on schwarz alternating methods for nonlinear elliptic pdes - CiteSeerX
ON SCHWARZ ALTERNATING METHODS FOR NONLINEAR ELLIPTIC PDES S. H. LUI
Abstract. The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an in nite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. This paper considers several Schwarz Alternating Methods for nonlinear elliptic problems. We show that Schwarz Alternating Methods can be imbedded in the framework of common techniques such as Banach and Schauder xed point methods and Global Inversion methods used to solve these nonlinear problems. Key words. domain decomposition, nonlinear elliptic PDE, Schwarz alternating method AMS subject classi cations. 65N55, 65J15
1. Introduction. The Schwarz Alternating Method was devised by H. A. Schwarz more than one hundred years ago to solve linear boundary value problems. It has garnered interest recently because of its potential as an ecient algorithm for parallel computers. See the fundamental work of Lions in [11] and [12]. The literature on this method for the linear boundary value problem is huge, see the recent reviews of Chan and Mathew [6] and Le Tallec [19], and the book of Smith, Bjorstad and Gropp [14]. The literature for nonlinear problems is rather sparse. Besides Lions' works, see also Cai and Dryja [3], Tai [15], Xu [20], Dryja and Hackbusch [7], Tai and Espedal [16], Tai and Xu [18], Tai and Tseng [17], and references therein. The e ectiveness of Schwarz methods for nonlinear problems has been demonstrated in many papers. See proceedings of the annual domain decomposition conferences beginning with [10]. We mention in particular the Newton-Krylov-Schwarz framework adopted in [4] and [5]. In this paper, we prove the convergence of the Schwarz sequence for some nonlinear elliptic partial di erential equations. We do not attempt to de ne the largest possible class of problems or give the weakest condition under which the Schwarz Alternating Method converges. The main aim is rather to illustrate that this remarkable method Hong Kong University of Science & Technology, Department of Mathematics, Clear Water Bay, Kowloon, Hong Kong ([email protected]). This work was in part supported by a grant from RGC CERG HKUST726/96E. 1
works for a wide variety of nonlinear elliptic PDEs. This paper is mostly concerned with multiplicative nonlinear Schwarz methods for two subdomains and they are discussed in the next section. In this class of methods, a nonlinear problem is solved in the rst subdomain followed by a nonlinear problem in the second subdomain. This is repeated until convergence to a desired accuracy has been reached. We shall consider three types of nonlinear elliptic PDEs, corresponding to three theories used to show existence of solutions to these PDEs. The three theories are the Banach xed point theory, Schauder xed point theory and the theory of Global Inversion. They are among the most well known tools in nonlinear analysis. Nonlinear PDEs are dicult to handle because of the in nite variety of nonlinearities and the possibility of an arbitrary number (including zero) of solutions. In addition, existence of a solution on the entire domain does not guarantee that the same PDE on a subdomain with a general boundary condition has a solution. There will probably be no single technique that can show existence/uniqueness for all nonlinear PDEs. It is indeed remarkable that Schwarz method is applicable to all these di erent types of PDEs mentioned above. In section three, we shall discuss three practical variants of Schwarz methods. The rst variant considers a sequence of functions resulting from the solution of linear versions of the given PDE while the second produces an \additive" Schwarz sequence which is suitable for parallel computation. The third variant is yet another Schwarz sequence where the subdomain problems can be computed in parallel. These three variants are applied to PDEs of the rst type, that is those whose solution is a xed point of a contracting operator. The rst Schwarz method for nonlinear problems is due to Lions [11]. He considers a functional I 2 C 1 (H01 ( ); IR) which is coercive, weakly lower semicontinuous, uniformly convex and bounded below. By making a correction alternately in each subdomain which minimizes the functional, he shows that the sequence converges to 2
the unique minimizer of the functional. Dryja and Hackbusch [7] study convergence of nonlinear subspace iterations for abstract nonlinear equations. They show that under weak assumptions, the nonlinear iteration converges locally with the same asymptotic speed as the corresponding linear iteration applied to the linearized problem. The paper of Tai and Espedal [16] considers monotone operators and proves the convergence of additive and multiplicative Schwarz sequences. Xu [20] gives convergence estimates for multigrid methods for nonlinear elliptic PDEs discretized by nite elements. We conclude this introduction with some notation. Let be a bounded, connected domain in IRN with a smooth boundary. Suppose = 1 [ 2 , where the subdomains
i are connected, have smooth boundaries and are overlapping. Let (u; v) denote the usual L2( ) inner product and kuk2 = (u; u). Denote the energy inner product in the Sobolev space H01 ( ) by [u; v] = norm on H ?1 ( ) by k k?1 with
kuk?1 =
R
ru rv
sup
and let kuk1 = [u; u]1=2. Denote the
v2H01 ( ); kvk1 =1
j[u; v]j:
Let 4i be the Laplacian operator considered as an operator from H01 ( i ) onto H ?1 ( i ),
i = 1; 2: The smallest eigenvalue of ?4 on is denoted by 1 while the smallest eigenvalue of ?4i is denoted by 1 ( i ); i = 1; 2. The collection of eigenvalues on is denoted by fj g1 j =1 . For notational convenience, we de ne 0 = ?1. We take overlapping to mean that H01 ( ) = H01 ( 1 )+ H01( 2 ). In this paper, a function in H01 ( i ) is considered as a function de ned on the whole domain by extension by zero. Let
Pi denote the orthogonal (with respect to the energy inner product) projection onto H01 ( i ), i = 1; 2. It is well known that d max(k(I ? P2 )(I ? P1 )k1 ; k(I ? P1 )(I ? P2 )k1 ) < 1: See Lions [11] and Bramble et al [2]. Throughout this paper, C will denote a positive constant which may not be the same at di erent occurrences. 3
2. Nonlinear Schwarz Method. In this section, we use the Schwarz method in conjunction with the methods of Banach and Schauder xed points and of Global Inversion to solve some nonlinear PDEs. Each subdomain problem is nonlinear.
2.1. Banach Fixed Point. The rst result is an adaptation of the variational approach of Lions [11] for linear problems to nonlinear problems. We assume the nonlinearity satis es a certain Lipschitz condition with a suciently small Lipschitz constant so that the method of proof for the linear problem still applies. We rst prove an existence and uniqueness result using the Banach xed point theorem. Proposition 1. Consider the equation
?4u = f (x; u; ru) on
with homogeneous Dirichlet boundary conditions. Assume f (x; u(x); ru(x)) 2 H ?1 ( ) for x 2 and u 2 H01 ( ) and satis es the condition
kf (x; u; ru) ? f (x; v; rv)k?1 c ku ? vk1 for u; v 2 H01 ( ) and some constant c < 1. Then, the equation has a unique solution in H01 ( ).
Proof: De ne the operator A : H01( ) ! H01( ) by A(u) = ?4?1f (x; u; ru). Then the equation is solved by nding a xed point of A. For u; v 2 H01 ( ), kA(u) ? A(v)k1 = k4?1(f (x; u; ru) ? f (x; v; rv))k1 = kf (x; u; ru) ? f (x; v; rv)k?1
c ku ? vk1 : By the Banach xed point theorem, A has a unique xed point in H01 ( ): 2 We remark that the Lipschitz constant above is optimal. For example, the equation
?4u = 1 u + g; 4
where g is not orthogonal in the L2 sense to an eigenfunction associated with 1 , has no solution. Theorem 1. Consider the equation
?4u = f (x; u; ru) on
(1)
with homogeneous Dirichlet boundary conditions. Assume f (x; u(x); ru(x)) 2 H ?1 ( ) for x 2 and u 2 H01 ( ) and satis es the condition
kf (x; u; ru) ? f (x; v; rv)k?1 c ku ? vk1 ; where u; v 2 H01 ( ); c is a constant such that c < 1 and p
d < 1 ? c2 ? c:
(2)
For n = 0; 1; 2; and some u(0) 2 H01 ( ), de ne the nonlinear Schwarz sequence as:
?4u(n+ 12 ) = f (x; u(n+ 21 ) ; ru(n+ 12 ) ) on 1 ; u(n+ 21 ) = u(n) on @ 1 ; ?4u(n+1) = f (x; u(n+1) ; ru(n+1) ) on 2 ; u(n+1) = u(n+ 21 ) on @ 2 : Then, the Schwarz sequence converges geometrically to the solution of (1) in the energy norm. Here, u(n+ 21 ) is considered as a function in H01 ( ) by de ning it to be u(n) on
n 1 and u(n+1) is de ned as u(n+ 21 ) on n 2 .
