Local error analysis of the interior penalty discontinuous Galerkin ...
Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems G. Kanschat
R. Rannacher
December 16, 2002
Summary. A local a priori and a posteriori analysis is developed for the Galerkin method with discontinuous finite elements for solving stationary diffusion problems. The main results are an optimal-order estimate for the point-wise error and a corre sponding a posteriori error bound. The proofs are based on weighted -norm error estimates for discrete Green functions as already known for the ‘continuous’ finite element method.
1
Introduction
This paper deals with the numerical approximation of second-order elliptic boundary value problems by interior penalty discontinuous Galerkin finite element method (in short dG method) on triangular or rectangular meshes; see Arnold [1] and Arnold et al. [2], and also [16]) In this Galerkin method the trial and test functions may be discontinuous across inter-element boundaries while continuity is enforced approximately via a penalty technique. We analyze the local convergence behavior of this method. Only weak assumptions are imposed on the meshes; they are allowed to be refined locally and the cells may be shifted against each other. The proof relies on the techniques for estimating discrete Green functions developed in Frehse and Rannacher [12] for the “continuous” Galerkin method (in short cG method. Finally, we apply the error estimation techniques in Becker and Rannacher [7] to obtain a posteriori error estimates. For simplicity, we concentrate on the Poisson equation,
(1)
with boundary conditions
!#"$ %&' ( )+* (2) on a polygonal domain -,/. . The boundary is assumed to be decomposed like 0 1 32 ) , and the boundary data 4 ) are smooth. We allow for such 5 Institut f¨ur Angewandte Mathematik, Universit¨at Heidelberg, INF 293, D-69120 Heidelberg, Germany
1
general nonhomogeneous boundary conditions in order to highlight the particular flexibility of the dG method. However, in order to avoid technical difficulties, we will regularity nevertheless assume that the weak solution of this problem possesses 6 which very much constrains the generality of this setting. We note that all results of this paper can be extended to more general equations in two and three dimensions with variable coefficients as well as to systems of such equations. The main result of this paper is the local a priori error estimate
7 8:;9 < = >@? 7BADCFE G I< H ?KJML NJMOQPRTSVUXW Y CZE JML NJ
8 9 V? ^ L J g Y E Ä g < ó > Ð n ¹ º ¹ º ? g ghji#k E g n Y < ó > Ð %#" #X¡#" j? g v l Ä J ó >¸L Ü J g YÅE Ä g < ó > ¹ Ü º ¹ Ü º ? g f ghji#k Y E g < ó > £%#" Ü #X¡#" Ü j? g v * 10
(37) and
The two terms
n
and
will be Ó estimated separately.
(iv) First, we estimate n . For defined in (9), we conclude from the interpolation estimates (13) and (14), observing Assumption 2, that
J g Y E gÐ `n Î J ¹ º J g Y E gn`Î JZ¡£" $J g ACFE g JXL N J g Ò Now, we split the summation over d > as follows: n f f ®ª*F*F*%´(Y ®ª*F*Z*%´$* Ò ghji#k g S U gh¡i¸k g+Ò Ö S U Ð n AC and therefore estimate On cells , ¦ b ; , we have ó > C E g n A f f E g JXL N J g NJ g Y C X J L Ò Ò ó gh¡i¸k gh¡i#k g+ Ö S U g S U > C C A E JXL NJ O P RTSVUXW J ó > Ð n J Y E JXL NJ * JXL
From this, we obtain
C C n A E G I< H ?ZJXL NJ O P RTSVUXW Y E XJ L NJ *
(38)
. Observing that ó > is constant on each cell , we have l [ > < ó > Ü Ü ?N f < L < ó > Ü ?ttL Ü ? g YÅE Ä g < ¹ ó > Ü º ¹ Ü º ? g ghji#k Â_ < ¹ ó > Ü º X¡#" Ü j? g Â_ < ¡#" < ó > Ü ?u# ¹ Ü º ? g v l f < ó > L Ü ML Ü ? g YÅE Ä g < ó > ¹ Ü º ¹ Ü º ? g ghji#k YÅE Ä g < ¹ ó > º Ü # ¹ Ü º ? g Â_ < ¹ ó > º Ü £F¡£" Ü j? g < ó > ¹ Ü º F%#" Ü ¡? g _ < ¹ ó > º ¹ #" Ü º ¹ Ü º ? g v *
(v) Next, we estimate
This leads us to
where
r
[ > < ó > Ü Ü È? Y
r
f l E g < ó > £¡#" #X¡#" j ? g Y _ < ¹ ó > £ F¡£" j ? g Ü Ü º Ü Ü Â g hji k Y < ó > ¹ Ü º X¡#" Ü j? g Y _ < ¹ ó > º ¹ #" Ü º ¹ Ü º ? g Y E Ä g < ¹ ó > º Ü # ¹ Ü º ? g v *
11
(39)
(v) We proceed with the first term on the right in (39). Using Galerkin orthogonality of Ü , we obtain > ó ?N > ó > ó ?t ?t*
[ < >Ü Ü [ < >Ü Þ < >Ü Ü ? Since ó > is constant on and the projection Þ > is local, we have Þ > < ó > Ü § > Ü Þ Ü ö Þ > ö , it follows that Hence, with the notation l [ > < ó > Ü Ü ?§ f < L < ó > ?ttL Ü ? g YÅE Ä g < ¹ ó > º ¹ Ü º ? g gh¡i¸k Â_ < ¹ ó > º X¡#" Ü j? g Â_ < ¡#" < ó > ?u# ¹ Ü º ?
ó > Þ > Ü
.
g v *
The four terms on the right hand side are now estimated separately using the trace inequalities (25), (26), and the interpolation estimates (13), (14). Furthermore, we employ the a priori bound (30) and the error estimate (30) for the Green function ö and the7 dual error ö ö > . Note that in order to 7 estimate the average values ¡#" g , it suffices to estimate each of the terms #" g separately. In virtue of Assumption 2, the weights ó > can be estimated by
7y¹ ó > 7 g ACZE g ó > g A CFE g 7 ó > 7 g º A Có and we use that L ö > g ñ . By (23), ó > . We will use a free parameter d< _ º . For the first term, we find
±± f < L < ó > ± gh¡i¸k
l _ E ?ttL Ü ? g ±± A C|f g J ó L ö J g Y ± gh¡i¸k A C G L Ü J g v
and, analogously, for the second term,
±± f E Ä < ¹ ó > ± ghji#k g AC3f ghji k AC3f ghji#k A C G @ Ð n ¹ ó >
¹ º J g Y E g Ä ZJ ó @> Ü º J g v L ¸ö º J g Y E g Ä JZ ó >@ ¹ Ü º J g v
E Y C *
12
The third term is estimated by ±± f < ¹ ó > º F¡£" Ü j? g ±±
± gh¡i¸k
±± f ± gh¡i k ACf gh¡i¸k ACf gh¡i¸k AC f gh¡i¸k AC Y
± l  ¹ ó > º ó @> £ Jó >
Y _ ¹ ó > º É ó J g J #> #" Ü J g
E Mrg Î J ó L J g VE gÐ n`Î ö l _ E g J ó >¸L Ü J g Y C G E L Ü J g Y E g J ó L ö J g
J ó L ö J g v
Finally, for the fourth term, we find ±± f < %#" < ó > ?u# ¹ Ü º ? g ±± A½f
± ghji#k
Y ó >@ ¹ º X¡#" Ü g v ±± ±
l _ J ó >#L
±
JZ ó > Ð n % #" < ó >
ghji#k J g Y E g J ó >¸L
AC|f ghji#k AC f l _ E g J ó L J g Y E Ä JZ ó @> ¹ ö Ü g ghji#k C A G ¹ Ü º J g
E Ä g JZ ó @> ¹ Ü º J g v
ºJg v
Combining these estimates yields 7 > ó
[ < > Ü Ü ? 7¸AC
Y
C G E £%#" Ü £F¡£" Ü j? g ±± AC f l J ó >#L Ü J g Y E g J ó >¸L ö J g v
± ghji#k
±
gh¡i¸k AC Y C G ¸L Ü J g Y g J ó ¸> L ö J g v
13
E The third term in r is the most critical one, since it does non contain a factor g . Therefore, we have to absorb this term into the other definite terms in using the stabilization parameter Ä . We recall that for } > d~ > ,
7 < ¡#" > # ¹ > ? g 7¸A C ¤ XJ L > J g Ò Y E Ä J ¹ > J g (41) } } º } Â Â g } º Ó Ä with defined in (9). Splitting the dual error like Ü Y Þ > Ü with ö Þ > ö , we have
< ó > ¹ Ü º F ¡£" Ü j ? g < ó > ¹ Ü º X ¡#"
¡? g Y < ó > ¹ Ü º X ¡#" Þ > Ü j ? g
The first term on the right is treated analogously as the other terms before leading to C the estimate ±± f < ó > ¹ Ü º X¡#" j? g ±± A Y E G Ü # X¡#" Þ > Ü ¡ ? g v *
Using relation (41), we conclude by a lengthy but standard calculation: 7 < ó > £¡£" > #X¡#" > j? g 7¸A < ó > g Y CFE g ó > g ?ZJZ¡#" > J g Þ Ü Þ Ü Þ Ü
with an arbitrary constant
E g f  g Y CFE g ó > g Z? JXL g C l A E ¤ _ Y J ó >¸L Þ > Ü J g É @ d'< _ º . Consequently,
f g h¡i#k
ó > £ ¡#" Þ > Ü # F%#" Þ > Ü j? g C ¤ l C f « _ Y J ó ¸> L Þ > Ü J g Y E g JXL Þ > Ü J g v * Â gh¡i¸k Ä É
JXL Þ > Ü J g A
follows
Ò Þ>Ü Jg C g Ò Y E g JXL > J g Ò v Þ Ü
E g  < ó > £¡#" Þ > Ü Ä C ¤ A½f «  ghji k Ä Y C|f gh¡i¸k
_ Y
É
C
JML Ü J g Y
JXL
J g
£F¡£" Þ > Ü j ? g
#L Ü J g l E Y J ó >¸L g JXL Ü J g 14
J g Y E g JXL
J g v *
Hence, observing the results of Lemma 1, we obtain
E g A C ¤ « j¡#" Þ > Ü # F%#" Þ > Ü j ? g Â Ä Ä
f g h¡i¸k
? Y
C G E º ¹ £" Ü ± ghji#k A!f l ghji#k A!f l ghji#k AC Y
¹ º Ü º ? g ±±± C E g ZJ Ü $J C J Ü J g Y C G E @£¡#" $ J g v Ü E g MJ L J g Y C J ó #> L J g Y E g J ó >#L J g v Ü Ü ö
and by
±± f E Ä < ¹ ó > º Ü £ ¹ Ü º ? g ±± ± ghji#k g ± E g l A!f ÄE g J ¹ ó > º J E g ZJ ó ¹ Ü º J g Y ZJ Ü $J g v ghji#k A!f l C E Ä < ó > ¹ ¹ ? g Y C J J g Y FC E g MJ L J g v g Ü º Ü º Ü Ü ghji#k AC Y C G < K ?t! d~ÝÿÀ~ >$ z
we have by Galerkin orthogonality
8 < ?N [ > < ?N [ > < >@?t Ü Ü z Ü z ï
for arbitrary ï > dà~ > . Integrating cell-wise by parts and reordering terms, we conclude by elementary but tedious calculation that, for ( z ï > ,
8 < ?§ [ > < * Ü Ü z )F>@? l f Ü ( ? g Y < £" º X ( j ? g
YÅE Ä g < ¹ > º ¹ (¡º ? g v *
@?t ( ? g Y Â_ < $- < >¸?tF ( j ? Â_ < - < >¸?tX¡#" ( YÅE Ä g ¹ (¡º ? g v 17
(47)
with the following notation of cell and edge residuals:
+À< -
$-
>¸?X g Y3 >$ n¹ ./ ¾ ¿ > º if · , 3Á , < >¸?X » ¯¼ = >$ if · 0 , if · ) n ¹ #"$ > ./ º if · , Á ¾¿ * < >¸?X » ¯¼ if · , 0 ) +#" >$ if · , )
From the error representation (47), we obtain the following error estimate: Theorem 2. For the error Ü the a posteriori error estimate
>
in the interior penalty scheme (7), there holds
RonpW RonpW m R W R W m RsrtW RruW 8 < ?N f lKm ge q g Y geq g Y geq gv (48) Ü ghji m RywxW RywoW with the cell residuals g and weight factors q g being defined by m RIn W RonpW g J +õ< >@?ZJ g q g J z ï > J g m R W EVÐ n`Î R W npÎ g g JX$- < >@?ZJ g q g E g J z ï > J g m RruW EVÐ rMÎ RsrtW rMÎ n g g J1- < >¸?ZJ g q g E g JF¡#" < z ï >@?u Ä E gÐ ¹ z ï > º J g for arbitrary ï > d~ > . Theorem 2 provides a posteriori estimates for arbitrary functionals of the error. This also includes the -error estimates. To see this, we take the special functional
8 < È? < tÈ?ZJ J Ð n * Ü Ü The corresponding dual solution z d~ satisfies d 6 < ? and the a priori bound JXL z J A C V (49) C where the stability constant only depends on the domain . From (48) and the interpolation estimates (13), (14), we infer that
l E m RonpW E m R W E m RruW `n Î J Ü J AC32C f g g Y g g Y g g v * gh¡i
(50)
This a posteriori error estimate is asymptotically optimal, too. Next, we state an a posteriori error estimate for the locally averaged error as considered in our a priori error analysis. In this case the dual solution is just the regularized Green 9 function, z ö ; introduced in the proof of Theorem 1. We have the estimate
7 8 9 ; < ? 7@A Ü
l m oR npW IR n W m R W R W m sR ruW sR rtW n < >¸? f geq g Y g(q g Y g(q g v gh¡i 18
(51)
where the residual terms mated as follows:
m yR wxW g
are as defined above and the weights
RonpW J ö ;9 ï >J g R W npÎ q g E g J ö ;9 ï RsrtW rtÎ q g E g JZ¡#" < ö
RswxW
can be esti-
q g
7
q g
> J g 9; >@u? E Ä ¹ ;9 > J g * ï g ö ï º
Numerical results 2
The a posteriori error estimate in Theorem 2 is tested at the same configuration as before; see Figure 3. The error is evaluated at the point [ < *54@u*64#? , which is the point with maximum error in Figure 2 (the value is taken from the solution in the upper right cell adjacent to this point) . We compare three types of “error estimators”. The first one, ¤ < >¸? , is obtained directly from the error representation (47), avoiding the use of triangle and H¨older inequalities:
8 < ?N Ü
¤ < >¸? f
g
R¤ W g *
(52)
7 gR ¤ W 7 : 7 gR ¤ W 7 *
The second estimator uses the local refinement indicators
n < >¸? f
The third one,
< >@?
g
g
RonpW
f g
, is given by Theorem 2 as described above:
< >¸? f g
R W g f
g
m m m ® goR npW q g oR npW Y gR W q g R W Y gR ruW q g sR ruW ´ * Ó
Ó
For practical evaluation of the error estimators, we solve the dual problem (45) on the current mesh with bi-quadraticÓ polynomials obtaining z d ~ > . We decided for exact computation in the higher order space to avoid additional error contributions. For a more efficient computation, z can be obtained by post-processing; see [8] for such strategies and their influence on the estimator. Ó The quality of the resulting approximate error estimators sured by the “effectivity index”: Ó
7 w < >¸? 7 7 8 ;9 < ? 7 * Þ98;: Ü RonpW Mesh adaptation is based on “error indicators” g
w < >¸?t$7 ã _ Â
, is mea-
R W
and g , respectively. For mesh refinement, a fixed fraction (here 20%) of the grid cells with largest indicator are refined.
19
Ó
Ü ¸? 4.288e-3 1.652e-3 6.528e-4 2.917e-4 1.401e-4 6.751e-5 3.373e-5 1.742e-5 9.068e-6 Ó
Ó
eff
Ü ¸? 6.127e-3 2.296e-3 9.431e-4 4.182e-4 2.000e-4 9.678e-5 4.875e-5 2.522e-5 1.324e-5
Þ
eff
1.42 1.39 1.44 1.43 1.43 1.43 1.44 1.45 1.46
Ó Ó Table 2: Point error Ü \< [ ? and Ü \< [ ? Ü \< [ ?KY ¤ < >¸? and the corresponding a poste >@? and n < >@? ( is maximum refinement level). riori estimators ¤ < Ó ¤ < >¸? Ü ¸? is asymptotically optimal while estima tor n < >¸? appears to be off by Ó a factor of aboutÓ < a . The error representation (52) suggests to consider 8 < >@? 8 < >¸?ÈY ¤ < >¸? as new approximation. This post-processing step can improve accuracy dramatically, as shown in Table 2. The mesh obtained in the eighth step of this iteration is shown in Figure 3. Table 2 presents the results obtained by the adaptive refinement process with a refined mesh shown in Figure 3. The error estimator ¤ < >¸? is asymptotically optimal while the estimator n < >¸? appears to be only suboptimal. In Table 3, we show results for estimator < >@? . Since this estimator involves H¨older and triangle inequalities on each cell, there is no cancellation between the different terms of the estimator. Therefore, the error is over-estimated by a factor between 4 and 5. The errors resulting from adaptive refinement —based on the estimator < >¸? here— are comparable to those of Table 2. Therefore, we conclude that both estimators are suited as refinement criteria. Finally, in Table 4, we compare the “error/mesh-ratio” 20
Ü
for uniform and
Figure 3: Initial coarse mesh and an adapted mesh (level computation. uniform refinement Ü
R. Rannacher
December 16, 2002
Summary. A local a priori and a posteriori analysis is developed for the Galerkin method with discontinuous finite elements for solving stationary diffusion problems. The main results are an optimal-order estimate for the point-wise error and a corre sponding a posteriori error bound. The proofs are based on weighted -norm error estimates for discrete Green functions as already known for the ‘continuous’ finite element method.
1
Introduction
This paper deals with the numerical approximation of second-order elliptic boundary value problems by interior penalty discontinuous Galerkin finite element method (in short dG method) on triangular or rectangular meshes; see Arnold [1] and Arnold et al. [2], and also [16]) In this Galerkin method the trial and test functions may be discontinuous across inter-element boundaries while continuity is enforced approximately via a penalty technique. We analyze the local convergence behavior of this method. Only weak assumptions are imposed on the meshes; they are allowed to be refined locally and the cells may be shifted against each other. The proof relies on the techniques for estimating discrete Green functions developed in Frehse and Rannacher [12] for the “continuous” Galerkin method (in short cG method. Finally, we apply the error estimation techniques in Becker and Rannacher [7] to obtain a posteriori error estimates. For simplicity, we concentrate on the Poisson equation,
(1)
with boundary conditions
!#"$ %&' ( )+* (2) on a polygonal domain -,/. . The boundary is assumed to be decomposed like 0 1 32 ) , and the boundary data 4 ) are smooth. We allow for such 5 Institut f¨ur Angewandte Mathematik, Universit¨at Heidelberg, INF 293, D-69120 Heidelberg, Germany
1
general nonhomogeneous boundary conditions in order to highlight the particular flexibility of the dG method. However, in order to avoid technical difficulties, we will regularity nevertheless assume that the weak solution of this problem possesses 6 which very much constrains the generality of this setting. We note that all results of this paper can be extended to more general equations in two and three dimensions with variable coefficients as well as to systems of such equations. The main result of this paper is the local a priori error estimate
7 8:;9 < = >@? 7BADCFE G I< H ?KJML NJMOQPRTSVUXW Y CZE JML NJ
8 9 V? ^ L J g Y E Ä g < ó > Ð n ¹ º ¹ º ? g ghji#k E g n Y < ó > Ð %#" #X¡#" j? g v l Ä J ó >¸L Ü J g YÅE Ä g < ó > ¹ Ü º ¹ Ü º ? g f ghji#k Y E g < ó > £%#" Ü #X¡#" Ü j? g v * 10
(37) and
The two terms
n
and
will be Ó estimated separately.
(iv) First, we estimate n . For defined in (9), we conclude from the interpolation estimates (13) and (14), observing Assumption 2, that
J g Y E gÐ `n Î J ¹ º J g Y E gn`Î JZ¡£" $J g ACFE g JXL N J g Ò Now, we split the summation over d > as follows: n f f ®ª*F*F*%´(Y ®ª*F*Z*%´$* Ò ghji#k g S U gh¡i¸k g+Ò Ö S U Ð n AC and therefore estimate On cells , ¦ b ; , we have ó > C E g n A f f E g JXL N J g NJ g Y C X J L Ò Ò ó gh¡i¸k gh¡i#k g+ Ö S U g S U > C C A E JXL NJ O P RTSVUXW J ó > Ð n J Y E JXL NJ * JXL
From this, we obtain
C C n A E G I< H ?ZJXL NJ O P RTSVUXW Y E XJ L NJ *
(38)
. Observing that ó > is constant on each cell , we have l [ > < ó > Ü Ü ?N f < L < ó > Ü ?ttL Ü ? g YÅE Ä g < ¹ ó > Ü º ¹ Ü º ? g ghji#k Â_ < ¹ ó > Ü º X¡#" Ü j? g Â_ < ¡#" < ó > Ü ?u# ¹ Ü º ? g v l f < ó > L Ü ML Ü ? g YÅE Ä g < ó > ¹ Ü º ¹ Ü º ? g ghji#k YÅE Ä g < ¹ ó > º Ü # ¹ Ü º ? g Â_ < ¹ ó > º Ü £F¡£" Ü j? g < ó > ¹ Ü º F%#" Ü ¡? g _ < ¹ ó > º ¹ #" Ü º ¹ Ü º ? g v *
(v) Next, we estimate
This leads us to
where
r
[ > < ó > Ü Ü È? Y
r
f l E g < ó > £¡#" #X¡#" j ? g Y _ < ¹ ó > £ F¡£" j ? g Ü Ü º Ü Ü Â g hji k Y < ó > ¹ Ü º X¡#" Ü j? g Y _ < ¹ ó > º ¹ #" Ü º ¹ Ü º ? g Y E Ä g < ¹ ó > º Ü # ¹ Ü º ? g v *
11
(39)
(v) We proceed with the first term on the right in (39). Using Galerkin orthogonality of Ü , we obtain > ó ?N > ó > ó ?t ?t*
[ < >Ü Ü [ < >Ü Þ < >Ü Ü ? Since ó > is constant on and the projection Þ > is local, we have Þ > < ó > Ü § > Ü Þ Ü ö Þ > ö , it follows that Hence, with the notation l [ > < ó > Ü Ü ?§ f < L < ó > ?ttL Ü ? g YÅE Ä g < ¹ ó > º ¹ Ü º ? g gh¡i¸k Â_ < ¹ ó > º X¡#" Ü j? g Â_ < ¡#" < ó > ?u# ¹ Ü º ?
ó > Þ > Ü
.
g v *
The four terms on the right hand side are now estimated separately using the trace inequalities (25), (26), and the interpolation estimates (13), (14). Furthermore, we employ the a priori bound (30) and the error estimate (30) for the Green function ö and the7 dual error ö ö > . Note that in order to 7 estimate the average values ¡#" g , it suffices to estimate each of the terms #" g separately. In virtue of Assumption 2, the weights ó > can be estimated by
7y¹ ó > 7 g ACZE g ó > g A CFE g 7 ó > 7 g º A Có and we use that L ö > g ñ . By (23), ó > . We will use a free parameter d< _ º . For the first term, we find
±± f < L < ó > ± gh¡i¸k
l _ E ?ttL Ü ? g ±± A C|f g J ó L ö J g Y ± gh¡i¸k A C G L Ü J g v
and, analogously, for the second term,
±± f E Ä < ¹ ó > ± ghji#k g AC3f ghji k AC3f ghji#k A C G @ Ð n ¹ ó >
¹ º J g Y E g Ä ZJ ó @> Ü º J g v L ¸ö º J g Y E g Ä JZ ó >@ ¹ Ü º J g v
E Y C *
12
The third term is estimated by ±± f < ¹ ó > º F¡£" Ü j? g ±±
± gh¡i¸k
±± f ± gh¡i k ACf gh¡i¸k ACf gh¡i¸k AC f gh¡i¸k AC Y
± l  ¹ ó > º ó @> £ Jó >
Y _ ¹ ó > º É ó J g J #> #" Ü J g
E Mrg Î J ó L J g VE gÐ n`Î ö l _ E g J ó >¸L Ü J g Y C G E L Ü J g Y E g J ó L ö J g
J ó L ö J g v
Finally, for the fourth term, we find ±± f < %#" < ó > ?u# ¹ Ü º ? g ±± A½f
± ghji#k
Y ó >@ ¹ º X¡#" Ü g v ±± ±
l _ J ó >#L
±
JZ ó > Ð n % #" < ó >
ghji#k J g Y E g J ó >¸L
AC|f ghji#k AC f l _ E g J ó L J g Y E Ä JZ ó @> ¹ ö Ü g ghji#k C A G ¹ Ü º J g
E Ä g JZ ó @> ¹ Ü º J g v
ºJg v
Combining these estimates yields 7 > ó
[ < > Ü Ü ? 7¸AC
Y
C G E £%#" Ü £F¡£" Ü j? g ±± AC f l J ó >#L Ü J g Y E g J ó >¸L ö J g v
± ghji#k
±
gh¡i¸k AC Y C G ¸L Ü J g Y g J ó ¸> L ö J g v
13
E The third term in r is the most critical one, since it does non contain a factor g . Therefore, we have to absorb this term into the other definite terms in using the stabilization parameter Ä . We recall that for } > d~ > ,
7 < ¡#" > # ¹ > ? g 7¸A C ¤ XJ L > J g Ò Y E Ä J ¹ > J g (41) } } º } Â Â g } º Ó Ä with defined in (9). Splitting the dual error like Ü Y Þ > Ü with ö Þ > ö , we have
< ó > ¹ Ü º F ¡£" Ü j ? g < ó > ¹ Ü º X ¡#"
¡? g Y < ó > ¹ Ü º X ¡#" Þ > Ü j ? g
The first term on the right is treated analogously as the other terms before leading to C the estimate ±± f < ó > ¹ Ü º X¡#" j? g ±± A Y E G Ü # X¡#" Þ > Ü ¡ ? g v *
Using relation (41), we conclude by a lengthy but standard calculation: 7 < ó > £¡£" > #X¡#" > j? g 7¸A < ó > g Y CFE g ó > g ?ZJZ¡#" > J g Þ Ü Þ Ü Þ Ü
with an arbitrary constant
E g f  g Y CFE g ó > g Z? JXL g C l A E ¤ _ Y J ó >¸L Þ > Ü J g É @ d'< _ º . Consequently,
f g h¡i#k
ó > £ ¡#" Þ > Ü # F%#" Þ > Ü j? g C ¤ l C f « _ Y J ó ¸> L Þ > Ü J g Y E g JXL Þ > Ü J g v * Â gh¡i¸k Ä É
JXL Þ > Ü J g A
follows
Ò Þ>Ü Jg C g Ò Y E g JXL > J g Ò v Þ Ü
E g  < ó > £¡#" Þ > Ü Ä C ¤ A½f «  ghji k Ä Y C|f gh¡i¸k
_ Y
É
C
JML Ü J g Y
JXL
J g
£F¡£" Þ > Ü j ? g
#L Ü J g l E Y J ó >¸L g JXL Ü J g 14
J g Y E g JXL
J g v *
Hence, observing the results of Lemma 1, we obtain
E g A C ¤ « j¡#" Þ > Ü # F%#" Þ > Ü j ? g Â Ä Ä
f g h¡i¸k
? Y
C G E º ¹ £" Ü ± ghji#k A!f l ghji#k A!f l ghji#k AC Y
¹ º Ü º ? g ±±± C E g ZJ Ü $J C J Ü J g Y C G E @£¡#" $ J g v Ü E g MJ L J g Y C J ó #> L J g Y E g J ó >#L J g v Ü Ü ö
and by
±± f E Ä < ¹ ó > º Ü £ ¹ Ü º ? g ±± ± ghji#k g ± E g l A!f ÄE g J ¹ ó > º J E g ZJ ó ¹ Ü º J g Y ZJ Ü $J g v ghji#k A!f l C E Ä < ó > ¹ ¹ ? g Y C J J g Y FC E g MJ L J g v g Ü º Ü º Ü Ü ghji#k AC Y C G < K ?t! d~ÝