6 and b = −c. Choosing c = 1 for example, means that f (x) = 1/6 − x + x2 .

Duality, Adjoints, Green’s Functions – p. 104/304

Adjoint operators - computation We solve the forward problem by integrating twice and using the boundary conditions at x = 0 to get a formula for the solution. The resulting solution satisfies the conditions at x = 1 as well.

Duality, Adjoints, Green’s Functions – p. 105/304

Adjoint operators - computation Example Since Z

Ω

(u∆v − v∆u) dx =

Z

∂Ω

∂v ∂u u −v ∂n ∂n

ds,

the Dirichlet and Neumann boundary value problems for the Laplacian are their own adjoints.

Duality, Adjoints, Green’s Functions – p. 106/304

Adjoint operators - computation Example Let Ω ⊂ R2 be bounded with a smooth boundary and let s = arclength along the boundary. Consider ( −∆u = f, x ∈ Ω, ∂u ∂u + ∂n ∂s = 0, x ∈ ∂Ω. Since Z Z (u∆v − v∆u) dx = Ω

∂v ∂v ∂u ∂u u − −v + ds, ∂n ∂s ∂n ∂s ∂Ω

the adjoint problem is ( −∆v = g, ∂v ∂v − ∂n ∂s = 0,

x ∈ Ω, x ∈ ∂Ω. Duality, Adjoints, Green’s Functions – p. 107/304

Green’s functions For simplicity, we consider ( Lu = f, suitable b.c. and i.v.,

x ∈ Ω, x ∈ ∂Ω,

where L is a linear differential operator. We specify the correct boundary and/or initial conditions so there is a unique solution.

Duality, Adjoints, Green’s Functions – p. 108/304

Green’s functions Definition The Green’s function satisfies ( L∗ φ(y, x) = δy (x), adjoint b.c. and i.v.,

x ∈ Ω, x ∈ ∂Ω,

where L∗ is the formal adjoint of L. We are solving the extended system,  Lu = f,    suitable b.c. and i.v.,  L∗ φ(y, x) = δy (x),    adjoint b.c. and i.v.,

x ∈ Ω, x ∈ ∂Ω, x ∈ Ω, x ∈ ∂Ω.

Duality, Adjoints, Green’s Functions – p. 109/304

Green’s functions Based on the bilinear identity, we obtain a representation formula for the value of the solution of the original problem at a point y ∈ Ω, u(y) = (u, δy ) = (y, L∗ φ) = (Lu, φ) = (f, φ).

The imposition of the adjoint boundary conditions is key here.

Duality, Adjoints, Green’s Functions – p. 110/304

Green’s functions Example For ( −∆u = f, u = 0,

x ∈ Ω, x ∈ ∂Ω,

φ solves ( −∆φ(y; x) = δy (x), φ(y; x) = 0,

x ∈ Ω, x ∈ ∂Ω,

and the bilinear identity reads Z u(y) = f (x)φ(y; x) dx. Ω

Duality, Adjoints, Green’s Functions – p. 111/304

Green’s functions

Duality, Adjoints, Green’s Functions – p. 112/304

Green’s functions Example For

φ solves

  ut − ∆u = f, u(t, x) = 0,   u(0, x) = 0,

x ∈ Ω, 0 < t ≤ T, x∂Ω, 0 < t ≤ T, x ∈ Ω,

  −φt − ∆φ = δ(s,y) (t, x), φ(t, x) = 0,   φ(T, x) = 0,

x ∈ Ω, T > t ≥ 0, x∂Ω, T > t ≥ 0, x ∈ Ω,

and the bilinear identity reads u(s, y) =

Z

0

T

Z

f (t, x)φ(s, y; t, x) dxdt.

Ω Duality, Adjoints, Green’s Functions – p. 113/304

Green’s functions Motivations: • The Green’s function is defined without reference to the

particular data f that determines a particular solution. The Green’s function depends on the operator and its properties. • The Green’s function generally has special properties

arising from the properties of the delta function, such as localized support and symmetry. • The Green’s function gives a “low-dimensional” snapshot of

the solution.

Duality, Adjoints, Green’s Functions – p. 114/304

Green’s functions Example The Green’s function for the Dirichlet problem for the Laplacian L = −∆ on the ball Ω of radius r centered at the origin in R3 is ( −1 r2 y −1 −1 |y − x| − r|y| |y|2 − x , y 6= 0, 1 × φ(y; x) = 4π |x|−1 − r−1 , y = 0, where |x| denotes the Euclidean norm of x The formula for the disk of radius r is  2 r y  |y| |y|2 −x  , y 6= 0, ln 1 r|y−x| φ(y; x) = × 2π  ln r , y = 0. |x| Duality, Adjoints, Green’s Functions – p. 115/304

Green’s functions Important issues: Existence, uniqueness, and smoothness of the Green’s function. Generally, everything goes as for the original problem except that the Green’s function may not be very smooth or may even be undefined at a point. The point of evaluation y must lie inside the domain Ω. The Green’s function often behaves badly in the limit of y approaching the boundary. By the way, the theory also extends to over-determined problems. However, if the original problem is under-determined, this approach fails. Duality, Adjoints, Green’s Functions – p. 116/304

Green’s functions Many expositions of the Green’s function avoid the adjoint. When the original problem has a unique solution and the original operator is the adjoint to the adjoint operator, the Green’s function φ for the original problem and the Green’s function for the adjoint problem φ∗ are related via φ(y; x) = φ∗ (x; y).

This is known as the reciprocity theorem. This makes it possible to introduce Green’s functions for some kinds of problems without talking about the adjoint. This depends on the dual space of the dual space of the original space being identifiable with the original space. Duality, Adjoints, Green’s Functions – p. 117/304

Green’s functions Including nonhomogeneous conditions is really a minor issue that usually “solves itself” without trouble. We carry out the analysis using the Green’s function for the homogeneous problem and some additional integrals involving data and the Green’s function will appear.

Duality, Adjoints, Green’s Functions – p. 118/304

Green’s functions Example Suppose the problem is ( −∆u = f, x ∈ Ω, u = g, x ∈ ∂Ω. We define the Green’s function as for the homogeneous problem, i.e., ( −∆φ(y; x) = δy (x), x ∈ Ω, φ(y; x) = 0, x ∈ ∂Ω.

Duality, Adjoints, Green’s Functions – p. 119/304

Green’s functions Evaluating the bilinear identity yields Z Z Z ∂u ∂φ ∂φ −φ ds = u ds. (u∆φ − φ∆u) dx = u ∂n ∂n ∂Ω ∂n Ω ∂Ω This yields ∂φ(y; s) u(y) = f (x)φ(y; x) dx − g(s) ds. ∂n Ω ∂Ω Z

Z

Duality, Adjoints, Green’s Functions – p. 120/304

Green’s functions Different representations are possible. Example For ( −∆u = f, u = 0,

x ∈ Ω, x ∈ ∂Ω,

We can define the Green’s function as the solution of ( −∆φ(y; x) = 0, x ∈ Ω, φ(y; x) = δy (x), x ∈ ∂Ω.

Duality, Adjoints, Green’s Functions – p. 121/304

Green’s functions Evaluating the bilinear identity yields

∂u ∂φ (u∆φ − φ∆u) dx = u −φ ds ∂n ∂n Ω ∂Ω Z Z ∂u ∂u ∂u ds = − δy ds = − (y). =− φ ∂n ∂n ∂Ω ∂Ω ∂n

Z

Z

This gives the value of the normal derivative of u at a point y on the boundary, Z ∂u (y) = − φ(y; x)f (x) dx. ∂n Ω

Duality, Adjoints, Green’s Functions – p. 122/304

Green’s functions

Duality, Adjoints, Green’s Functions – p. 123/304

A Posteriori Error Analysis and Adaptive Error Control

Duality, Adjoints, Green’s Functions – p. 124/304

A generalization of the Green’s function The Green’s function yields u(y) = (u, δy ) = (φ(y; x), f (x)).

The value of a function at a point is an example of a linear functional. There are frequently other quantities of interest that can be expressed as functionals. The Riesz Representation theorem suggests these functionals have the form (u, ψ) for a suitable distribution ψ .

Duality, Adjoints, Green’s Functions – p. 125/304

A generalization of the Green’s function Some useful choices of ψ include: • We can construct ψ = δc to get the average value

H

c e(s) ds

of the error over a curve c in Rn , n = 2, 3, and ψ = δs to get the average value of the error over a plane surface s in R3 .

Duality, Adjoints, Green’s Functions – p. 126/304

A generalization of the Green’s function This choice requires a certain smoothness, which is given by Theorem Sobolev If s > k + n/2, then there is a constant C such that for f ∈ H s (Rn ), max sup |Dα (x)| ≤ Ckf ks . |α|≤k x∈Rn

This implies that the derivatives of f of order k and less are continuous. The Sobolev theorem shows that if k ≤ n and s > k/2, restricting a function in H s to a submanifold of (co)dimension k is well-defined.

Duality, Adjoints, Green’s Functions – p. 127/304

A generalization of the Green’s function Some useful choices of ψ include: • We can construct ψ = δc to get the average value

H

c e(s) ds

of the error over a curve c in Rn , n = 2, 3, and ψ = δs to get the average value of the error over a plane surface s in R3 . • We can obtain errors in derivatives using dipoles in a similar

way. • ψ = χω /|ω| gives the error in the average value over a

subset ω ⊂ Ω, where χω is the characteristic function of ω . • Choosing ψ to be the residual of the finite element

approximation sometimes yields the energy norm of the error. • ψ = χω e/kekω , where e is the error of the finite element

discretization, gives the L2 (ω) norm of the error. Only some of these data ψ have spatially local support. Duality, Adjoints, Green’s Functions – p. 128/304

A generalization of the Green’s function We consider ( Lu = f, suitable b.c. and i.v.,

x ∈ Ω, x ∈ ∂Ω,

where L is a linear differential operator. The boundary and/or initial conditions should insure there is a unique solution.

Duality, Adjoints, Green’s Functions – p. 129/304

A generalization of the Green’s function Definition The generalized Green’s function corresponding to the quantity of interest (u, ψ) satisfies ( L∗ φ(y, x) = ψ(x), x ∈ Ω, adjoint b.c. and i.v., x ∈ ∂Ω, where L∗ is the formal adjoint of L.

Duality, Adjoints, Green’s Functions – p. 130/304

Discretization by the finite element method We concentrate on ( Lu = f, u = 0,

x ∈ Ω, x ∈ ∂Ω,

where L(D, x)u = −∇ · a(x)∇u + b(x) · ∇u + c(x)u(x),

with u : Rn → R, a is a n × n matrix function of x, b is a n-vector function of x, and c is a function of x.

Duality, Adjoints, Green’s Functions – p. 131/304

Discretization by the finite element method We assume • Ω ⊂ Rn , n = 2, 3, is a smooth or polygonal domain • a = (aij ), where ai,j are continuous in Ω for 1 ≤ i, j ≤ n and

there is a a0 > 0 such that v > av ≥ a0 for all v ∈ Rn \ {0} and x∈Ω • b = (bi ) where bi is continuous in Ω • c and f are continuous in Ω

Duality, Adjoints, Green’s Functions – p. 132/304

Discretization by the finite element method The associated variational formulation reads Find u ∈ H01 (Ω) such that A(u, v) = (a∇u, ∇v)+(b·∇u, v)+(cu, v) = (f, v) for all v ∈ H01 (Ω).

Duality, Adjoints, Green’s Functions – p. 133/304

Discretization by the finite element method We construct a piecewise polygonal approximation of ∂Ω whose nodes lie on ∂Ω and which is contained inside Ω. This forms the boundary of a convex polygonal domain Ωh . Th denotes a simplex triangulation of Ωh that is locally quasi-uniform. hK denotes the length of the longest edge of K ∈ Th and h is the mesh function with h(x) = hK for x ∈ K .

We also use h to denote maxK hK .

Duality, Adjoints, Green’s Functions – p. 134/304

Discretization by the finite element method

Th

hK

K

U=0

Discretization of a domain Ω with curved boundaries. Duality, Adjoints, Green’s Functions – p. 135/304

Discretization by the finite element method Vh denotes the space of functions that are • continuous on Ω • piecewise linear on Ωh with respect to Th • zero on the boundary ∂Ωh and Ω \ Ωh

Vh ⊂ H01 (Ω), and for smooth functions, the error of interpolation into Vh is O(h2 ) in k k

Definition The finite element method is: Compute U ∈ Vh such that A(U, v) = (f, v) for all v ∈ Vh .

Duality, Adjoints, Green’s Functions – p. 136/304

An a posteriori analysis for an algebraic equation Estimate the error e = x − X of a numerical solution X of Ax = b

where b, x ∈ Rn and A is a n × n matrix. The computable residual error of X is R = AX − b

Goal: Find a relation between the residual and the error. The residual can be small even if the error is large.

Duality, Adjoints, Green’s Functions – p. 137/304

An a posteriori analysis for an algebraic equation Since the residual error of the true solution is zero Ae = −R.

To obtain an estimate on a quantity of interest, we introduce the generalized Green’s vector solving the adjoint problem A> φ = ψ,

where ψ is any unit vector. This gives an a posteriori estimate |(e, ψ)| = |(e, A> φ)| = |(Ae, φ)| = |(R, φ)|.

Duality, Adjoints, Green’s Functions – p. 138/304

An a posteriori analysis for an algebraic equation A posteriori error bound |(e, ψ)| ≤ kφk kRk.

Definition kφk is called the stability factor e kRk kx k , ψ ≤ cond ψ (A) kbk ,

where the “weak” condition number is

cond ψ (A) = kφk kAk = kA−> ψk kAk

Duality, Adjoints, Green’s Functions – p. 139/304

An a posteriori analysis for a finite element method Goal: Estimate the error in a quantity of interest computed from the finite element solution U . We use a generalized Green’s function φ solving the adjoint problem corresponding to a special choice of data ψ . The error in the desired information may not have much to do with the error in some global norm. It may be computationally infeasible as well as very inefficient to attempt to control the error in a global norm when all that is desired is accuracy in some quantities of interest.

Duality, Adjoints, Green’s Functions – p. 140/304

An a posteriori analysis for a finite element method We assume that the information can be represented as (u, ψ). Definition The generalized Green’s function φ solves the weak adjoint problem : Find φ ∈ H01 (Ω) such that A∗ (v, φ) = (∇v, a∇φ)−(v, div (bφ))+(v, cφ) = (v, ψ) for all v ∈ H01 (Ω),

corresponding to the adjoint problem L∗ (D, x)φ = ψ .

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An a posteriori analysis for a finite element method (e, ψ) = (∇e, a∇φ) − (e, div (bφ)) + (e, cφ) = (a∇e, ∇φ) + (b · ∇e, φ) + (ce, φ) = (a∇u, ∇φ) + (b · ∇u, φ) + (cu, φ) − (a∇U, ∇φ) − (b · ∇U, φ) − (cU, φ) = (f, φ) − (a∇U, ∇φ) − (b · ∇U, φ) − (cU, φ).

Duality, Adjoints, Green’s Functions – p. 142/304

An a posteriori analysis for a finite element method Definition πh φ denotes an approximation of φ in Vh Theorem The error in the quantity of interest computed from the finite element solution satisfies the error representation, (e, ψ) = (f, φ−πh φ)−(a∇U, ∇(φ−πh φ))−(b·∇U, φ−πh φ)−(cU, φ−πh φ),

where φ is the generalized Green’s function corresponding to data ψ .

Duality, Adjoints, Green’s Functions – p. 143/304

An a posteriori analysis for a finite element method We use the error representation by approximating φ using a relatively high order finite element method. Good results are obtained using the space Vh2 Definition The approximate generalized Green’s function satisfies: Compute Φ ∈ Vh2 such that A∗ (v, Φ) = (∇v, a∇Φ)−(v, div (bΦ))+(v, cΦ) = (v, ψ) for all v ∈ Vh2 .

The approximate error representation is (e, ψ) ≈ (f, Φ − πh Φ) − (a∇U, ∇(Φ − πh Φ)) − (b · ∇U, Φ − πh Φ) − (cU, Φ − πh Φ). Duality, Adjoints, Green’s Functions – p. 144/304

An a posteriori analysis for a finite element method Examples are computed using FETkLab. • Runs under MATLAB • Solves general nonlinear elliptic systems on general

domains in two space dimensions • Implements the a posteriori error estimate for 16

simultaneous adjoint data ψi • Uses bisection or red-green quadrisection to refine

elements • Only those elements whose element indicators are larger

than the mean plus one standard deviation of all of the element indicators in that level are refined.

Duality, Adjoints, Green’s Functions – p. 145/304

An a posteriori analysis for a finite element method Example An elliptic problem on the unit square Ω = (0, 1) × (0, 1) ( −∆u = 200 sin(10πx) sin(10πy), (x, y) ∈ Ω, u(x, y) = 0, (x, y) ∈ ∂Ω The solution is highly oscillatory u(x, y) = sin(10πx) sin(10πy).

Duality, Adjoints, Green’s Functions – p. 146/304

An a posteriori analysis for a finite element method

1 u0 -1 x

y

Highly oscillatory solution.

Duality, Adjoints, Green’s Functions – p. 147/304

An a posteriori analysis for a finite element method We estimate the error in the average value by choosing ψ ≡ 1. The generalized Green’s function is approximated on the same mesh using a piecewise quadratic finite element function. We plot the ratios error/estimate for a wide variety of meshes.

Duality, Adjoints, Green’s Functions – p. 148/304

An a posteriori analysis for a finite element method 3

error/estimate

error/estimate

1.2

0.8

0.4

0.0 102

103 Number of elements

104

2

1

0 0.0

0.1

1.0

10.0

100.0

percent error

Plots of the error/estimate ratio.

Duality, Adjoints, Green’s Functions – p. 149/304

Adaptive error control Goal of adaptive error control: Generate a mesh with a relatively small number of elements such that for a given tolerance TOL and data ψ , |(e, ψ)| ≤ TOL. This cannot be verified in practice.

Duality, Adjoints, Green’s Functions – p. 150/304

Adaptive error control Rewrite the error representation elementwise: X Z (e, ψ) ≈ (f −b·∇U −cU )(Φ−πh Φ)−a∇U ·∇(Φ−πh Φ) dx. K∈Th

K

Definition The mesh acceptance criterion is Z X (f − b · ∇U − cU )(Φ − πh Φ) − a∇U · ∇(Φ − πh Φ) dx ≤ TOL K K∈Th

If this is satisfied, then the numerical solution is deemed acceptable.

Duality, Adjoints, Green’s Functions – p. 151/304

Adaptive error control When the mesh criterion is not satisfied, we have to “enrich” the discretization. The cancellation among the contributions from each element make the enrichment decision problematic. There is currently no theory or practical method for accommodating cancellation of errors in an adaptive error control in a way that achieves true optimality of efficiency.

Duality, Adjoints, Green’s Functions – p. 152/304

Adaptive error control A standard approach is to formulate the discretization selection problem as an optimization problem. This requires an estimate consisting of a sum over elements of positive quantities. Inserting norms gives X Z (f −b·∇U −cU )(Φ−πh Φ)−a∇U ·∇(Φ−πh Φ) dx. |(e, ψ)| ≤ K∈Th

K

If the acceptance criterion fails, the mesh is refined to achieve X Z (f −b·∇U −cU )(Φ−πh Φ)−a∇U ·∇(Φ−πh Φ) dx ≤ TOL. K∈Th

K

Duality, Adjoints, Green’s Functions – p. 153/304

Adaptive error control The calculus of variations yields “Principle of Equidistribution” The element contributions on a nearly optimal mesh are roughly equal across the elements. Definition Two element acceptance criteria for the element indicators are TOL max (f − b · ∇U − cU )(Φ − πh Φ) − a∇U · ∇(Φ − πh Φ) . , K |Ω|

or Z

(f − b · ∇U − cU )(Φ − πh Φ) − a∇U · ∇(Φ − πh Φ) dx . TOL , M K

where M is the number of elements in Th .

Duality, Adjoints, Green’s Functions – p. 154/304

Adaptive error control Computing a mesh using these criteria is usually performed by a “compute-estimate-mark-refine” adaptive strategy. We start with a coarse mesh then successively refine some fraction of the elements on which the element acceptance criterion fails.

Duality, Adjoints, Green’s Functions – p. 155/304

Adaptive error control Example  −∆u =   −400((x−.5)2 +(y−.5)2 )   4800 2 2 1 − 400((x − .5) + (y − .5) ) e , π  (x, y) ∈ Ω,    u(x, y) = 0, (x, y) ∈ ∂Ω,

f is an approximation of a delta function at (.5, .5).

We control the error in the average value using a tolerance of .05%.

Duality, Adjoints, Green’s Functions – p. 156/304

Adaptive error control

350

0.08

0

0 Solution

Generalized Green’s Function 0 -10 -20 -30

Mesh

Element Contributions

Results for the initial mesh. Duality, Adjoints, Green’s Functions – p. 157/304

Adaptive error control

140

0.08

0

0 Solution

Generalized Green’s Function 0 -10 -20 -30

Mesh

Element Contributions

Results after 10 iterations. Duality, Adjoints, Green’s Functions – p. 158/304

Adaptive error control 1.10

error/estimate

1.05 1.00 0.95 0.90 0

50

100 percent error

150

200

The error/estimate ratio.

Duality, Adjoints, Green’s Functions – p. 159/304

Further analysis on the a posteriori error estimate We derive a more common form of the estimate for the simple problem with L(u) = −∆u. The error representation formula is Z Z (e, ψ) = f (φ − πh φ) dx − ∇U · ∇(φ − πh φ) dx. Ω

Ω

We break up the second integral on the right as Z X Z ∇U · ∇(φ − πh φ) dx = ∇U · ∇(φ − πh φ) dx. Ω

K∈Th

K

Duality, Adjoints, Green’s Functions – p. 160/304

Further analysis on the a posteriori error estimate Using Green’s formula, Z

K

∇U · ∇(φ − πh φ) dx = −

Z

∆U (φ − πh φ) dx K Z + ∇U · n∂K (φ − πh φ) ds, ∂K

where the last term is a line integral and n∂K denotes the outward normal to ∂K . Summing over all of the elements, the boundary integrals give two contributions from each element edge, computed in opposite directions.

Duality, Adjoints, Green’s Functions – p. 161/304

Further analysis on the a posteriori error estimate If K1 , K2 ∈ Th share a common edge σ1 ⊂ ∂K1 = σ2 ⊂ ∂K2 , then Z Z ∇U |K1 · nσ1 (φ − πh φ) ds + ∇U |K2 · nσ2 (φ − πh φ) ds σ1 σ2 Z = ∇U |K1 · nσ1 (φ − πh φ) ds σ1 Z − ∇U |K2 · nσ1 (φ − πh φ) ds σ1 Z [∇U ] · nσ1 (φ − πh φ) ds, =− σ1

where [U ] = ∇U |K2 − ∇U |K1 denotes the “jump” in ∇U across σ1 in the direction of the normal n∂K1 .

Duality, Adjoints, Green’s Functions – p. 162/304

Further analysis on the a posteriori error estimate An alternate error representation,: (e, ψ) = −

X Z

K∈Th

(∆U + f )(φ − πh φ) dx

K

1 − 2

Z

∂K

[∇U ] · n∂K (φ − πh φ) ds .

Duality, Adjoints, Green’s Functions – p. 163/304

Further analysis on the a posteriori error estimate Definition The residual and corresponding adjoint or dual weights are ! ! k∆U + f kK kφ − πh φkK RK = , WK = . −1/2 1/2 kh [∇U ]k∂K /2 kh (φ − πh φ)k∂K

Theorem The error of the finite element approximation is bounded by |(e, ψ)| ≤

X

RK · WK .

K∈Th

Duality, Adjoints, Green’s Functions – p. 164/304

Further analysis on the a posteriori error estimate We derive a priori bounds on the residual and dual weights. There is a constant C independent of the mesh such that RK ≤ C|K|1/2 ,

where |K| denote the area of K ∈ Th . The bound on the first component follows from k∆U + f kK = kf kK ≤ maxΩ |f | × |K|1/2 .

Duality, Adjoints, Green’s Functions – p. 165/304

Further analysis on the a posteriori error estimate For the second component, consider an integral over the common edge σ between two elements K1 and K2 , k[∇U ]kσ = k∇U |K2 −∇U |K1 kσ ≤ k∇U |K2 −∇u|σ kσ +k∇u|σ −∇U |K1 kσ .

By a trace inequality, the standard energy norm convergence result, and a standard elliptic regularity result, 1/2

1/2

k∇U |Ki − ∇u|σ kσ ≤ k∇U − ∇ukKi k∇U − ∇uk1,Ki 1/2

1/2

≤ Ckhuk2,Ki kuk2,Ki ≤ Ckh1/2 f kKi ,

for i = 1, 2. The local quasi-uniformity of the mesh implies 1 −1/2 kh [∇U ]k∂K ≤ C max |f | × |K|1/2 . Ω 2 Duality, Adjoints, Green’s Functions – p. 166/304

Further analysis on the a posteriori error estimate Clearly, the convergence of the Galerkin approximation is strongly influenced by the dual weights φ − πh φ, i.e. by the approximation properties of Vh and the smoothness of φ. This reflects the importance of the cancellation of errors inherent to the Galerkin method.

Duality, Adjoints, Green’s Functions – p. 167/304

The Effective Domain of Influence and Solution Decomposition

Duality, Adjoints, Green’s Functions – p. 168/304

The Effective Domain of Influence A characteristic property of elliptic partial differential equations is a global domain of influence. A local perturbation of data near one point affects the solution throughout the domain of the problem. Elliptic problems also have the property that the strength of the effect of a localized perturbation on a solution decays significantly with the distance from the support of the perturbation, at least in some directions. It turns out that this property also has profound consequences for the numerical solution of elliptic problems.

Duality, Adjoints, Green’s Functions – p. 169/304

The Effective Domain of Influence Example Consider the Green’s function for the Dirichlet problem for the Laplacian −∆u = f on the ball Ω of radius r centered at the origin in R3 . If the data f is perturbed by a smooth function δf , the perturbation in the value of the solution δu(y) is given by Z δu(y) = φ(y; x)δf (x) dx, y ∈ Ω. Ω

Duality, Adjoints, Green’s Functions – p. 170/304

The Effective Domain of Influence If δf has compact support supp(δf ) ⊂ Ω, 2 r y |y − x| ≤ 2 − x , |y|

x ∈ supp(δf ), y ∈ Ω \ supp(δf ).

We conclude that |δu(y)| ≤

max |δf | × volume of supp(δf ) × 1 + 4π × the distance from y to supp(δf )

r |y|

.

Duality, Adjoints, Green’s Functions – p. 171/304

Poisson’s equation in a disk Ω denotes the disk of radius r centered at the origin in R2

Consider the Dirichlet problem for the Laplacian L = −∆u = f . Suppose that ω is a small region contained in Ω located well away from ∂Ω. We wish to estimate the error e = u − U in the energy norm kek1,ω in ω .

Duality, Adjoints, Green’s Functions – p. 172/304

Poisson’s equation in a disk

ω Ω

Duality, Adjoints, Green’s Functions – p. 173/304

Poisson’s equation in a disk We evaluate the norm weakly kek1,ω =

sup

(e, ψ).

ψ∈H −1 (ω) kψk−1,ω =1

The supremum is achieved for some ψ ∈ H −1 (ω). We extend ψ to H −1 (Ω) by setting it to zero in Ω \ ω . To discuss orders of convergence, we use the a posteriori bound.

Duality, Adjoints, Green’s Functions – p. 174/304

Poisson’s equation in a disk G(x; y) denotes the Green’s function for the Laplacian on Ω, so Z Z φ(x) = G(x; y)ψ(y) dy = G(x; y)ψ(y) dy. Ω

ω

Recall,  2 r x  ln |x| |x|2 −y ,

1 × G(x; y) = 2π  ln

r|x−y|

r |y|

,

x 6= 0, x = 0.

Duality, Adjoints, Green’s Functions – p. 175/304

Poisson’s equation in a disk Case 1: For y ∈ ω , G(x; y) is a smooth function of x for x ∈ Ω \ ω and so is φ. We assume that δ > 0 satisfies • It is small enough that ωδ = {x ∈ Ω : dist (x, ω) ≤ δ} is

contained in Ω • It is large enough that for K ⊂ Ω \ ωδ , the union N (K) of K

and the elements bordering K does not intersect ω

Duality, Adjoints, Green’s Functions – p. 176/304

Poisson’s equation in a disk

∂Ω ∂ωδ ∂ω

For K ⊂ Ω \ ωδ , we let πh be the Lagrange nodal interpolant with respect to Th , so that kφ − πh φkK ≤ C

X

kh2 Dα φkK .

|α|=2

Duality, Adjoints, Green’s Functions – p. 177/304

Poisson’s equation in a disk Case 2: φ is not so smooth in ω . For K ∩ ωδ 6= ∅, πh is the Scott-Zhang interpolant, for which kφ − πh φkK ≤ C|hφ|1,N (K) ,

for a mesh-independent constant C .

Duality, Adjoints, Green’s Functions – p. 178/304

Poisson’s equation in a disk The second component of WK is bounded similarly after using a trace theorem, 1/2

1/2

kh1/2 (φ − πh φ)k∂K ≤ kφ − πh φkN (K) kh(φ − πh φ)k1,N (K) ,

and the local quasi-uniformity of the mesh.

Duality, Adjoints, Green’s Functions – p. 179/304

Poisson’s equation in a disk We conclude, Theorem For any δ > 0 small enough that ωδ ⊂ Ω but large enough that N (K) ∩ ω = ∅ for K ⊂ Ω \ ωδ , there is a constant C such that kek1,ω ≤

X

X

K⊂Ω\ωδ |α|=2

2

α

1/2

Ckh D φkK |K|

+

X

C|hφ|1,N (K) |K|1/2 .

K∩ωδ 6=∅

Duality, Adjoints, Green’s Functions – p. 180/304

Poisson’s equation in a disk We estimate the quantities in the a posteriori bound. To handle the first sum, we use kDxα φk2K =

Z Z K

ω

Dxα G(x, y)ψ(y) dy

Z

2

dx

kDxα G(x, ·)k21,ω kψk2−1,ω dx K Z Z XZ Z = |Dxα Dyβ G(x, y)|2 dydx + |Dxα G(x, y)|2 dydx.

≤

|β|=1 K

ω

K

ω

Duality, Adjoints, Green’s Functions – p. 181/304

Poisson’s equation in a disk There is a constant C such that |Dxα Dyβ G(x, y)| ≤

C , 2 |x − y|

x 6= y ∈ Ω, |α| = 2, |β| ≤ 1.

We conclude there is a constant C independent of the mesh such that for K ⊂ Ω \ ωδ , kφ − πh φkK

Ch2K 1/2 ≤ |K| . 2 dist (K, ω)

Duality, Adjoints, Green’s Functions – p. 182/304

Poisson’s equation in a disk To handle the second sum on the right, we use the basic stability estimate, kφk1,Ω ≤ kψk−1,Ω = kψk−1,ω = 1. If we assume a uniform (small) size hK = h for elements such that K ∩ ωδ 6= ∅, we obtain X

K∩ωδ 6=∅

ChK |φ|1,N (K) ≤ Chkφk1,Ω

C = Ch ≤ |ωδ |

X

h|K|.

K∩ωδ 6=∅

Duality, Adjoints, Green’s Functions – p. 183/304

Poisson’s equation in a disk We conclude Theorem For any δ > 0 small enough that ωδ ⊂ Ω but large enough that N (K) ∩ ω = ∅ for K ⊂ Ω \ ωδ , there is a constant C such that kek1,ω ≤

X

K⊂Ω\ωδ

Ch2K |K| + 2 dist (K, ω)

X

Ch|K|.

K∩ωδ 6=∅

Duality, Adjoints, Green’s Functions – p. 184/304

Poisson’s equation in a disk We apply the “Principle of Equidistribution” The element indicators are Ch2K /dist (K, ω)2 respectively Ch, and in an optimal mesh, h2K ≈h 2 dist (K, ω)

or

hK ≈ h1/2 × dist (K, ω),

K ⊂ Ω \ ωδ .

Duality, Adjoints, Green’s Functions – p. 185/304

Poisson’s equation in a disk The decay of influence inherent to the Laplacian on the disk means that away from the region ω where we estimate the norm, we can choose elements asymptotically larger than the element size used in ωδ . Moreover, the elements can increase in size as the distance to ωδ increases ωδ is an the effective domain of influence for the error in the energy norm in ω .

Duality, Adjoints, Green’s Functions – p. 186/304

Poisson’s equation in a disk Definition An effective domain of influence corresponding to the data ψ is the region ωψ in which the corresponding elements must be significantly smaller in size than the elements used in the complement Ω \ ωψ in order to satisfy the adaptive goal. Equivalently, if Th comprises uniformly sized elements, then the effective domain of influence comprises those elements on which the element indicators, are substantially larger than those in the complement.

Duality, Adjoints, Green’s Functions – p. 187/304

A decomposition of the solution Often, the goal of solving a differential equation is to compute several pieces of information. The problem of computing multiple quantities of interest also arises naturally when the data ψ for the adjoint problem does not have spatially localized support (averages and norms). We cannot expect a significant localization effect from the decay of influence when the support of the data for the adjoint problem is not spatially localized. If the data ψ has the property that the adjoint weight φ − πh φ has a more-or-less uniform size throughout Ω, then the degree of non-uniformity in an adapted mesh depends largely on the spatial variation of the residual.

Duality, Adjoints, Green’s Functions – p. 188/304

A decomposition of the solution We use a partition of unity to “localize” a problem in which supp (ψ) does not have local support. Definition Suppose that {Ωi }N i=1 is a finite open cover of Ω. A Lipschitz partition of unity subordinate to {Ωi } is a collection of functions {pi }N i=1 with the properties supp (pi ) ⊂ Ωi ,

1 ≤ i ≤ N,

N X

pi (x) = 1,

x ∈ Ω,

i=1

pi is continuous on Ω and differentiable on Ωi , kpi kL∞ (Ω) ≤ C and k∇pi kL∞ (Ωi ) ≤ C/diam (Ωi ),

1 ≤ i ≤ N, 1 ≤ i ≤ N,

where C is a constant and diam (Ωi ) is the diameter of Ωi . Duality, Adjoints, Green’s Functions – p. 189/304

A decomposition of the solution Corresponding to a partition of unity {pi }, ψ ≡

PN

i=1 ψpi ,

Definition The quantities {(U, ψpi )} corresponding to the data {ψi = ψpi } are called the localized information corresponding to the partition of unity. We consider the problem of estimating the error in the localized information for 1 ≤ i ≤ N .

Duality, Adjoints, Green’s Functions – p. 190/304

A decomposition of the solution We obtain a finite element solution via: ˆi ∈ Vˆi such that A(U ˆi , v) = (f, v) for all v ∈ Vˆi , Compute U

where Vˆi is a space of continuous, piecewise linear functions on a locally quasi-uniform simplex triangulation Ti of Ω obtained by local refinement of an initial coarse triangulation T0 of Ω. {Vˆi } is globally defined and the “localized” problem is solved over the entire domain

We hope that this will require a locally refined mesh because the corresponding data has localized support.

Duality, Adjoints, Green’s Functions – p. 191/304

A decomposition of the solution A partition of unity approximation in the sense of Babuška and ˆi , 1 ≤ i ≤ N , where χi is the characteristic Melenk uses Ui = χi U function of Ωi . The local approximation Ui is in the local finite element space Vi = χi Vˆi . Definition The partition of unity approximation is defined by PN Up = i=1 Ui pi , which is in the partition of unity finite element space ) (N N X X vi pi : vi ∈ Vi . Vi p i = Vp = i=1

i=1

Duality, Adjoints, Green’s Functions – p. 192/304

A decomposition of the solution The partition of unity approximation recovers the full convergence properties of an approximation of the original solution. Note Up =

N X i=1

Ui pi =

N X i=1

ˆ i pi ≡ χi U

N X

ˆ i pi . U

i=1

ˆi outside of Ωi are immaterial in forming the The values of Ui or U global partition of unity approximation.

Duality, Adjoints, Green’s Functions – p. 193/304

A decomposition of the solution We use the generalized Green’s function satisfying the adjoint problem: Find φi ∈ H01 (Ω) such that A∗ (v, φi ) = (v, ψi ) for all v ∈ H01 (Ω). We expand the global error (u − Up , ψ) =

N X

(u − Ui )pi , ψ .

i=1

We estimate each summand on the right as ˆi , ψi ) = A∗ (u − U ˆi , φi ) (u − Ui )pi , ψ = (u − U

ˆi , ∇φi ) − (b · ∇U ˆi , φi ) − (cU ˆi , φi ). = (f, φi ) − (a∇U

Duality, Adjoints, Green’s Functions – p. 194/304

A decomposition of the solution Letting πi φi denote an approximation of φi in Vˆi , Theorem The error of the partition of unity finite element solution Up satisfies the error representation,

(u − Up , ψ) =

N X

ˆi , ∇(φi − πi φi )) (f, φi − πi φi ) − (a∇U

i=1

ˆ ˆ − (b · ∇Ui , φi − πi φi ) − (cUi , φi − πi φi ) .

Duality, Adjoints, Green’s Functions – p. 195/304

A decomposition of the solution Definition The approximate generalized Green’s functions solve: Compute Φi ∈ Vi2 such that A∗ (v, Φi ) = (v, ψi ) for all v ∈ Vi2 ,

1 ≤ i ≤ N,

where Vi2 is the space of continuous, piecewise quadratic functions with respect to Ti . The approximate error representations are ˆi , ψi ) ≈ (f, Φi − πi Φi ) − (a∇U ˆi , ∇(Φi − πi Φi )) (u − U ˆi , Φi − πi Φi ) − (cU ˆi , Φi − πi Φi ). − (b · ∇U

Duality, Adjoints, Green’s Functions – p. 196/304

A decomposition of the solution If the localized error satisfies TOL ˆ u − Ui , ψi ≤ , N

1 ≤ i ≤ N,

then |(u − Up , ψ)| ≤ TOL.

This justifies treating the N “localized” problems independently in terms of mesh refinement.

Duality, Adjoints, Green’s Functions – p. 197/304

Computation of multiple quantities of interest We present an algorithm for computing multiple quantities of interest efficiently using knowledge of the effective domains of influence of the corresponding Green’s functions. We assume that the information is specified as {(U, ψi )}N i=1 for a set of N functions {ψi }N i=1 . These data might arise as particular goals or via localization through a partition of unity. We assume that the goal is to compute the information associated to ψi so that the error is smaller than a tolerance TOLi for 1 ≤ i ≤ N .

Duality, Adjoints, Green’s Functions – p. 198/304

Computation of multiple quantities of interest Two approaches: Approach 1: A Global Computation Find one triangulation such that the corresponding finite element solution satisfies |(e, ψi )| ≤ TOLi , for 1 ≤ i ≤ N . Approach 2: A Decomposed Computation Find N independent triangulations and finite element solutions Ui so that the errors satisfy |(ei , ψi )| ≤ TOLi , for 1 ≤ i ≤ N. The Global Computation can be implemented with a straightforward modification of the standard adaptive strategy in which the N corresponding mesh acceptance criteria are checked on each element and if any of the N criteria fail, the element is marked for refinement.

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Computation of multiple quantities of interest If the correlation, i.e., overlap, between the effective domains of influence associated to the N data {ψi } is relatively small and the effective domains of influence are relatively small subsets of Ω, then each individual solution in the Decomposed Computation will require significantly fewer elements than the solution in the Global Computation to achieve the desired accuracy. This can yield significant computational advantage in terms of lowering the maximum memory requirement to solve the problem.

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Computation of multiple quantities of interest Decreasing the maximum memory required to solve a problem can be significant in at least two situations. • If the individual solutions in the Decomposed Computation

are computed in parallel and they require significantly fewer elements than the Global Computation, we can expect to see significant speedup. • In an environment with limited memory capabilities,

decomposing a Global Computation requiring a large number of elements into a set of significantly smaller computations can greatly increase the accuracy that can be achieved and/or decrease the time required.

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Computation of multiple quantities of interest If the effective domains of influence associated to the N data {ψ} have relatively large intersections, then the individual solutions in the Decomposed Computation will require roughly the same number of elements as the solution for the Global Computation. In this case, there is little to be gained in using the Decomposed Computation.

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Computation of multiple quantities of interest In general, some of the N effective domains of influence associated to data {ψi } in the Decomposed Computation will correlate significantly and the rest will have low correlation. We can optimize the use of resources by combining computations for data whose associated domains of influence have significant correlation and treating the rest independently.

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Computation of multiple quantities of interest Determining the Solution Decomposition 1. Discretize Ω by an initial coarse triangulation T0 and compute an initial finite element solution U0 . 2. Estimate the error in each quantity (U0 , ψi ) by solving the N approximate adjoint problems and computing the estimate. 3. Using the element indicators associated to identify the effective domains of influence for the data {ψi } in terms of the mesh T0 and significant correlations between the effective domains of influence. 4. Decide on the number of approximate solutions to be computed and the subset of information to be computed from each solution. 5. Compute the approximate solutions independently using adaptive error control aimed at computing the specified quantity or quantities of interest accurately. Duality, Adjoints, Green’s Functions – p. 204/304

Identifying significant correlations The key issue in the proposed algorithm is identifying the effective domains of influence and recognizing significant correlation, or overlap, between different effective domains of influence. Mesh refinement decisions are based on the sizes of the element indicators ˆi − cU ˆi )(Φi − πi Φi ) − a∇U ˆi · ∇(Φi − πi Φi ) Ei |K = max (f − b · ∇U K

or

Ei |K =

Z

K

ˆi − cU ˆi )(Φi − πi Φi ) − a∇U ˆi · ∇(Φi − πi Φi ) dx, (f − b · ∇U

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Identifying significant correlations Definition We let Ei (x) denote the piecewise constant element error indicator function associated to data ψi with Ei (x) ≡ Ei |K for K ∈ T0 . Identifying the effective domain of influence means finding a set of elements on which the element error indicators are significantly larger than on the complement. Identifying significant correlation between the effective domains of influence of two data entails showing that the effective domains of influence have a significant number of elements in common.

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Identifying significant correlations Pattern matching uses Definition The (cross-)correlation of two functions f ∈ Lp (Ω) and g ∈ Lq (Ω), defined as: Z (f ◦ g)(y) = f (x)g(y + x) dx, Ω

which is an L1 (Ω) function. The correlation indicator c(f, g) is a functional of the correlation function (f ◦ g). We treat the element error indicator functions {Ei } as signal functions. Duality, Adjoints, Green’s Functions – p. 207/304

Identifying significant correlations Discounting translation or rotation, correlation of Ei and Ej reduces to the L2 -inner-product: Z (Ei ◦ Ej )(0) = Ei (x)Ej (x) dx = (Ei , Ej )Ω . Ω

Definition The correlation indicator c(Ei , Ej ) is c(Ei , Ej ) = |(Ei ◦ Ej )(0)| = (Ei , Ej )Ω .

Duality, Adjoints, Green’s Functions – p. 208/304

Identifying significant correlations Definition We say that the effective domain of influence associated to ψi is significantly correlated to the domain of influence associated to ψj if two conditions hold.

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Identifying significant correlations Definition We say that the effective domain of influence associated to ψi is significantly correlated to the domain of influence associated to ψj if two conditions hold. Condition 1 The correlation of Ei and Ej is larger than a fixed fraction of the norm of Ej , or c(Ei , Ej ) Correlation Ratio 1 = ≥ γ1 , 2 kEj k

for some fixed 0 ≤ γ1 ≤ 1. This means that the projection of Ei onto Ej is sufficiently large.

Duality, Adjoints, Green’s Functions – p. 210/304

Ej

Element

Ei Ej

Element

Element Indicator

Ei

Element Indicator

Element Indicator

Identifying significant correlations

Ei Ej Element

Three examples of significant correlation of Ei with Ej . Plotted are the element indicator functions Ei (x), Ej (x) versus the element number.

Duality, Adjoints, Green’s Functions – p. 211/304

Identifying significant correlations Definition We say that the effective domain of influence associated to ψi is significantly correlated to the domain of influence associated to ψj if two conditions hold. Condition 2 The component of Ej orthogonal to Ei is smaller than a fixed fraction of the norm of Ej , or

c(Ej ,Ei )

Ej − kEi k2 Ei ≤ γ2 , Correlation Ratio 2 = kEj k for some fixed 0 ≤ γ2 ≤ 1. This corrects for the difficulties that arise when Ei is much larger than Ej . Duality, Adjoints, Green’s Functions – p. 212/304

Element Indicator

Identifying significant correlations

Ej Ei Element

An example in which the second condition fails.

Duality, Adjoints, Green’s Functions – p. 213/304

Identifying significant correlations 3

E1

E6

E2

E7

E3

E8

E4

E9

2 E5

1

0 1

9

18 Element Number

27

36

Plots of nine element indicator functions Ei versus the element number. Duality, Adjoints, Green’s Functions – p. 214/304

Identifying significant correlations With γ1 = .9 and γ2 = .7: E1 with E8 E2 with E6 , E7 E3 with E1 , E8

E4 with none E5 with E2 , E6 E6 with none

E7 with none E8 with none E9 with none

We emphasize that the initial identification of significant correlation between effective domains of influence of various Green’s functions in a computation is carried out on a coarse initial partition of the domain and hence is relatively inexpensive.

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Examples We solve elliptic problems using adaptive mesh refinement to achieve accuracy in a set of quantities of interest using a Global Computation and a Decomposed Computation. The results suggest that the individual solutions in the Decomposed Computation require significantly fewer elements to achieve the desired accuracy than the Global Computation in a variety of situations.

Duality, Adjoints, Green’s Functions – p. 216/304

Examples As a relatively universal measure of the gain from using the Decomposed Computation, we use Definition The Final Element Ratio is the number of elements in the final mesh refinement level required to achieve accuracy in the specified quantities of interest in the Global Computation to the maximum number of elements in the final mesh refinement levels for the individual computations in the Decomposed Computation. Generally, we expect the gain in efficiency to scale super-linearly with the Final Element Ratio. We compute the Final Element Ratio using solutions that are have roughly the same accuracy. Duality, Adjoints, Green’s Functions – p. 217/304

Examples Example 1 We solve ( 1 (x, y) ∈ Ω, − 10π 2 ∆u(x) = sin(πx) sin(πy), u(x, y) = 0, (x, y) ∈ ∂Ω, on the Ω = [0, 8] × [0, 8]. The solution is u(x, y) = 5 sin(πx) sin(πy). Goal: Control the error in the average value of u by choosing ψ ≡ 1/|Ω| = 1/64.

Duality, Adjoints, Green’s Functions – p. 218/304

Examples Global Computation: Mesh is adapted the error in the average value of u is smaller than the error tolerance of 5%. Initial mesh is 10 × 10 elements and after five refinement levels, we end up with 3505 elements, achieving an error of .022.

Duality, Adjoints, Green’s Functions – p. 219/304

Examples Initial Mesh

Final Mesh

Initial and final meshes.

Duality, Adjoints, Green’s Functions – p. 220/304

Examples

Numerical solutions.

Duality, Adjoints, Green’s Functions – p. 221/304

Examples Estimates, errors, and error/estimate ratios: Level Elements Estimate 1 100 .1567 2 211 .1157 3 585 .3063 4 1309 .1159 5 3505 .02163

Error .1534 .1224 .3078 .1166 .02148

Ratio .9786 1.058 1.005 1.006 .9975

Duality, Adjoints, Green’s Functions – p. 222/304

Examples

1 2

3 4

6

7

10

11

5

8

9

12

2

3

14

15

1

4

13

16

Partitions for the decomposition.

Duality, Adjoints, Green’s Functions – p. 223/304

Examples The partition of unity yields four data {ψ1 , ψ2 , ψ3 , ψ4 }. ˆ1 , · · · , U ˆ4 } We compute the four localized approximations {U using the same initial mesh as before.

The Correlation Ratios indicates that all four localized solutions should be computed independently.

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Examples We obtain acceptable results using the tolerance of 5%. Data Level Elements Estimate ψ1 3 618 .01242 ψ2 3 575 −.0009109 ψ3 3 618 .01242 ψ4 3 575 −.0009109 This yields a partition of unity solution Up with accuracy .023. Using the Decomposed Computation yields a Final Element Ratio of ≈ 5.7.

Duality, Adjoints, Green’s Functions – p. 225/304

Examples ^ Final Mesh for U1

Final Mesh for ^ U2

Two of the final meshes.

Duality, Adjoints, Green’s Functions – p. 226/304

Examples

The generalized Green’s functions for the global average error and the localized data ψ2 .

Duality, Adjoints, Green’s Functions – p. 227/304

Examples A Decomposed Computation using a partition of unity on 16 equal-sized regions yields significant correlations: E2 with E3 E4 with E3

E5 with E8 E7 with E8

E10 with E9 E12 with E9

E13 with E14 E15 with E14

This computation yields a Final Element Ratio of ≈ 2.6

Duality, Adjoints, Green’s Functions – p. 228/304

Examples Example 2 We solve  −∇ · (1.1 + sin(πx) sin(πy))∇u(x, y)     = −3 cos2 (πx) + 4 cos2 (πx) cos2 (πx) 2 (πy), (x, y) ∈ Ω,  +2.2 sin(πx) sin(πy) + 2 − 3 cos    u(x, y) = 0, (x, y) ∈ ∂Ω,

where Ω = [0, 2] × [0, 2].

The exact solution is u(x, y) = sin(πx) sin(πy).

Duality, Adjoints, Green’s Functions – p. 229/304

Examples We compute the average error using ψ1 ≡ 1/4 and four point values using ψ2 ≈ δ(.5,.5) , ψ3 ≈ δ(.5,1.5) , ψ4 ≈ δ(1.5,1.5) , and ψ5 ≈ δ(1.5,.5) . We use 400 −400((x−cx )2 +(y−cy )2 ) ˆ e δ(cx ,cy ) = π

to approximate the delta function δ(cx ,cy ) .

Duality, Adjoints, Green’s Functions – p. 230/304

Examples Global Computation: We use a tolerance of 2% starting with an 8 × 8 mesh.

Lev. Elt’s 1 64 2 201 3 763 4 2917

ψ1 Est. Err. Rat. .035 .035 1.0 .0088 .0089 1.0 .0027 .0027 1.0 .00044 .00044 1.0

ψ2 Est. Err. Rat. .090 .29 3.3 .042 .082 1.9 .020 .020 .99 .0050 .00504 1.0

Duality, Adjoints, Green’s Functions – p. 231/304

Examples Global Computation: We use a tolerance of 2% starting with an 8 × 8 mesh.

Lev. Elt’s 1 64 2 201 3 763 4 2917

ψ3 Est. Err. Rat. .24 .022 .091 .0024 .014 6.0 .0020 .0020 1.0 .0049 .00504 1.0

Duality, Adjoints, Green’s Functions – p. 232/304

Examples Initial Mesh

Final Mesh

Initial and final meshes. Final mesh has 2917 elements.

Duality, Adjoints, Green’s Functions – p. 233/304

Examples Decomposed Computation: Use data {ψ1 , · · · , ψ5 } independently using tolerances of 2%. ˆ1 ; 9 × 9 for U ˆ2 and We vary the initial meshes; using 7 × 7 for U ˆ4 ; and 12 × 12 for U ˆ3 and U ˆ5 . U

Data Level Elements Estimate ψ1 3 409 −.0004699 ψ2 4 1037 −.007870 ψ3 2 281 −.005571 ψ4 4 1037 −.007870 ψ5 2 281 −.005571 The Final Element Ratio is ≈ 2.8.

Duality, Adjoints, Green’s Functions – p. 234/304

Examples ^ Final Mesh for U1

^ Final Mesh for U 2

^ Final Mesh for U 3

ˆ1 , U ˆ2 , U ˆ 3 }. Final meshes for {U

Duality, Adjoints, Green’s Functions – p. 235/304

Examples Example 3 We solve ( −∆u = 16(y − y 2 + x − x2 ) u(x, y) = 0,

(x, y) ∈ Ω, (x, y) ∈ ∂Ω,

where Ω = [0, 1] × [0, 1]. The exact solution is u(x, y) = 8x(1 − x)y(1 − y).

Duality, Adjoints, Green’s Functions – p. 236/304

Examples We estimate the error in the average value of u. Since the domain is small and the solution and the generalized Green’s function are very smooth, the gain from decomposing the solution is greatly reduced. Beginning with a 4 × 4 mesh and using a tolerance of 1%, we obtain a sufficiently accurate solution using a Global Computation after five refinements. The final mesh uses 885 elements and produces an error of .0008699.

Duality, Adjoints, Green’s Functions – p. 237/304

Examples If we partition the domain using four equal regions there are no substantial correlations. Computing the four solutions independently in the Decomposed Computation yields a Final Element Ratio of around 1.5.

Duality, Adjoints, Green’s Functions – p. 238/304

Examples If we use sixteen equal regions, there are a number of substantial correlations. Correlation Ratio 1 for E1 Correlation Ratio 2 for E1 Correlation Ratio 1 for E2 Correlation Ratio 2 for E2

on E2 on E2 on E1 on E1

= .98, = .44, = .82, = .44.

Duality, Adjoints, Green’s Functions – p. 239/304

Examples ˆ1 corresponding to the localized data ψ1 using a Computing U tolerance of 1% requires 367 elements. ˆ2 requires a mesh 494 elements. Computing U

Combining these two computations by using data equal to the sum of the two partition functions for the regions Ω1 and Ω2 requires 496 elements. ˆ1 and U ˆ2 independently We gain almost nothing by computing U

Duality, Adjoints, Green’s Functions – p. 240/304

Examples Final Mesh for ^ U1

Final Mesh for ^ U2

^ +U ^ " Final Mesh for "U 1 2

Final meshes.

Duality, Adjoints, Green’s Functions – p. 241/304

Examples We consider the error in the average value and the point values at (.25, .25) and (.5, .5). We use a partition of unity decomposition for the error in the average to get data {ψ1 , · · · , ψ4 }. We let ψ5 ≈ δ(.25,.25) and ψ6 ≈ δ(.5,.5) . We compare the correlation indicators on initial meshes ranging from 16 to 144 or 400 uniformly sized elements.

Duality, Adjoints, Green’s Functions – p. 242/304

Examples 1.0

1.0

Corr. Rat. 1: E1 on E2 Corr. Rat. 2: E1 on

E2

Corr. Rat. 1: E2 on

E1 E1

Corr. Rat. 2: E2 on

0.5

0.0

Corr. Rat. 1: E1 on E3 Corr. Rat. 2: E1 on

E3

Corr. Rat. 1: E3 on

E1 E1

Corr. Rat. 2: E3 on

0.5

0.0

0

50 100 Number of Elements

150

0

50 100 Number of Elements

150

Correlation Ratios versus number of elements.

Duality, Adjoints, Green’s Functions – p. 243/304

Examples 4

10

Corr. Rat. 1: E1 on E5

5

Corr. Rat. 2: E1 on

E5

Corr. Rat. 1: E5 on

E1 E1

Corr. Rat. 2: E5 on

Corr. Rat. 1: E1 on E6 2

Corr. Rat. 2: E1 on

E6

Corr. Rat. 1: E6 on

E1 E1

Corr. Rat. 2: E6 on

0

0

0

100

200 Number of Elements

300

400

0

100

200 Number of Elements

300

400

Correlation Ratios versus number of elements.

Duality, Adjoints, Green’s Functions – p. 244/304

Examples 1.2

3

Corr. Rat. 2: E5 on

E6

Corr. Rat. 1: E6 on

E5 E5

Corr. Rat. 2: E6 on

Corr. Rat. 1: E5 on

Corr. Rat. 2: E3 on

Corr. Rat. 2: E5 on

E5

E3 E3

2

Corr. Rat. 1: E5 on E6 0.6

Corr. Rat. 1: E3 on E5

0.0

1

0

0

50 100 Number of Elements

150

0

100

200 Number of Elements

300

400

Correlation Ratios versus number of elements.

Duality, Adjoints, Green’s Functions – p. 245/304

Examples In general, we find that all Correlation Ratios converge to a limit as the number of elements increases. Generally, the second Correlation Ratio varies relatively little as the mesh density increases for all data. The first Correlation Ratio between data representing a partition of unity decomposition also varies relatively little. The first Correlation Ratio varies quite a bit on coarse meshes when one of the data is an approximate delta function.

Duality, Adjoints, Green’s Functions – p. 246/304

Examples The determination that two effective domains of influence are not closely correlated is relatively robust with respect to the density of the mesh. There is a mild tendency to combine computations that are more efficiently treated independently if the correlation indicators are computed on very coarse meshes.

Duality, Adjoints, Green’s Functions – p. 247/304

Examples Example 4 We solve  2 2 −∇ · .05 + tanh 10(x − 5) + 10(y − 1) ! ∇u     −100 + · ∇u = 1, (x, y) ∈ Ω,  0    u(x, y) = 0, (x, y) ∈ ∂Ω,

where Ω = [0, 10] × [0, 2].

Duality, Adjoints, Green’s Functions – p. 248/304

Diffusion

Examples

1

10 x

2 y

0

Plot of the diffusion coefficient.

Duality, Adjoints, Green’s Functions – p. 249/304

Examples We use an initial mesh of 80 elements and an error tolerance of T OL = .04%.

Level Elements Estimate 1 80 −.0005919 2 193 −.001595 3 394 −.0009039 4 828 −.0003820 5 1809 −.0001070 6 3849 −.00004073 7 9380 −.00001715 8 23989 −.000007553

Duality, Adjoints, Green’s Functions – p. 250/304

Examples

Final Mesh.

Duality, Adjoints, Green’s Functions – p. 251/304

Examples Pe=1000

Pe=.1

Plots of the meshes for the original problem with P e = 1000 and a problem with P e = .1.

Duality, Adjoints, Green’s Functions – p. 252/304

Examples

11

12

13

14

15

16

17

18

19

1

2

3

4

5

6

7

8

9

20 10

Domains for the partition of unity.

Duality, Adjoints, Green’s Functions – p. 253/304

Examples Significant Correlations: E3 with E4 E13 with E14

E6 with E7 E16 with E17

E7 with E6 E17 with E16

E9 with E8 E19 with E18

E10 with E8 , E9 E20 with E18 , E19

Note, there are no significant correlations in the “cross-wind” direction.

Duality, Adjoints, Green’s Functions – p. 254/304

Examples Solutions are symmetric across y = 1. Data ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9 ψ10

TOL Level Elements Estimate .04% 7 7334 −6.927 × 10−7 .04% 7 8409 −5.986 × 10−7 .04% 7 7839 −5.189 × 10−7 .04% 7 7177 −5.306 × 10−7 .04% 7 7301 −4.008 × 10−7 .02% 7 6613 −2.471 × 10−7 .02% 7 4396 −2.938 × 10−7 .02% 7 4248 −1.656 × 10−7 .02% 7 3506 −1.221 × 10−7 .02% 7 1963 −5.550 × 10−8

The Final Element Ratio is ≈ 2.9. Duality, Adjoints, Green’s Functions – p. 255/304

Examples ^ Final Mesh for U1

^ Final Mesh for U5

^ ˆ1 and U ˆ5 . Plots of the final meshes for the localized solutions U

Duality, Adjoints, Green’s Functions – p. 256/304

Examples ^ Final Mesh for U9

ˆ9 . Plots of the final mesh for the localized solution U

Duality, Adjoints, Green’s Functions – p. 257/304

Examples x 10 -4

x 10 -4

6

6

4

4

2

2

10

0

8 6

2 y

4

1

2 0 0

x

10

0

8

2 y

4

1

6 x

2 0 0

Plots of the generalized Green’s functions corresponding to ψ11 (left) and ψ19 (right).

Duality, Adjoints, Green’s Functions – p. 258/304

Examples The effective domains of influence may not be spatially compactly-shaped.

Duality, Adjoints, Green’s Functions – p. 259/304

Examples We combine some of the localized computations by solving for localized solutions corresponding to summing the two of the partition of unity data. Data ψ3 + ψ4 ψ6 + ψ7 ψ8 + ψ9 ψ9 + ψ10

TOL Level Elements Estimate .04% 7 8330 −9.8884 × 10−7 .02% 7 5951 −5.897 × 10−7 .02% 7 4406 −3.486 × 10−7 .02% 7 3202 −2.243 × 10−7

The solutions for ψ3 + ψ4 and ψ8 + ψ9 use a few more elements than required for either of the original localized solutions. The solutions for ψ6 + ψ7 and ψ9 + ψ10 use less than the maximum required for the individual localized solutions. Duality, Adjoints, Green’s Functions – p. 260/304

Examples Example 5 We solve ( 2 2 1 −5((x−.5) +(y−2.5) ) , (x, y) ∈ Ω, − π2 ∆u = 2 + 4e u(x, y) = 0, (x, y) ∈ ∂Ω, where Ω is the “square annulus” Ω = [0, 3] × [0, 3] \ [1, 2] × [1, 2]. The initial mesh has 48 elements and the error tolerance is 1%.

Duality, Adjoints, Green’s Functions – p. 261/304

Examples Level Elements Estimate 1 48 −5.168 2 125 −1.584 3 380 −.6879 4 894 −.3029 5 2075 −.1435

Duality, Adjoints, Green’s Functions – p. 262/304

Examples Initial Mesh

Final Mesh

Initial and Final Meshes.

Duality, Adjoints, Green’s Functions – p. 263/304

Examples

6

1.6

5

1.2

4 3

0.8

2

0.4

1

3

3

0 1

1 x

2

3 0

2 y

2

0 1

1

2 x

3

y

0

Solution and Generalized Green’s Function on the Final Mesh.

Duality, Adjoints, Green’s Functions – p. 264/304

Examples

7

6

8 1

5 4

2

3

The partition of unity has 8 square subdomains.

Duality, Adjoints, Green’s Functions – p. 265/304

Examples There are no significant correlations. Data Level Elements Estimate ψ1 5 1082 −.01935 ψ2 5 1101 −.01399 ψ3 5 1144 −.01540 ψ4 5 1107 −.01360

Duality, Adjoints, Green’s Functions – p. 266/304

Examples There are no significant correlations. Data Level Elements Estimate ψ5 5 1104 −.01436 ψ6 5 1110 −.01587 ψ7 5 1074 −.02529 ψ8 5 1098 −.01660 Final Element Ratio ≈ 1.8.

Duality, Adjoints, Green’s Functions – p. 267/304

Examples ^ Final Mesh for U3

^ Final Mesh for U4

Final meshes.

Duality, Adjoints, Green’s Functions – p. 268/304

Examples ^ Final Mesh for U6

^ Final Mesh for U7

Final meshes.

Duality, Adjoints, Green’s Functions – p. 269/304

Examples 1.4

1

1 0.6 3 2

0 1 x

2

1 3

0

y

3

0.2 0 1

1 x

2

3

2 y

0

Plots of some Generalized Green’s Functions.

Duality, Adjoints, Green’s Functions – p. 270/304

Nonlinear Problems

Duality, Adjoints, Green’s Functions – p. 271/304

A nonlinear algebraic equation The problem is to estimate the error e = x − X of a numerical solution X of a system of nonlinear algebraic equations, f (x) = b

where the data b, nonlinearity f , and the solution x all have the same dimension. The residual error is R = f (X) − b

which immediately gives f (x) − f (X) = −R.

The error is related to the residual through a nonlinear equation.

Duality, Adjoints, Green’s Functions – p. 272/304

A nonlinear algebraic equation The mean value theorem for integrals: Z

f (x) − f (X) =

1

f 0 (sx + (1 − s)X) (x − X) ds

0

where f 0 is the Jacobian matrix of f . We get ˜ = −R Ae

with A˜ =

Z

1

f 0 (sx + (1 − s)X) ds.

0

This is the linear problem obtained by linearizing f around an average of x and −X .

Duality, Adjoints, Green’s Functions – p. 273/304

A nonlinear algebraic equation We obtain ˜ φ)| = |(R, φ)|. |(e, ψ)| = |(e, A˜> φ)| = |(Ae,

To obtain a computable adjoint problem, we can try to replace x by X in the definition of A˜, A˜ → A =

Z

1

f 0 (sX + (1 − s)X) ds = f 0 (X),

0

But, what is the effect on the estimate?

Duality, Adjoints, Green’s Functions – p. 274/304

Defining the adjoint to a nonlinear operator We assume that the Banach spaces X and Y are Sobolev spaces and use ( , ) for the L2 inner product, and so forth. We define the adjoint for a specific kind of nonlinear operator. We assume f is a nonlinear map from X into Y , where the domain D(f ) is a convex set. Definition A subset A of a vector space is convex if for any a, b ∈ A, the set of points on the “line segment” joining a and b, i.e., {sa + (1 − s)b| 0 ≤ s ≤ 1} is contained in A.

Duality, Adjoints, Green’s Functions – p. 275/304

Defining the adjoint to a nonlinear operator We choose u inside D(f ) and define F (e) = f (u + e) − f (u),

where we think of e as representing an “error”, i.e., e = U − u. The domain of F is D(F ) = {v ∈ X| v + u ∈ D(f )}.

We assume that D(F ) is independent of e and dense in X . Note that 0 ∈ D(F ) and F (0) = 0. We also assume that D(F ) is a vector subspace of X .

Duality, Adjoints, Green’s Functions – p. 276/304

Defining the adjoint to a nonlinear operator Two reasons to work with functions of this form: • This is the kind of nonlinearity that arises when estimating

the error of a numerical solution or studying the effects of perturbations. • Nonlinear problems typically do not enjoy the global

solvability that characterizes linear problems, only a local solvability.

Duality, Adjoints, Green’s Functions – p. 277/304

Defining the adjoint to a nonlinear operator The first definition is based on the bilinear identity. Definition An operator A∗ (e) with domain D(A∗ ) ⊂ Y ∗ and range in X ∗ is an adjoint operator corresponding to F if (F (e), w) = (e, A∗ (e)w)

for all e ∈ D(F ), w ∈ D(A∗ ).

This is an adjoint operator associated with F , not the adjoint operator to F .

Duality, Adjoints, Green’s Functions – p. 278/304

Defining the adjoint to a nonlinear operator Example Suppose that F can be represented as F (e) = A(e)e, where A(e) is a linear operator with D(F ) ⊂ D(A). For a fixed e ∈ D(F ), define the adjoint of A satisfying (A(e)w, v) = (w, A∗ (e)v) for all w ∈ D(A), v ∈ D(A∗ ).

Substituting w = e shows this defines an adjoint of F as well. If there are several such linear operators A, then there will be several different possible adjoints.

Duality, Adjoints, Green’s Functions – p. 279/304

Defining the adjoint to a nonlinear operator Example Let (t, x) ∈ Ω = (0, 1) × (0, 1), with X = X ∗ = Y = Y ∗ = L2 denoting the space of periodic functions in t and x, with period equal to 1. Consider a periodic problem ∂e ∂e F (e) = +e + ae = f ∂t ∂x

where a > 0 is a constant and the domain of F is the set of continuously differentiable functions.

Duality, Adjoints, Green’s Functions – p. 280/304

Defining the adjoint to a nonlinear operator We can write F (e) = Ai (e)e where ∂v ∂v A1 (e)v = +e + av ∂t ∂x ∂e ∂v + a+ v A2 (e)v = ∂t ∂x

∂v 1 ∂(ev) A3 (e)v = + + av. ∂t 2 ∂x

Duality, Adjoints, Green’s Functions – p. 281/304

Defining the adjoint to a nonlinear operator The adjoints are ∂w ∂(ew) − + aw =− ∂t ∂x ∂w ∂e ∗ A2 (e)w = − + a+ w ∂t ∂x ∂w e ∂w ∗ − + aw. A3 (e)w = − ∂t 2 ∂x A∗1 (e)w

Duality, Adjoints, Green’s Functions – p. 282/304

Defining the adjoint to a nonlinear operator We base the second definition of an adjoint on the integral mean value theorem. We assume that the original nonlinearity is Frechet differentiable. The integral mean value theorem states f (U ) = f (u) +

Z

1

f 0 (u + se) ds e

0

where e = U − u and f 0 is the Frechet derivative of f .

Duality, Adjoints, Green’s Functions – p. 283/304

Defining the adjoint to a nonlinear operator We rewrite this as F (e) = f (U ) − f (u) = A(e)e

with A(e) =

Z

1

f 0 (u + se) ds.

0

Note that we can apply the integral mean value theorem to F : A(e) =

Z

1

F 0 (se) ds.

0

To be precise, we should discuss the smoothness of F .

Duality, Adjoints, Green’s Functions – p. 284/304

Defining the adjoint to a nonlinear operator Definition For a fixed e, the adjoint operator A∗ (e), defined in the usual way for the linear operator A(e), is said to be an adjoint for F . Example Continuing the previous example,

∂v ∂e ∂v +e + a+ F (e)v = ∂t ∂x ∂x 0

v.

After some technical analysis of the domains of the operators involved, ∂w e ∂w ∗ A (e)w = − − + aw. ∂t 2 ∂x This coincides with the third adjoint computed above.

Duality, Adjoints, Green’s Functions – p. 285/304

Analysis of a space-time finite element method We study a system of D reaction-diffusion equations consisting of d parabolic equations and D − d ordinary equations for the RD valued function u = (ui ):  +  u˙ i − ∇ · (i (u, x, t)∇ui ) = fi (u, x, t), (x, t) ∈ Ω × R , 1 ≤ i ≤ D, (x, t) ∈ ∂Ω × R+ , 1 ≤ i ≤ d, ui (x, t) = 0,   u(x, 0) = u0 (x), x ∈ Ω, where there is a constant > 0 such that

i (u, x, t) ≥ for 1 ≤ i ≤ d and i (u, x, t) ≡ 0 for the rest.

Duality, Adjoints, Green’s Functions – p. 286/304

Analysis of a space-time finite element method Assumptions: • Ω is a convex polygonal domain in R2 with boundary ∂Ω • = (i ) and f = (fi ) have smooth second derivatives

u˙ i denotes the time derivative of ui

We write i (u, x, t) = i (u) and f (u, x, t) = f (u).

Duality, Adjoints, Green’s Functions – p. 287/304

Analysis of a space-time finite element method We use up and uo to denote the parts of u associated to the parabolic and ordinary differential equations respectively. upi = ui for 1 ≤ i ≤ d and upi = 0 for d < i ≤ D uo = u − up

The presence of ordinary differential equations in the system has strong consequences for the smoothness of solutions.

Duality, Adjoints, Green’s Functions – p. 288/304

Analysis of a space-time finite element method Discretization of the space-time domain: We partition [0, ∞) as 0 = t0 < t1 < t2 < · · · < tn < . . . , denoting each time interval by In = (tn−1 , tn ] and time step by kn = tn − tn−1 . To each interval In , we associate a triangulation Tn of Ω arranged so the union of the elements in Tn is Ω.

Duality, Adjoints, Green’s Functions – p. 289/304

Time

Analysis of a space-time finite element method

Ω

Space

A space-time discretization. Duality, Adjoints, Green’s Functions – p. 290/304

Analysis of a space-time finite element method As usual, we assume the intersection of any two elements in a space triangulation is either a common edge, node, or is empty. We also assume that the smallest angle of any triangle in a triangulation is bounded below by a fixed constant hn denotes the piecewise constant mesh function hn |K = diam(K) for K ∈ Tn and h denotes the global mesh function, where h|In = hn . k denotes the piecewise constant function that is kn on In .

Duality, Adjoints, Green’s Functions – p. 291/304

Analysis of a space-time finite element method The approximations are polynomials in time and piecewise polynomials in space on each space-time “slab” Sn = Ω × In . Vn ⊂ (H01 (Ω))d × (H 1 (Ω))D−d denotes the space of piecewise linear continuous vector-valued functions v(x) ∈ RD defined on Tn , where the first d components of v are zero on ∂Ω.

We define Wnq

= w(x, t) : w(x, t) =

q X j=0

j

t vj (x), vj ∈ Vn , (x, t) ∈ Sn .

W q denotes the space of functions defined on the space-time domain Ω × R+ such that v|Sn ∈ Wnq for n ≥ 1.

Duality, Adjoints, Green’s Functions – p. 292/304

Analysis of a space-time finite element method Functions in W q are generally discontinuous across the discrete time levels. [w]n = wn+ − wn− , where wn± = lims→tn ± w(s), denotes the jump across tn Pn is the L2 projection onto Vn , i.e., Pn : L2 (Ω) → Vn is defined by (Pn v, w) = (v, w) for all w ∈ Vn . P is defined by P = Pn on Sn . πn : L2 (In ) → P q (In ) denotes the L2 projection onto the piecewise polynomial functions in time, where P q (In ) is the space of polynomials of degree q or less on In . π is defined by π = πn on Sn .

Duality, Adjoints, Green’s Functions – p. 293/304

Analysis of a space-time finite element method Definition The continuous Galerkin cG(q) approximation U ∈ W q satisfies U0− = P0 u0 and for n ≥ 1, the Galerkin orthogonality relation Z t Z tn n    (U˙ i , vi ) + (i (U )∇Ui , ∇vi ) dt = (fi (U ), vi ) dt  tn−1

   U + = P U − . n n−1 n−1

for all v ∈

tn−1 Wnq−1 ,

1 ≤ i ≤ D,

U is continuous across time nodes over which there is no mesh change.

Duality, Adjoints, Green’s Functions – p. 294/304

Analysis of a space-time finite element method Definition The discontinuous Galerkin dG(q) approximation U ∈ W q satisfies U0− = P0 u0 and for n ≥ 1, Z

tn

tn−1

+ ˙ (Ui , vi ) + (i (U )∇Ui , ∇vi ) dt + [Ui ]n−1 , vi =

Z

tn

(fi (U ), vi ) dt

tn−1

for all v ∈ Wnq , 1 ≤ i ≤ D.

Duality, Adjoints, Green’s Functions – p. 295/304

Analysis of a space-time finite element method Example We discretize the scalar problem  +,  u ˙ − ∆u = f (u), (x, t) ∈ Ω × R  u(x, t) = 0, (x, t) ∈ ∂Ω × R+ ,   u(x, 0) = u0 (x), x ∈ Ω, using the dG(0) method.

~ n denote the Mn vector of nodal values with respect to We let U n the nodal basis {ηn,i }M i=1 for Vn on In .

Duality, Adjoints, Green’s Functions – p. 296/304

Analysis of a space-time finite element method Bn : Bn

An : An

= ηn,i , ηn,j for 1 ≤ i, j ≤ Mn and Bn,n−1 : Bn,n−1 ij = ηn,i , ηn−1,j for 1 ≤ i ≤ Mn , 1 ≤ j ≤ Mn−1 denotes the mass matrices ij

ij

= ∇ηn,i , ∇ηn,j denotes the stiffness matrices.

Un satisfies

~ n − F~ (U − )kn = Bn,n−1 U ~ n−1 , Bn + kn An U n

n ≥ 1,

where (F~ (Un− ))i = (f (Un− ), ηn,i ).

Duality, Adjoints, Green’s Functions – p. 297/304

Analysis of a space-time finite element method Use of quadrature yields standard difference schemes Example If Mn is constant and the lumped mass quadrature is used to evaluate the coefficients of Bn and Bn,n−1 = Bn , then the resulting set of equations for the dG(0) approximation is the same as the equations for the nodal values of the backward Euler difference scheme. The dG(0) method is related to the backward Euler method, the cG(1) method is related to the Crank-Nicolson scheme, and the dG(1) method is related to the third order sub-diagonal Padé difference scheme.

Duality, Adjoints, Green’s Functions – p. 298/304

Analysis of a space-time finite element method Under general assumptions, the cG(q) and dG(q) have order of accuracy q + 1 in time and 2 in space at any point. In addition, they enjoy a superconvergence property in time at time nodes. The dG(q) method will have order of accuracy 2q + 1 in time and the cG(q) method will have order 2q in time at time nodes for sufficiently smooth solutions. The discontinuous Galerkin method has stability properties that make it well suited for the solution of dissipative problems. The continuous Galerkin method is “energy” preserving and is well-suited for problems with a conserved quantity.

Duality, Adjoints, Green’s Functions – p. 299/304

Analysis of a space-time finite element method We use the second definition of the adjoint, so use ¯i = ¯i (u, U ) =

Z

0

1

1

i us + U (1 − s) ds,

∂j us + U (1 − s))∇(ui s + Ui (1 − s) ds, 0 ∂ui Z 1 ∂fj ¯ ¯ fij = fij (u, U ) = (us + U (1 − s) ds. 0 ∂ui

β¯ij = β¯ij (u, U ) =

Z

Typically, ¯ and f¯ are piecewise continuous with respect to t and continuous, H 1 functions in space while β¯ is discontinuous in time and space.

Duality, Adjoints, Green’s Functions – p. 300/304

Analysis of a space-time finite element method The adjoint problem is  PD PD ¯ ˙ ¯ −φi − ∇ · ¯i ∇φi + j=1 βji · ∇φj − j=1 fij φj = ψi ,     (x, t) ∈ Ω × (t , 0], 1 ≤ i ≤ D, n

 φi (x, t) = 0,    φ(x, tn ) = 0,

(x, t) ∈ ∂Ω × (tn , 0], 1 ≤ i ≤ d, x ∈ Ω,

Example In the case of the scalar problem with constant diffusion, the adjoint problem is  ˙ ¯  −φ − ∆φ − f φ = ψ, (x, t) ∈ Ω × (tn , 0], φ(x, t) = 0, (x, t) ∈ ∂Ω × (tn , 0],   φ(x, tn ) = 0, x ∈ Ω. Duality, Adjoints, Green’s Functions – p. 301/304

Analysis of a space-time finite element method Example In the case of one parabolic equation with nonlinear diffusion coupled to one ordinary differential equation, the dual problem is  ˙ 1 − ∇ · ¯1 ∇φ1 + β¯11 ∇φ1 − f¯11 φ1 − f¯12 φ2 = ψ1 , − φ    −φ˙ + β¯ ∇φ − f¯ φ − f¯ φ = ψ , 2 12 1 21 1 22 2 2  φ1 (x, t) = 0,    φ(x, tn ) = 0.

Duality, Adjoints, Green’s Functions – p. 302/304

Analysis of a space-time finite element method For the cG method, we obtain tn

Z

Z +

0 tn

0

(e, ψ) dt = (e+ (0), φ(0)) ˙ (U , πP φ−φ)+((U )∇U, ∇(πP φ−φ))−(f (U ), πP φ−φ) dt.

For the dG method, we obtain Z

tn

(e, ψ) dt = (e (0), φ(0)) +

0

Z +

0

−

tn

n X j=1

[U ]j−1 , (πP φ −

φ)+ j−1

˙ (U , πP φ−φ)+((U )∇U, ∇(πP φ−φ))−(f (U ), πP φ−φ) dt. Duality, Adjoints, Green’s Functions – p. 303/304

Collaborators Sean Eastman, Colorado State University Mike Holst, UCSD Claes Johnson, Chalmers Mats Larson, Umea University Duane Mikulencak, Georgia Tech David Neckels, Colorado State University Tim Wildey, Colorado State University Roy Williams, Caltech

Duality, Adjoints, Green’s Functions – p. 304/304

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