Proof: By the above proposition, (1) has a unique solution u. Next, we show that the Schwarz sequence is well-de ned. Let u(n+ 21 ) = u(n) + v1 where v1 2 H01 ( 1 ). The de ning equation for u(n+ 21 ) can be written as (3)
?4v1 = 4u(n) + h(x; v1 ; rv1 ) on 1 ;
where h(x; v1 ; rv1 ) = f (x; u(n) + v1 ; ru(n) + rv1 ). Assume u(n) 2 H01 ( ). For
v; w 2 H01 ( 1 ),
kh(x; v; rv) ? h(x; w; rw)kH ?1 ( 1 ) = kf (x; u(n) + v; ru(n) + rv) ? f (x; u(n) + w; ru(n) + rw)kH ?1 ( 1 ) c kv ? wk1 : 5
By Proposition 1 (applied to 1 ), (3) has a unique solution in H01 ( 1 ) and thus u(n+ 21 ) exists and is unique. This holds similarly for u(n+1) . We are now ready to show convergence of the Schwarz sequence. For any v1 2
H01 ( 1 ), [u(n+ 21 ) ? u(n) ; v1 ] = (f (x; u(n+ 21 ) ; ru(n+ 12 ) ); v1 ) ? [u(n); v1 ]
= (f (x; u; ru); v1 ) + f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru); v1 ? [u(n) ; v1 ]
= [u; v1 ] + f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru); v1 ? [u(n); v1 ]
= [P1 (u ? u(n) ); v1 ] + f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru); v1 : Noting that u(n+ 21 ) ? u(n) 2 H01 ( 1 ), the last equation can also be written as
u(n+ 12 ) ? u(n) = P1 (u ? u(n) ) ? 41?1 f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru)
or
e(n+ 21 ) = (I ? P1 )e(n) ? 4?1 1 f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru) ;
(4)
where e(n) = u(n) ? u. Similarly, ?
e(n+1) = (I ? P2 )e(n+ 21 ) ? 4?2 1 f (x; u(n+1) ; ru(n+1) ) ? f (x; u; ru) : We have
e(n+1) = (I ? P2 )(I ? P1 )e(n) ? (I ? P2 )4?1 1 f (x; u(n+ 21 ) ; ru(n+ 21 ) ) ? f (x; u; ru) ? ?
4?2 1 f (x; u(n+1) ; ru(n+1) ) ? f (x; u; ru) :
(5)
From (5), we obtain ?
kP2 e(n+1) k1 = k4?2 1 f (x; u(n+1); ru(n+1) ) ? f (x; u; ru) k1 cke(n+1)k1 : Noting that (I ? P1 )e(n) = (I ? P1 )e(n+ 21 ) from (4), we get, upon applying (I ? P2 ) to (5),
k(I ? P2 )e(n+1) k1 (k(I ? P2 )(I ? P1 )k1 + c) ke(n+ 21 ) k1: 6
Thus
ke(n+1)k21 = kP2 e(n+1) k21 + k(I ? P2 )e(n+1) k21 c2 ke(n+1)k21 + (k(I ? P2 )(I ? P1 )k1 + c)2 ke(n+ 21 ) k21 or
ke(n+1)k1 pke(n+ 21 ) k1 where
p = pc + d 2 : 1?c A similar expression to (5) is ?
e(n+ 12 ) = (I ? P1 )(I ? P2 )e(n? 21 ) ? (I ? P1 )4?2 1 f (x; u(n); ru(n) ) ? f (x; u; ru) ?
4?1 1 f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru) : From this we obtain
ke(n+ 21 ) k1 pke(n)k1 and hence
ke(n+1)k1 p2 ke(n)k1 : Thus the Schwarz sequence converges geometrically if p < 1 or equivalently, d
sup?1 je(n) j. We may de ne
kn =
2
sup?1 je(n) j < 1: sup 2 je(n) j 13
Lemma 1. Let fxn g be an unbounded sequence of real numbers. Suppose it has
the property that if fxn g is any subsequence such that xn ! 1 as j ! 1 then j
j
xn ?1 ! 1 as j ! 1. Then j
lim x = 1: n!1 n
Proof: Since the original sequence is unbounded, there must exist some subsequence fxn g which goes to in nity as j ! 1. Hence fxn ?1 g also goes to in nity as j ! 1. By the same property, xn ?2 ! 1 as j ! 1, etc. Given any positive , there are numbers N0 ; N1 ; N2 ; such that j
j
j
ni > N0 ) xn > 1 ni ? 1 > N1 ) xn ?1 > 1 i
i
ni ? 2 > N2 ) xn ?2 > 1 .. .
i
Suppose the conclusion of the lemma is false. Then there is some subsequence fxk g j
which is bounded by M , say. Take = M ?1 . Then there is some j and l so that
kj = ni ? l with ni ? l > Nl . Thus xk = xn ?l > 1 = M j
i
which is a contradiction. This completes the proof. It is natural to inquire whether the rather strong condition on the nonlinearity,
@f=@u 0, is really necessary. We believe that any restriction on f leading to a unique solution would also do. However, without any conditions on f , the quasilinear equation may have multiple solutions and some numerical evidence suggests that the Schwarz sequence does not converge. We tried several examples for which there are at least two distinct solutions. We monitor ku(n+ 21 ) ? u(n)k in 1 \ 2 and nd that it oscillates and does not seem to converge. 14
2.3. Global Inversion. Next, we show that the Schwarz method can be applied to a certain class of semilinear elliptic problem whose solution can be shown to be unique using the Global Inversion Theorem. Previous theories are not applicable because the equations have a term linear in the unknown. A strong assumption on the nonlinearity can imply existence and uniqueness of the solution. We shall consider two cases corresponding to resonance and non-resonance. For completeness, we rst prove the existence and uniqueness result for the nonresonance case. A similar version can be found in Ambrosetti and Prodi [1]. Proposition 3. Consider the semilinear equation
(11)
?4u = u + f (x; u) + g on
with homogeneous Dirichlet boundary conditions. Here 2 IR is given with 6= j for all j and f 2 C 1 ( ; IR) and satis es the conditions
(12)
kf (x; vn )k?1 ! 0 whenever kv k ! 1 n 1 kvn k1
and
(13)
k?1 < + fu (x; t) < k
for every x 2 and t 2 IR and for some k 2 IN. The function g is assumed to be in
L2 ( ). Then the equation (11) has a unique solution in H01 ( ).
Proof: We use the Global Inversion Theorem (see [1] for instance) to show this result. Let F : H01 ( ) ! H01 ( ) with F (u) = u + 4?1u + 4?1f (x; u) for u 2 H01 ( ). Now F is continuous and we need to show that it is proper and locally invertible. For the former, suppose hn 2 H01 ( ) and un 2 H01 ( ) with F (un ) = hn ! h 2 H01 ( ). We need to show that un has a convergent subsequence. Now suppose fkunk1g is unbounded. Then there is some subsequence which we still label by n such that
kunk1 ! 1 as n ! 1. Let zn = un =kunk1 . Then ?1 un ) : zn + 4?1 zn = kuhnk ? 4 kuf (x; n 1 n k1 15
Certainly, the right-hand side of the above goes to zero in the L2 norm as n ! 1. Thus fzng is bounded in H 2 and must have a convergent subsequence converging to some nonzero z strongly in the energy norm. From the above equation, we obtain the contradiction that z is an eigenfunction of ?4 with corresponding eigenvalue . Thus
fkunk1g must be bounded. This implies that f4?1un = hn ? un ? 4?1f (x; un )g is a bounded sequence in L2 ( ). Thus fung is bounded in H 2 and hence must have a strongly convergent subsequence in H01 ( ). To show that F is locally invertible, we simply note that the linear problem (I + 4?1 + 4?1fu (x; v))w = 0;
v; w 2 H01 ( );
rewritten in a more familiar form
?4w = ( + fu (x; v))w
(14)
has only the trivial solution because of the assumption on + fu . The Fredholm Alternative implies that F 0 (u) is invertible and by the Inverse Function Theorem, F is locally invertible. We nally can conclude from the Global Inversion Theorem that the semilinear elliptic equation (11) has a unique solution. 2 Note that (12) is satis ed when, for instance, f is a bounded function. Note also that if f also depends on ru, then (12) contains an extra term involving rw and more assumptions are required to conclude that the equation corresponding to (14) has only the trivial solution. For j = 1; 2, let k kH 1 ( ) be the norm induced by the inner product j
(v; w)H
1 ( j )
Z
j
rv rw + vw:
The following lemma is useful. Lemma 2. Let j = 1 or 2 and 2 IR; 6= i ( j ); 8i. Then for w 2 H?1 ( j )
fv 2 H 1 ( j ); v = 0 on @ j \ @ g, k(Pj + 4j?1 )wkH 1 ( ) C kPj wkH 1 ( ) : j
16
j
Here Pj + 4?j 1 is considered as an operator from H?1 ( j ) onto H01 ( j ).
Proof: Let f i g be an orthonormal basis (with respect to the inner product (; )H 1 ( ) ) of eigenfunctions of ?4j with corresponding eigenvalues i ( j ). Let j
w=
1 X i=1
ci i + v
where ci 2 IR and v is in the orthogonal complement of H01 ( j ) in H?1 ( j ) (i.e., v 2 H01 ( j )? where H?1 ( j ) = H01 ( j ) H01 ( j )? ). Note that 4?j 1 is a shorthand
for 4?j 1Rj where Rj : H?1 ( j ) ! H ?1 ( j ) is the restriction operator de ned by
z 2 H?1 ( j ); y 2 H01 ( j ):
(Rj z )(y) = (z; y)H 1 ( ) ; j
By de nition, Rj v = 0. Thus
k(Pj + 4?1 )wk2 1
i
ci 1 ? ( ) i j H 1 ( ) 1 2 X = c2i 1 ? (
i j) i=1 2 1 ? (
kPj wk2H 1 ( ) ; c j)
H ( j ) =
j
1
X
i=1
j
j
where c ( j ) is an eigenvalue of ?4j closest to in the sense that 1
1 ? ; 8i: ? (
) ( ) i j
c j
2 Theorem 3. Consider the semilinear elliptic equation as in Proposition 3 except
that (13) is replaced by
(15)
+ fu (x; t) 0; t 2 IR:
For n = 0; 1; 2; and any u(0) 2 H01 ( ), de ne the Schwarz sequence as:
?4u(n+ 21 ) = u(n+ 12 ) + f (x; u(n+ 21 ) ) + g on 1 ; u(n+ 21 ) = u(n) on @ 1 ; ?4u(n+1) = u(n+1) + f (x; u(n+1)) + g on 2 ; u(n+1) = u(n+ 21 ) on @ 2 : 17
Then the Schwarz sequence converges geometrically to the unique solution of the semilinear elliptic equation (11) in the L1 norm.
Proof: Because of (15), the Schwarz sequence is well de ned by Proposition 3. In the rst part of the proof, we show that the Schwarz sequence is bounded. For any v1 2 H01 ( 1 ), [u(n+ 12 ) ? u(n); v1 ] = (u(n+ 12 ) + f (n+ 12 ) + g; v1 ) ? [u(n) ; v1 ] where f (n+ 12 ) = f (x; u(n+ 21 ) ). Noting that u(n+ 12 ) ? u(n) 2 H01 ( 1 ), we have (I + 4?1 1)u(n+ 21 ) = ?4?1 1(f (n+ 12 ) + g) + (I ? P1 )u(n) ; or (P1 + 41?1)u(n+ 21 ) = ?4?1 1(f (n+ 12 ) + g) and (I ? P1 )u(n+ 21 ) = (I ? P1 )u(n): Applying lemma 2 to the rst term, noting that 6= i ( 1 ); 8i, we obtain
kP1 u(n+ 21 ) k1 C (1 + kf (n+ 21 ) k?1 ): Assume fku(n+ 21 ) k1 g is unbounded. Then there is some subsequence which we label by nj + 12 such that ku(n + 21 ) k1 ! 1 as j ! 1. Now j
ku(n + 12 ) k1 kP1 u(n + 12 ) k1 + k(I ? P1 )u(n + 12 ) k1 j
j
j
C (1 + kf (n + 12 ) k?1 ) + k(I ? P1 )u(n ) k1 :
(16)
j
j
Thus (n ) 1 o(1) + k(I ?(nP1+)u1 ) k1 : j
ku
j
2
k1
This shows that fku(n ) k1g must also be unbounded. By lemma 1, the entire sequences j
(not just subsequences) fku(n)k1 g and fku(n+ 21 ) k1 g go to in nity as n ! 1. Applying lemma 2 to (P2 + 4?2 1 )u(n) = ?4?2 1(f (n) + g); 18
we have
kP2 u(n) k1 C (1 + kf (n)k?1 ):
(17) Also,
k(I ? P1 )u(n) k1 k(I ? P1 )P2 u(n) k1 + k(I ? P1 )(I ? P2 )u(n) k1 C (1 + kf (n) k?1) + dku(n)k1 : Using this result in (16), we obtain
ku(n+ 21 ) k1 o(1) ku(n+ 21 ) k1 + o(1) + d ku(n)k1 ku(n)k1 or
ku(n+ 21 ) k1 d + o(1): ku(n)k1 In a parallel development, we also have
ku(n+1)k1 d + o(1): ku(n+ 12 ) k1 Combining the above equations, we have
ku(n+1)k1 d2 + o(1); ku(n)k1 contradicting that fku(n)k1 g is unbounded since d < 1. Thus fku(n+ 21 ) k1 g is bounded. If fku(n)k1g is unbounded, then from (17),
ku(n+1)k1 kP2 u(n+1)k1 + k(I ? P2)u(n+1) k1 C (1+ kf (n+1)k?1 )+ k(I ? P2)u(n+ 21 ) k1 : Hence,
u(n+ 21 ) k1 1 o(1) + k(I ? P(2n)+1)
ku
k1
which is a contradiction since fku(n+ 21 ) k1g is bounded. This shows that fku(n)k1g must also be bounded.
19
Since the Schwarz sequences are bounded in k k1 , there exist u0 ; u1; u2 2 H01 ( ) and a subsequence labeled by nj such that u(n ) * u0 ; u(n + 21 ) * u1 and u(n +1) * j
j
j
u2 (weak convergence in the energy norm). We now show that the subsequences actually converge strongly in the energy norm. By compactness of the restriction map
H 1 ( 1 ) ,! L2 (@ 1 ), we have u(n + 21 ) ! u1 in L2 (@ 1 ). Since fu(n + 21 ) +f (n + 12 ) +gg is bounded in the L2 norm, fu(n + 21 ) g is bounded in H 2 ( 1 ) from the de ning equation j
j
j
j
of u(n + 21 ) . By extracting a further subsequence if necessary, u(n + 21 ) converges to u1 j
j
strongly in H 1 ( 1 ). Similarly, u(n ) ! u0 and u(n +1) ! u2 strongly in H 1 ( 2 ). j
j
Hence, u(n ) ; u(n + 21 ) and u(n +1) converge strongly in H01 ( ). Note that u0 and u2 j
j
j
are weak solutions of semilinear equation (11) on 2 while u1 is a weak solution on
1 . Finally, as in the proof of the previous theorem, we apply the strong maximum principle to show convergence in the L1 norm of the iterates to the solution to the semilinear equation (11) on .
2
For the above semilinear equation, we made the strong assumption (15) so that the maximum principle can be applied in the nal step of the proof. It is unknown whether the Schwarz iteration with the weaker condition (13) converges. Next we consider the resonance problem for the above semilinear equation. See [1] for a proof. Proposition 4. Consider the semilinear equation
?4u = 1 u + f (x; u) + g on
(18)
with homogeneous Dirichlet boundary conditions. Here f 2 C 1 ( ; IR) and satis es the following conditions: 1. 9M such that jf (x; s)j M; 8x 2 ; s 2 IR: 2. lims!1 f (x; s) = f 2 IR; 8x 2 : 3. f?
R
1
R
< ? g 1 < f+
R
1 ,
where 1 is the positive eigenfunction of
?4 corresponding to the principal eigenvalue 1 . 20
4. fu (x; s) 6= 0; 8x 2 ; s 2 IR: 5. 1 + fu (x; t) < 2 ; 8x 2 ; t 2 IR: The function g is assumed to be in L2 ( ). Then, the equation (18) has a unique solution in H01 ( ). Theorem 4. Consider the hypotheses as in the above Proposition except that the
5th condition is replaced by: 8x 2 and s 2 IR, 1 + fu (x; s) 0. In addition, assume the subdomains are proper subsets of so that 1 < min(1 ( 1 ); 1 ( 2 )). For n = 0; 1; 2; and any u(0) 2 H01 ( ), de ne the Schwarz sequence as:
?4u(n+ 12 ) = 1 u(n+ 21 ) + f (x; u(n+ 21 ) ) + g on 1 ; u(n+ 21 ) = u(n) on @ 1 ; ?4u(n+1) = 1 u(n+1) + f (x; u(n+1) ) + g on 2 ; u(n+1) = u(n+ 21 ) on @ 2 : Then the Schwarz sequence converges geometrically to the unique solution of the semilinear elliptic equation (18) in the L1 norm.
Proof: The Schwarz sequence is well de ned by Proposition 3 because of the assumption that the subdomains are proper subsets of . The rest of the proof follows exactly as in the proof of Theorem 3.
2
3. Other Schwarz Methods. In the last section, each subdomain problem is nonlinear. We now consider iterations where linear problems are solved in each subdomain. This is of great importance because in practice, we always like to avoid solving nonlinear problems. One way is in the framework of Newton's method. Write a model semilinear problem as G(u) u ? 4?1f (x; u) = 0 for u 2 H01 ( ). Suppose it has a solution u and suppose that k4?1fu (x; u)k < 1, then for initial guess u(0) suciently close to u, the Newton iterates u(n) de ned by (19)
u(n+1) = u(n) ? Gu (u(n) )?1 G(u(n) )
converge to u. Note that the assumption means that Gu = I ? 4?1fu has a bounded inverse in a neighborhood of u. Now each linear problem (19) can be solved using the classical Schwarz Alternating Method. We take three di erent approaches. 21
In this section, we consider the equation
?4u = f (x; u; ru) on
(20)
with homogeneous Dirichlet boundary conditions. Assume f (x; u(x); ru(x)) 2 H ?1 ( ) for x 2 and u 2 H01 ( ) and satis es the condition
kf (x; u; ru) ? f (x; v; rv)k?1 c ku ? vk1 ; where u; v 2 H01 ( ); c is a constant such that c < 1.
3.1. Linear Schwarz Method. In the linear Schwarz sequence de ned below, each subdomain problem is linear.
p
Theorem 5. Assume d < 1 ? c2 ? c. For n = 0; 1; 2; and any u(0) 2 H01 ( ),
de ne the linear Schwarz sequence by
?4u(n+ 12 ) = f (x; u(n); ru(n) ) on 1 ; u(n+ 21 ) = u(n) on @ 1 ; ?4u(n+1) = f (x; u(n+ 21 ) ; ru(n+ 12 ) ) on 2 ; u(n+1) = u(n+ 21 ) on @ 2 : Then, the Schwarz sequence converges to the solution of (20) in the energy norm.
Proof: In a similar manner as in the proof of Theorem 1, we obtain (21)
?
u(n+ 21 ) ? u(n) = P1 (u ? u(n) ) ? 4?1 1 f (x; u(n) ; ru(n) ) ? f (x; u; ru) :
and
u(n+1) ? u(n+ 21 ) = P2 (u ? u(n+ 12 ) ) ? 4?2 1 f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru) : De ne e(n) = u(n) ? u, we have ?
e(n+1) = (I ? P2 )(I ? P1 )e(n) ? (I ? P2 )4?1 1 f (x; u(n) ; ru(n)) ? f (x; u; ru) ?
42?1 f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru) : ?
From (21), e(n+ 21 ) = (I ? P1 )e(n) ? 4?1 1 f (x; u(n) ; ru(n) ) ? f (x; u; ru) . 22
We have the estimate
ke(n+1)k21 = kP2 e(n+1) k21 + k(I ? P2 )e(n+1) k21 c2 ke(n+ 21 ) k21 + k(I ? P2 )e(n+ 12 ) k21 ?
c2 ke(n+ 21 ) k21 + k(I ? P2 )(I ? P1 )e(n) ? (I ? P2 )4?1 1 f (x; u(n) ; ru(n)) ? f (x; u; ru) k21 c2 ke(n+ 21 ) k21 + (d + c)2 ke(n)k21 : We rewrite the above as
ke(n+1)1 k21 A ke(n+ 21 ) k21 ; where A = c2 (c + d)2 : 1 0 ke(n)k21 ke(n+ 2 ) k21
Similarly,
ke(n+ 21 ) k21 A ke(n)1k21 : ke(n) k21 ke(n? 2 ) k21
ke(n+1)1 k21 A2 ke(n)1k21 : ke(n+ 2 ) k21 ke(n? 2 ) k21
Thus
A sucient condition for the convergence of these vectors to zero is that the eigenvalues of A2 have magnitude less than one or equivalently that the spectral radius of A is less than one. The eigenvalues of A are p
c2 c4 + 4(c + d)2 : 2
Clearly, the eigenvalue with the larger magnitude is the one with the plus sign in the above expression. Thus the Schwarz sequence converges in the energy norm provided p
c2 + c4 + 4(c + d)2 < 1 or d < p1 ? c2 ? c: 2
2
3.2. Additive Schwarz Method. One of the main motivations for studying Schwarz methods is their suitability for parallel computers. The algorithms discussed thus far are not ideal for parallel computers because the subdomain problems must 23
be solved sequentially. We now show convergence of an additive Schwarz method in which subdomain problems can be solved concurrently. The additive Schwarz method was rst proposed by Dryja and Widlund [8] for linear elliptic PDEs. Theorem 6. For n = 0; 1; 2; and any u(0) 2 H01 ( ), de ne the sequences
?4d(n+ 21 ) = f (x; u(n); ru(n) ) + 4u(n) on 1 ?4d(n+1) = f (x; u(n); ru(n) ) + 4u(n) on 2 ; for d(n+ 21 ) 2 H01 ( 1 ) and d(n+1) 2 H01 ( 2 ). De ne the additive Schwarz sequence by
u(n+1) = u(n) +!(d(n+ 21 ) +d(n+1) ) where ! is a relaxation parameter with 0 < ! < 1=2. Assume kI ? !(P1 + P2 )k1 + 2!c < 1. Then, the additive Schwarz sequence converges geometrically to the solution of (20) in the energy norm.
Proof: The analysis is similar to before and we only record the key equations. From the de ning equations of d(n+ 21 ) and d(n+1) , we have ?
?
d(n+ 12 ) = ?P1 e(n) ? 4?1 1 f (x; u(n) ; ru(n) ) ? f (x; u; ru)
d(n+1) = ?P2 e(n) ? 4?2 1 f (x; u(n) ; ru(n) ) ? f (x; u; ru) : Substituting into the de nition of u(n+1) , we obtain ?
e(n+1) = (I ? !(P1 + P2 ))e(n) ? !(4?1 1 + 4?2 1) f (x; u(n) ; ru(n) ) ? f (x; u; ru) : Hence
ke(n+1)k1 kI ? !(P1 + P2 )k1 ke(n)k1 + 2!cke(n)k1: When 0 < ! < 1=2; kI ? !(P1 + P2 )k1 < 1 and the result follows. 2 Note that the subdomain problems are linear and can be solved concurrently. Roughly speaking, d(n+ 21 ) and d(n+1) are corrections to the iterate u(n) in the subdomains 1 and 2 , respectively, and the right-hand sides of the de ning equations for these corrections are the residuals of u(n) in the subdomains. If f is independent of u, then this reduces to the classical additive Schwarz method with a relaxation parameter.
24
3.3. Parallel Schwarz Method. Other variations are also possible. We give one more which we called the parallel Schwarz method because the subdomain problems can also be solved in parallel. The proof is similar to previous ones and is omitted. Theorem 7. Assume d < 1 ? 2c. For n = 0; 1; 2; and any u(0) de ne u(? 21 ) = u(0) and de ne the parallel Schwarz sequence by
2 H01 ( ),
?4u(n+ 12 ) = f (x; u(n) ; ru(n) ) on 1 ; u(n+ 21 ) = u(n) on @ 1 ?4u(n+1) = f (x; u(n? 21 ) ; ru(n? 21 ) ) on 2 ; u(n+1) = u(n? 12 ) on @ 2 : Then, the parallel Schwarz sequence converges to the solution of (20) in the energy norm.
4. Discussions and Conclusion. In this paper, we showed how Schwarz Alternating Methods can be imbedded within the framework of Banach and Schauder xed point theories and Global Inversion theory to construct solutions of nonlinear elliptic PDEs. We also give other versions of these methods where a linear problem is solved in each subdomain and these linear problems can be computed in parallel. We treated homogeneous boundary conditions in this paper. For the problems considered here, non-homogeneous boundary conditions can also be handled. If the boundary condition is u = h 2 H 1=2 (@ ), then the procedure is to extend h to be a function in H 1 ( ) in a bounded way and to make a change of variable w = u ? h so that w satis es the perturbed PDE
?4w = F (x; w; rw) + 4h with homogeneous boundary conditions, where F (x; w; rw) = f (x; w + h; r(w + h)). For PDEs of Theorem 1, it is trivial to check that F satis es the same Lipschitz condition as f . For PDEs of Theorem 2, @F=@w = @f=@w and
jF (x; w; rw)j = jf (x; w + h; r(w + h))j 25
C (1 + jr(w + h)j ) C (1 + (jrwj + jrhj) ) C [1 + 2 (jrwj + jrhj )] C (1 + jrwj ) and hence Theorem 2 can be applied to the perturbed PDE. We can also check that Theorem 3 can be applied to the perturbed PDE. This procedure, of course, does not work for all nonlinear PDEs. Future work include some numerical experiments and extending our results to the case of multiple subdomains. For the PDEs of Theorem 1, this is manageable. In particular, for the additive Schwarz sequence, this is trivial. If there are m subdomains, as long as
kI ? !(P1 + + Pm )k1 + m!c < 1; the sequence converges. Here, 0 < ! < 1=K where K is the minimum number of colors needed to color the subdomains in such a way that overlapping subdomains are assigned di erent colors. The extension for the other classes of PDEs is not at all obvious principally because of the diculty to apply the maximum principle in these cases. It is desirable to use tools other than the maximum principle for these classes of PDEs. Besides possibly weakening the hypotheses required, they may allow extension to multiple subdomains.
Acknowledgment. I thank Professor Juncheng Wei for helpful discussions and Professor Martin Gander for reading an early draft. I also thank the referees for carefully reading the original manuscript and suggesting numerous improvements. REFERENCES [1] A. Ambrosetti and G. Prodi. A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge, 1993. [2] J. H. Bramble, J. E. Pasciak, J. Wang, and J. Xu. Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comput., 57:1{21, 1991. 26
[3] X. C. Cai and M. Dryja. Domain decomposition methods for monotone nonlinear elliptic problems. In D. Keyes and J. Xu, editors, Domain decomposition methods in scienti c and engineering computing, pages 335{360, Providence, R.I., 1994. AMS. [4] X. C. Cai, W. D. Gropp, D. E. Keyes, R. G. Melvin, and D. P. Young. Parallel NewtonKrylov-Schwarz algorithms for the transonic full potential equation. SIAM J. Sci. Comput., 19:245{265, 1998. [5] X. C. Cai, D. E. Keyes, and V. Venkatakrishnan. Newton-Krylov-Schwarz: An implicit solver for cfd. In R. Glowinski et al., editor, Proceedings of the Eight International Conference on Domain Decomposition Methods in Science and Engineering, New York, 1997. Wiley. [6] T. F. Chan and T. P. Mathew. Domain decomposition algorithms. Acta Numerica, pages 61{143, 1994. [7] M. Dryja and W. Hackbusch. On the nonlinear domain decomposition method. BIT, pages 296{311, 1997. [8] M. Dryja and O. B. Widlund. An additive variant of the Schwarz alternating method for the case of many subregions. Technical report, Courant Institute Technical Report 339, 1987. [9] L. C. Evans. Partial Di erential Equations. AMS, Providence, RI, 1998. [10] R. Glowinski, G. H. Golub, G. A. Meurant, and J. Periaux. First Int. Symp. on Domain Decomposition Methods. SIAM, Philadelphia, 1988. [11] P. L. Lions. On the Schwarz alternating method I. In R. Glowinski, G. H. Golub, G. A. Meurant, and J. Periaux, editors, First Int. Symp. on Domain Decomposition Methods, pages 1{42, Philadelphia, 1988. SIAM. [12] P. L. Lions. On the Schwarz alternating method II. In T. F. Chan, R. Glowinski, J. Periaux, and O. Widlund, editors, Second Int. Conference on Domain Decomposition Methods, pages 47{70, Philadelphia, 1989. SIAM. [13] L. Nirenberg. Topics in Nonlinear Functional Analysis. Courant Institute of Mathematical Sciences, New York University, New York, 1974. [14] B. F. Smith, P. Bjorstad, and W. D. Gropp. Domain Decomposition : Parallel Multilevel Algorithms for Elliptic Partial Di erential Equations. Cambridge University Press, New York, 1996. [15] X. C. Tai. Domain decomposition for linear and nonlinear elliptic problems via function or space decomposition. In D. Keyes and J. Xu, editors, Domain decomposition methods in scienti c and engineering computing, pages 335{360, Providence, R.I., 1994. AMS. [16] X. C. Tai and M. Espedal. Rate of convergence of some space decomposition methods for linear and nonlinear problems. SIAM J. Numer. Anal., 35:1558{1570, 1998. [17] X. C. Tai and P. Tseng. Convergence rate analysis of an asynchronous space decomposition method for convex minimization. Report 120, Department of Mathematics, University of Bergen, 1998. [18] X. C. Tai and J. Xu. Global convergence of subspace correction methods for convex optimization problems. Report 114, Department of Mathematics, University of Bergen, 1998. [19] P. Le Tallec. Domain decomposition methods in computational mechanics. Computational Mechanics Advances, 1:121{220, 1994. [20] J. Xu. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal., 33:1759{1777, 1996.
27
Abstract. The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an in nite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. This paper considers several Schwarz Alternating Methods for nonlinear elliptic problems. We show that Schwarz Alternating Methods can be imbedded in the framework of common techniques such as Banach and Schauder xed point methods and Global Inversion methods used to solve these nonlinear problems. Key words. domain decomposition, nonlinear elliptic PDE, Schwarz alternating method AMS subject classi cations. 65N55, 65J15
1. Introduction. The Schwarz Alternating Method was devised by H. A. Schwarz more than one hundred years ago to solve linear boundary value problems. It has garnered interest recently because of its potential as an ecient algorithm for parallel computers. See the fundamental work of Lions in [11] and [12]. The literature on this method for the linear boundary value problem is huge, see the recent reviews of Chan and Mathew [6] and Le Tallec [19], and the book of Smith, Bjorstad and Gropp [14]. The literature for nonlinear problems is rather sparse. Besides Lions' works, see also Cai and Dryja [3], Tai [15], Xu [20], Dryja and Hackbusch [7], Tai and Espedal [16], Tai and Xu [18], Tai and Tseng [17], and references therein. The e ectiveness of Schwarz methods for nonlinear problems has been demonstrated in many papers. See proceedings of the annual domain decomposition conferences beginning with [10]. We mention in particular the Newton-Krylov-Schwarz framework adopted in [4] and [5]. In this paper, we prove the convergence of the Schwarz sequence for some nonlinear elliptic partial di erential equations. We do not attempt to de ne the largest possible class of problems or give the weakest condition under which the Schwarz Alternating Method converges. The main aim is rather to illustrate that this remarkable method Hong Kong University of Science & Technology, Department of Mathematics, Clear Water Bay, Kowloon, Hong Kong ([email protected]). This work was in part supported by a grant from RGC CERG HKUST726/96E. 1
works for a wide variety of nonlinear elliptic PDEs. This paper is mostly concerned with multiplicative nonlinear Schwarz methods for two subdomains and they are discussed in the next section. In this class of methods, a nonlinear problem is solved in the rst subdomain followed by a nonlinear problem in the second subdomain. This is repeated until convergence to a desired accuracy has been reached. We shall consider three types of nonlinear elliptic PDEs, corresponding to three theories used to show existence of solutions to these PDEs. The three theories are the Banach xed point theory, Schauder xed point theory and the theory of Global Inversion. They are among the most well known tools in nonlinear analysis. Nonlinear PDEs are dicult to handle because of the in nite variety of nonlinearities and the possibility of an arbitrary number (including zero) of solutions. In addition, existence of a solution on the entire domain does not guarantee that the same PDE on a subdomain with a general boundary condition has a solution. There will probably be no single technique that can show existence/uniqueness for all nonlinear PDEs. It is indeed remarkable that Schwarz method is applicable to all these di erent types of PDEs mentioned above. In section three, we shall discuss three practical variants of Schwarz methods. The rst variant considers a sequence of functions resulting from the solution of linear versions of the given PDE while the second produces an \additive" Schwarz sequence which is suitable for parallel computation. The third variant is yet another Schwarz sequence where the subdomain problems can be computed in parallel. These three variants are applied to PDEs of the rst type, that is those whose solution is a xed point of a contracting operator. The rst Schwarz method for nonlinear problems is due to Lions [11]. He considers a functional I 2 C 1 (H01 ( ); IR) which is coercive, weakly lower semicontinuous, uniformly convex and bounded below. By making a correction alternately in each subdomain which minimizes the functional, he shows that the sequence converges to 2
the unique minimizer of the functional. Dryja and Hackbusch [7] study convergence of nonlinear subspace iterations for abstract nonlinear equations. They show that under weak assumptions, the nonlinear iteration converges locally with the same asymptotic speed as the corresponding linear iteration applied to the linearized problem. The paper of Tai and Espedal [16] considers monotone operators and proves the convergence of additive and multiplicative Schwarz sequences. Xu [20] gives convergence estimates for multigrid methods for nonlinear elliptic PDEs discretized by nite elements. We conclude this introduction with some notation. Let be a bounded, connected domain in IRN with a smooth boundary. Suppose = 1 [ 2 , where the subdomains
i are connected, have smooth boundaries and are overlapping. Let (u; v) denote the usual L2( ) inner product and kuk2 = (u; u). Denote the energy inner product in the Sobolev space H01 ( ) by [u; v] = norm on H ?1 ( ) by k k?1 with
kuk?1 =
R
ru rv
sup
and let kuk1 = [u; u]1=2. Denote the
v2H01 ( ); kvk1 =1
j[u; v]j:
Let 4i be the Laplacian operator considered as an operator from H01 ( i ) onto H ?1 ( i ),
i = 1; 2: The smallest eigenvalue of ?4 on is denoted by 1 while the smallest eigenvalue of ?4i is denoted by 1 ( i ); i = 1; 2. The collection of eigenvalues on is denoted by fj g1 j =1 . For notational convenience, we de ne 0 = ?1. We take overlapping to mean that H01 ( ) = H01 ( 1 )+ H01( 2 ). In this paper, a function in H01 ( i ) is considered as a function de ned on the whole domain by extension by zero. Let
Pi denote the orthogonal (with respect to the energy inner product) projection onto H01 ( i ), i = 1; 2. It is well known that d max(k(I ? P2 )(I ? P1 )k1 ; k(I ? P1 )(I ? P2 )k1 ) < 1: See Lions [11] and Bramble et al [2]. Throughout this paper, C will denote a positive constant which may not be the same at di erent occurrences. 3
2. Nonlinear Schwarz Method. In this section, we use the Schwarz method in conjunction with the methods of Banach and Schauder xed points and of Global Inversion to solve some nonlinear PDEs. Each subdomain problem is nonlinear.
2.1. Banach Fixed Point. The rst result is an adaptation of the variational approach of Lions [11] for linear problems to nonlinear problems. We assume the nonlinearity satis es a certain Lipschitz condition with a suciently small Lipschitz constant so that the method of proof for the linear problem still applies. We rst prove an existence and uniqueness result using the Banach xed point theorem. Proposition 1. Consider the equation
?4u = f (x; u; ru) on
with homogeneous Dirichlet boundary conditions. Assume f (x; u(x); ru(x)) 2 H ?1 ( ) for x 2 and u 2 H01 ( ) and satis es the condition
kf (x; u; ru) ? f (x; v; rv)k?1 c ku ? vk1 for u; v 2 H01 ( ) and some constant c < 1. Then, the equation has a unique solution in H01 ( ).
Proof: De ne the operator A : H01( ) ! H01( ) by A(u) = ?4?1f (x; u; ru). Then the equation is solved by nding a xed point of A. For u; v 2 H01 ( ), kA(u) ? A(v)k1 = k4?1(f (x; u; ru) ? f (x; v; rv))k1 = kf (x; u; ru) ? f (x; v; rv)k?1
c ku ? vk1 : By the Banach xed point theorem, A has a unique xed point in H01 ( ): 2 We remark that the Lipschitz constant above is optimal. For example, the equation
?4u = 1 u + g; 4
where g is not orthogonal in the L2 sense to an eigenfunction associated with 1 , has no solution. Theorem 1. Consider the equation
?4u = f (x; u; ru) on
(1)
with homogeneous Dirichlet boundary conditions. Assume f (x; u(x); ru(x)) 2 H ?1 ( ) for x 2 and u 2 H01 ( ) and satis es the condition
kf (x; u; ru) ? f (x; v; rv)k?1 c ku ? vk1 ; where u; v 2 H01 ( ); c is a constant such that c < 1 and p
d < 1 ? c2 ? c:
(2)
For n = 0; 1; 2; and some u(0) 2 H01 ( ), de ne the nonlinear Schwarz sequence as:
?4u(n+ 12 ) = f (x; u(n+ 21 ) ; ru(n+ 12 ) ) on 1 ; u(n+ 21 ) = u(n) on @ 1 ; ?4u(n+1) = f (x; u(n+1) ; ru(n+1) ) on 2 ; u(n+1) = u(n+ 21 ) on @ 2 : Then, the Schwarz sequence converges geometrically to the solution of (1) in the energy norm. Here, u(n+ 21 ) is considered as a function in H01 ( ) by de ning it to be u(n) on
n 1 and u(n+1) is de ned as u(n+ 21 ) on n 2 .
Proof: By the above proposition, (1) has a unique solution u. Next, we show that the Schwarz sequence is well-de ned. Let u(n+ 21 ) = u(n) + v1 where v1 2 H01 ( 1 ). The de ning equation for u(n+ 21 ) can be written as (3)
?4v1 = 4u(n) + h(x; v1 ; rv1 ) on 1 ;
where h(x; v1 ; rv1 ) = f (x; u(n) + v1 ; ru(n) + rv1 ). Assume u(n) 2 H01 ( ). For
v; w 2 H01 ( 1 ),
kh(x; v; rv) ? h(x; w; rw)kH ?1 ( 1 ) = kf (x; u(n) + v; ru(n) + rv) ? f (x; u(n) + w; ru(n) + rw)kH ?1 ( 1 ) c kv ? wk1 : 5
By Proposition 1 (applied to 1 ), (3) has a unique solution in H01 ( 1 ) and thus u(n+ 21 ) exists and is unique. This holds similarly for u(n+1) . We are now ready to show convergence of the Schwarz sequence. For any v1 2
H01 ( 1 ), [u(n+ 21 ) ? u(n) ; v1 ] = (f (x; u(n+ 21 ) ; ru(n+ 12 ) ); v1 ) ? [u(n); v1 ]
= (f (x; u; ru); v1 ) + f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru); v1 ? [u(n) ; v1 ]
= [u; v1 ] + f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru); v1 ? [u(n); v1 ]
= [P1 (u ? u(n) ); v1 ] + f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru); v1 : Noting that u(n+ 21 ) ? u(n) 2 H01 ( 1 ), the last equation can also be written as
u(n+ 12 ) ? u(n) = P1 (u ? u(n) ) ? 41?1 f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru)
or
e(n+ 21 ) = (I ? P1 )e(n) ? 4?1 1 f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru) ;
(4)
where e(n) = u(n) ? u. Similarly, ?
e(n+1) = (I ? P2 )e(n+ 21 ) ? 4?2 1 f (x; u(n+1) ; ru(n+1) ) ? f (x; u; ru) : We have
e(n+1) = (I ? P2 )(I ? P1 )e(n) ? (I ? P2 )4?1 1 f (x; u(n+ 21 ) ; ru(n+ 21 ) ) ? f (x; u; ru) ? ?
4?2 1 f (x; u(n+1) ; ru(n+1) ) ? f (x; u; ru) :
(5)
From (5), we obtain ?
kP2 e(n+1) k1 = k4?2 1 f (x; u(n+1); ru(n+1) ) ? f (x; u; ru) k1 cke(n+1)k1 : Noting that (I ? P1 )e(n) = (I ? P1 )e(n+ 21 ) from (4), we get, upon applying (I ? P2 ) to (5),
k(I ? P2 )e(n+1) k1 (k(I ? P2 )(I ? P1 )k1 + c) ke(n+ 21 ) k1: 6
Thus
ke(n+1)k21 = kP2 e(n+1) k21 + k(I ? P2 )e(n+1) k21 c2 ke(n+1)k21 + (k(I ? P2 )(I ? P1 )k1 + c)2 ke(n+ 21 ) k21 or
ke(n+1)k1 pke(n+ 21 ) k1 where
p = pc + d 2 : 1?c A similar expression to (5) is ?
e(n+ 12 ) = (I ? P1 )(I ? P2 )e(n? 21 ) ? (I ? P1 )4?2 1 f (x; u(n); ru(n) ) ? f (x; u; ru) ?
4?1 1 f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru) : From this we obtain
ke(n+ 21 ) k1 pke(n)k1 and hence
ke(n+1)k1 p2 ke(n)k1 : Thus the Schwarz sequence converges geometrically if p < 1 or equivalently, d
sup?1 je(n) j. We may de ne
kn =
2
sup?1 je(n) j < 1: sup 2 je(n) j 13
Lemma 1. Let fxn g be an unbounded sequence of real numbers. Suppose it has
the property that if fxn g is any subsequence such that xn ! 1 as j ! 1 then j
j
xn ?1 ! 1 as j ! 1. Then j
lim x = 1: n!1 n
Proof: Since the original sequence is unbounded, there must exist some subsequence fxn g which goes to in nity as j ! 1. Hence fxn ?1 g also goes to in nity as j ! 1. By the same property, xn ?2 ! 1 as j ! 1, etc. Given any positive , there are numbers N0 ; N1 ; N2 ; such that j
j
j
ni > N0 ) xn > 1 ni ? 1 > N1 ) xn ?1 > 1 i
i
ni ? 2 > N2 ) xn ?2 > 1 .. .
i
Suppose the conclusion of the lemma is false. Then there is some subsequence fxk g j
which is bounded by M , say. Take = M ?1 . Then there is some j and l so that
kj = ni ? l with ni ? l > Nl . Thus xk = xn ?l > 1 = M j
i
which is a contradiction. This completes the proof. It is natural to inquire whether the rather strong condition on the nonlinearity,
@f=@u 0, is really necessary. We believe that any restriction on f leading to a unique solution would also do. However, without any conditions on f , the quasilinear equation may have multiple solutions and some numerical evidence suggests that the Schwarz sequence does not converge. We tried several examples for which there are at least two distinct solutions. We monitor ku(n+ 21 ) ? u(n)k in 1 \ 2 and nd that it oscillates and does not seem to converge. 14
2.3. Global Inversion. Next, we show that the Schwarz method can be applied to a certain class of semilinear elliptic problem whose solution can be shown to be unique using the Global Inversion Theorem. Previous theories are not applicable because the equations have a term linear in the unknown. A strong assumption on the nonlinearity can imply existence and uniqueness of the solution. We shall consider two cases corresponding to resonance and non-resonance. For completeness, we rst prove the existence and uniqueness result for the nonresonance case. A similar version can be found in Ambrosetti and Prodi [1]. Proposition 3. Consider the semilinear equation
(11)
?4u = u + f (x; u) + g on
with homogeneous Dirichlet boundary conditions. Here 2 IR is given with 6= j for all j and f 2 C 1 ( ; IR) and satis es the conditions
(12)
kf (x; vn )k?1 ! 0 whenever kv k ! 1 n 1 kvn k1
and
(13)
k?1 < + fu (x; t) < k
for every x 2 and t 2 IR and for some k 2 IN. The function g is assumed to be in
L2 ( ). Then the equation (11) has a unique solution in H01 ( ).
Proof: We use the Global Inversion Theorem (see [1] for instance) to show this result. Let F : H01 ( ) ! H01 ( ) with F (u) = u + 4?1u + 4?1f (x; u) for u 2 H01 ( ). Now F is continuous and we need to show that it is proper and locally invertible. For the former, suppose hn 2 H01 ( ) and un 2 H01 ( ) with F (un ) = hn ! h 2 H01 ( ). We need to show that un has a convergent subsequence. Now suppose fkunk1g is unbounded. Then there is some subsequence which we still label by n such that
kunk1 ! 1 as n ! 1. Let zn = un =kunk1 . Then ?1 un ) : zn + 4?1 zn = kuhnk ? 4 kuf (x; n 1 n k1 15
Certainly, the right-hand side of the above goes to zero in the L2 norm as n ! 1. Thus fzng is bounded in H 2 and must have a convergent subsequence converging to some nonzero z strongly in the energy norm. From the above equation, we obtain the contradiction that z is an eigenfunction of ?4 with corresponding eigenvalue . Thus
fkunk1g must be bounded. This implies that f4?1un = hn ? un ? 4?1f (x; un )g is a bounded sequence in L2 ( ). Thus fung is bounded in H 2 and hence must have a strongly convergent subsequence in H01 ( ). To show that F is locally invertible, we simply note that the linear problem (I + 4?1 + 4?1fu (x; v))w = 0;
v; w 2 H01 ( );
rewritten in a more familiar form
?4w = ( + fu (x; v))w
(14)
has only the trivial solution because of the assumption on + fu . The Fredholm Alternative implies that F 0 (u) is invertible and by the Inverse Function Theorem, F is locally invertible. We nally can conclude from the Global Inversion Theorem that the semilinear elliptic equation (11) has a unique solution. 2 Note that (12) is satis ed when, for instance, f is a bounded function. Note also that if f also depends on ru, then (12) contains an extra term involving rw and more assumptions are required to conclude that the equation corresponding to (14) has only the trivial solution. For j = 1; 2, let k kH 1 ( ) be the norm induced by the inner product j
(v; w)H
1 ( j )
Z
j
rv rw + vw:
The following lemma is useful. Lemma 2. Let j = 1 or 2 and 2 IR; 6= i ( j ); 8i. Then for w 2 H?1 ( j )
fv 2 H 1 ( j ); v = 0 on @ j \ @ g, k(Pj + 4j?1 )wkH 1 ( ) C kPj wkH 1 ( ) : j
16
j
Here Pj + 4?j 1 is considered as an operator from H?1 ( j ) onto H01 ( j ).
Proof: Let f i g be an orthonormal basis (with respect to the inner product (; )H 1 ( ) ) of eigenfunctions of ?4j with corresponding eigenvalues i ( j ). Let j
w=
1 X i=1
ci i + v
where ci 2 IR and v is in the orthogonal complement of H01 ( j ) in H?1 ( j ) (i.e., v 2 H01 ( j )? where H?1 ( j ) = H01 ( j ) H01 ( j )? ). Note that 4?j 1 is a shorthand
for 4?j 1Rj where Rj : H?1 ( j ) ! H ?1 ( j ) is the restriction operator de ned by
z 2 H?1 ( j ); y 2 H01 ( j ):
(Rj z )(y) = (z; y)H 1 ( ) ; j
By de nition, Rj v = 0. Thus
k(Pj + 4?1 )wk2 1
i
ci 1 ? ( ) i j H 1 ( ) 1 2 X = c2i 1 ? (
i j) i=1 2 1 ? (
kPj wk2H 1 ( ) ; c j)
H ( j ) =
j
1
X
i=1
j
j
where c ( j ) is an eigenvalue of ?4j closest to in the sense that 1
1 ? ; 8i: ? (
) ( ) i j
c j
2 Theorem 3. Consider the semilinear elliptic equation as in Proposition 3 except
that (13) is replaced by
(15)
+ fu (x; t) 0; t 2 IR:
For n = 0; 1; 2; and any u(0) 2 H01 ( ), de ne the Schwarz sequence as:
?4u(n+ 21 ) = u(n+ 12 ) + f (x; u(n+ 21 ) ) + g on 1 ; u(n+ 21 ) = u(n) on @ 1 ; ?4u(n+1) = u(n+1) + f (x; u(n+1)) + g on 2 ; u(n+1) = u(n+ 21 ) on @ 2 : 17
Then the Schwarz sequence converges geometrically to the unique solution of the semilinear elliptic equation (11) in the L1 norm.
Proof: Because of (15), the Schwarz sequence is well de ned by Proposition 3. In the rst part of the proof, we show that the Schwarz sequence is bounded. For any v1 2 H01 ( 1 ), [u(n+ 12 ) ? u(n); v1 ] = (u(n+ 12 ) + f (n+ 12 ) + g; v1 ) ? [u(n) ; v1 ] where f (n+ 12 ) = f (x; u(n+ 21 ) ). Noting that u(n+ 12 ) ? u(n) 2 H01 ( 1 ), we have (I + 4?1 1)u(n+ 21 ) = ?4?1 1(f (n+ 12 ) + g) + (I ? P1 )u(n) ; or (P1 + 41?1)u(n+ 21 ) = ?4?1 1(f (n+ 12 ) + g) and (I ? P1 )u(n+ 21 ) = (I ? P1 )u(n): Applying lemma 2 to the rst term, noting that 6= i ( 1 ); 8i, we obtain
kP1 u(n+ 21 ) k1 C (1 + kf (n+ 21 ) k?1 ): Assume fku(n+ 21 ) k1 g is unbounded. Then there is some subsequence which we label by nj + 12 such that ku(n + 21 ) k1 ! 1 as j ! 1. Now j
ku(n + 12 ) k1 kP1 u(n + 12 ) k1 + k(I ? P1 )u(n + 12 ) k1 j
j
j
C (1 + kf (n + 12 ) k?1 ) + k(I ? P1 )u(n ) k1 :
(16)
j
j
Thus (n ) 1 o(1) + k(I ?(nP1+)u1 ) k1 : j
ku
j
2
k1
This shows that fku(n ) k1g must also be unbounded. By lemma 1, the entire sequences j
(not just subsequences) fku(n)k1 g and fku(n+ 21 ) k1 g go to in nity as n ! 1. Applying lemma 2 to (P2 + 4?2 1 )u(n) = ?4?2 1(f (n) + g); 18
we have
kP2 u(n) k1 C (1 + kf (n)k?1 ):
(17) Also,
k(I ? P1 )u(n) k1 k(I ? P1 )P2 u(n) k1 + k(I ? P1 )(I ? P2 )u(n) k1 C (1 + kf (n) k?1) + dku(n)k1 : Using this result in (16), we obtain
ku(n+ 21 ) k1 o(1) ku(n+ 21 ) k1 + o(1) + d ku(n)k1 ku(n)k1 or
ku(n+ 21 ) k1 d + o(1): ku(n)k1 In a parallel development, we also have
ku(n+1)k1 d + o(1): ku(n+ 12 ) k1 Combining the above equations, we have
ku(n+1)k1 d2 + o(1); ku(n)k1 contradicting that fku(n)k1 g is unbounded since d < 1. Thus fku(n+ 21 ) k1 g is bounded. If fku(n)k1g is unbounded, then from (17),
ku(n+1)k1 kP2 u(n+1)k1 + k(I ? P2)u(n+1) k1 C (1+ kf (n+1)k?1 )+ k(I ? P2)u(n+ 21 ) k1 : Hence,
u(n+ 21 ) k1 1 o(1) + k(I ? P(2n)+1)
ku
k1
which is a contradiction since fku(n+ 21 ) k1g is bounded. This shows that fku(n)k1g must also be bounded.
19
Since the Schwarz sequences are bounded in k k1 , there exist u0 ; u1; u2 2 H01 ( ) and a subsequence labeled by nj such that u(n ) * u0 ; u(n + 21 ) * u1 and u(n +1) * j
j
j
u2 (weak convergence in the energy norm). We now show that the subsequences actually converge strongly in the energy norm. By compactness of the restriction map
H 1 ( 1 ) ,! L2 (@ 1 ), we have u(n + 21 ) ! u1 in L2 (@ 1 ). Since fu(n + 21 ) +f (n + 12 ) +gg is bounded in the L2 norm, fu(n + 21 ) g is bounded in H 2 ( 1 ) from the de ning equation j
j
j
j
of u(n + 21 ) . By extracting a further subsequence if necessary, u(n + 21 ) converges to u1 j
j
strongly in H 1 ( 1 ). Similarly, u(n ) ! u0 and u(n +1) ! u2 strongly in H 1 ( 2 ). j
j
Hence, u(n ) ; u(n + 21 ) and u(n +1) converge strongly in H01 ( ). Note that u0 and u2 j
j
j
are weak solutions of semilinear equation (11) on 2 while u1 is a weak solution on
1 . Finally, as in the proof of the previous theorem, we apply the strong maximum principle to show convergence in the L1 norm of the iterates to the solution to the semilinear equation (11) on .
2
For the above semilinear equation, we made the strong assumption (15) so that the maximum principle can be applied in the nal step of the proof. It is unknown whether the Schwarz iteration with the weaker condition (13) converges. Next we consider the resonance problem for the above semilinear equation. See [1] for a proof. Proposition 4. Consider the semilinear equation
?4u = 1 u + f (x; u) + g on
(18)
with homogeneous Dirichlet boundary conditions. Here f 2 C 1 ( ; IR) and satis es the following conditions: 1. 9M such that jf (x; s)j M; 8x 2 ; s 2 IR: 2. lims!1 f (x; s) = f 2 IR; 8x 2 : 3. f?
R
1
R
< ? g 1 < f+
R
1 ,
where 1 is the positive eigenfunction of
?4 corresponding to the principal eigenvalue 1 . 20
4. fu (x; s) 6= 0; 8x 2 ; s 2 IR: 5. 1 + fu (x; t) < 2 ; 8x 2 ; t 2 IR: The function g is assumed to be in L2 ( ). Then, the equation (18) has a unique solution in H01 ( ). Theorem 4. Consider the hypotheses as in the above Proposition except that the
5th condition is replaced by: 8x 2 and s 2 IR, 1 + fu (x; s) 0. In addition, assume the subdomains are proper subsets of so that 1 < min(1 ( 1 ); 1 ( 2 )). For n = 0; 1; 2; and any u(0) 2 H01 ( ), de ne the Schwarz sequence as:
?4u(n+ 12 ) = 1 u(n+ 21 ) + f (x; u(n+ 21 ) ) + g on 1 ; u(n+ 21 ) = u(n) on @ 1 ; ?4u(n+1) = 1 u(n+1) + f (x; u(n+1) ) + g on 2 ; u(n+1) = u(n+ 21 ) on @ 2 : Then the Schwarz sequence converges geometrically to the unique solution of the semilinear elliptic equation (18) in the L1 norm.
Proof: The Schwarz sequence is well de ned by Proposition 3 because of the assumption that the subdomains are proper subsets of . The rest of the proof follows exactly as in the proof of Theorem 3.
2
3. Other Schwarz Methods. In the last section, each subdomain problem is nonlinear. We now consider iterations where linear problems are solved in each subdomain. This is of great importance because in practice, we always like to avoid solving nonlinear problems. One way is in the framework of Newton's method. Write a model semilinear problem as G(u) u ? 4?1f (x; u) = 0 for u 2 H01 ( ). Suppose it has a solution u and suppose that k4?1fu (x; u)k < 1, then for initial guess u(0) suciently close to u, the Newton iterates u(n) de ned by (19)
u(n+1) = u(n) ? Gu (u(n) )?1 G(u(n) )
converge to u. Note that the assumption means that Gu = I ? 4?1fu has a bounded inverse in a neighborhood of u. Now each linear problem (19) can be solved using the classical Schwarz Alternating Method. We take three di erent approaches. 21
In this section, we consider the equation
?4u = f (x; u; ru) on
(20)
with homogeneous Dirichlet boundary conditions. Assume f (x; u(x); ru(x)) 2 H ?1 ( ) for x 2 and u 2 H01 ( ) and satis es the condition
kf (x; u; ru) ? f (x; v; rv)k?1 c ku ? vk1 ; where u; v 2 H01 ( ); c is a constant such that c < 1.
3.1. Linear Schwarz Method. In the linear Schwarz sequence de ned below, each subdomain problem is linear.
p
Theorem 5. Assume d < 1 ? c2 ? c. For n = 0; 1; 2; and any u(0) 2 H01 ( ),
de ne the linear Schwarz sequence by
?4u(n+ 12 ) = f (x; u(n); ru(n) ) on 1 ; u(n+ 21 ) = u(n) on @ 1 ; ?4u(n+1) = f (x; u(n+ 21 ) ; ru(n+ 12 ) ) on 2 ; u(n+1) = u(n+ 21 ) on @ 2 : Then, the Schwarz sequence converges to the solution of (20) in the energy norm.
Proof: In a similar manner as in the proof of Theorem 1, we obtain (21)
?
u(n+ 21 ) ? u(n) = P1 (u ? u(n) ) ? 4?1 1 f (x; u(n) ; ru(n) ) ? f (x; u; ru) :
and
u(n+1) ? u(n+ 21 ) = P2 (u ? u(n+ 12 ) ) ? 4?2 1 f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru) : De ne e(n) = u(n) ? u, we have ?
e(n+1) = (I ? P2 )(I ? P1 )e(n) ? (I ? P2 )4?1 1 f (x; u(n) ; ru(n)) ? f (x; u; ru) ?
42?1 f (x; u(n+ 21 ) ; ru(n+ 12 ) ) ? f (x; u; ru) : ?
From (21), e(n+ 21 ) = (I ? P1 )e(n) ? 4?1 1 f (x; u(n) ; ru(n) ) ? f (x; u; ru) . 22
We have the estimate
ke(n+1)k21 = kP2 e(n+1) k21 + k(I ? P2 )e(n+1) k21 c2 ke(n+ 21 ) k21 + k(I ? P2 )e(n+ 12 ) k21 ?
c2 ke(n+ 21 ) k21 + k(I ? P2 )(I ? P1 )e(n) ? (I ? P2 )4?1 1 f (x; u(n) ; ru(n)) ? f (x; u; ru) k21 c2 ke(n+ 21 ) k21 + (d + c)2 ke(n)k21 : We rewrite the above as
ke(n+1)1 k21 A ke(n+ 21 ) k21 ; where A = c2 (c + d)2 : 1 0 ke(n)k21 ke(n+ 2 ) k21
Similarly,
ke(n+ 21 ) k21 A ke(n)1k21 : ke(n) k21 ke(n? 2 ) k21
ke(n+1)1 k21 A2 ke(n)1k21 : ke(n+ 2 ) k21 ke(n? 2 ) k21
Thus
A sucient condition for the convergence of these vectors to zero is that the eigenvalues of A2 have magnitude less than one or equivalently that the spectral radius of A is less than one. The eigenvalues of A are p
c2 c4 + 4(c + d)2 : 2
Clearly, the eigenvalue with the larger magnitude is the one with the plus sign in the above expression. Thus the Schwarz sequence converges in the energy norm provided p
c2 + c4 + 4(c + d)2 < 1 or d < p1 ? c2 ? c: 2
2
3.2. Additive Schwarz Method. One of the main motivations for studying Schwarz methods is their suitability for parallel computers. The algorithms discussed thus far are not ideal for parallel computers because the subdomain problems must 23
be solved sequentially. We now show convergence of an additive Schwarz method in which subdomain problems can be solved concurrently. The additive Schwarz method was rst proposed by Dryja and Widlund [8] for linear elliptic PDEs. Theorem 6. For n = 0; 1; 2; and any u(0) 2 H01 ( ), de ne the sequences
?4d(n+ 21 ) = f (x; u(n); ru(n) ) + 4u(n) on 1 ?4d(n+1) = f (x; u(n); ru(n) ) + 4u(n) on 2 ; for d(n+ 21 ) 2 H01 ( 1 ) and d(n+1) 2 H01 ( 2 ). De ne the additive Schwarz sequence by
u(n+1) = u(n) +!(d(n+ 21 ) +d(n+1) ) where ! is a relaxation parameter with 0 < ! < 1=2. Assume kI ? !(P1 + P2 )k1 + 2!c < 1. Then, the additive Schwarz sequence converges geometrically to the solution of (20) in the energy norm.
Proof: The analysis is similar to before and we only record the key equations. From the de ning equations of d(n+ 21 ) and d(n+1) , we have ?
?
d(n+ 12 ) = ?P1 e(n) ? 4?1 1 f (x; u(n) ; ru(n) ) ? f (x; u; ru)
d(n+1) = ?P2 e(n) ? 4?2 1 f (x; u(n) ; ru(n) ) ? f (x; u; ru) : Substituting into the de nition of u(n+1) , we obtain ?
e(n+1) = (I ? !(P1 + P2 ))e(n) ? !(4?1 1 + 4?2 1) f (x; u(n) ; ru(n) ) ? f (x; u; ru) : Hence
ke(n+1)k1 kI ? !(P1 + P2 )k1 ke(n)k1 + 2!cke(n)k1: When 0 < ! < 1=2; kI ? !(P1 + P2 )k1 < 1 and the result follows. 2 Note that the subdomain problems are linear and can be solved concurrently. Roughly speaking, d(n+ 21 ) and d(n+1) are corrections to the iterate u(n) in the subdomains 1 and 2 , respectively, and the right-hand sides of the de ning equations for these corrections are the residuals of u(n) in the subdomains. If f is independent of u, then this reduces to the classical additive Schwarz method with a relaxation parameter.
24
3.3. Parallel Schwarz Method. Other variations are also possible. We give one more which we called the parallel Schwarz method because the subdomain problems can also be solved in parallel. The proof is similar to previous ones and is omitted. Theorem 7. Assume d < 1 ? 2c. For n = 0; 1; 2; and any u(0) de ne u(? 21 ) = u(0) and de ne the parallel Schwarz sequence by
2 H01 ( ),
?4u(n+ 12 ) = f (x; u(n) ; ru(n) ) on 1 ; u(n+ 21 ) = u(n) on @ 1 ?4u(n+1) = f (x; u(n? 21 ) ; ru(n? 21 ) ) on 2 ; u(n+1) = u(n? 12 ) on @ 2 : Then, the parallel Schwarz sequence converges to the solution of (20) in the energy norm.
4. Discussions and Conclusion. In this paper, we showed how Schwarz Alternating Methods can be imbedded within the framework of Banach and Schauder xed point theories and Global Inversion theory to construct solutions of nonlinear elliptic PDEs. We also give other versions of these methods where a linear problem is solved in each subdomain and these linear problems can be computed in parallel. We treated homogeneous boundary conditions in this paper. For the problems considered here, non-homogeneous boundary conditions can also be handled. If the boundary condition is u = h 2 H 1=2 (@ ), then the procedure is to extend h to be a function in H 1 ( ) in a bounded way and to make a change of variable w = u ? h so that w satis es the perturbed PDE
?4w = F (x; w; rw) + 4h with homogeneous boundary conditions, where F (x; w; rw) = f (x; w + h; r(w + h)). For PDEs of Theorem 1, it is trivial to check that F satis es the same Lipschitz condition as f . For PDEs of Theorem 2, @F=@w = @f=@w and
jF (x; w; rw)j = jf (x; w + h; r(w + h))j 25
C (1 + jr(w + h)j ) C (1 + (jrwj + jrhj) ) C [1 + 2 (jrwj + jrhj )] C (1 + jrwj ) and hence Theorem 2 can be applied to the perturbed PDE. We can also check that Theorem 3 can be applied to the perturbed PDE. This procedure, of course, does not work for all nonlinear PDEs. Future work include some numerical experiments and extending our results to the case of multiple subdomains. For the PDEs of Theorem 1, this is manageable. In particular, for the additive Schwarz sequence, this is trivial. If there are m subdomains, as long as
kI ? !(P1 + + Pm )k1 + m!c < 1; the sequence converges. Here, 0 < ! < 1=K where K is the minimum number of colors needed to color the subdomains in such a way that overlapping subdomains are assigned di erent colors. The extension for the other classes of PDEs is not at all obvious principally because of the diculty to apply the maximum principle in these cases. It is desirable to use tools other than the maximum principle for these classes of PDEs. Besides possibly weakening the hypotheses required, they may allow extension to multiple subdomains.
Acknowledgment. I thank Professor Juncheng Wei for helpful discussions and Professor Martin Gander for reading an early draft. I also thank the referees for carefully reading the original manuscript and suggesting numerous improvements. REFERENCES [1] A. Ambrosetti and G. Prodi. A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge, 1993. [2] J. H. Bramble, J. E. Pasciak, J. Wang, and J. Xu. Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comput., 57:1{21, 1991. 26
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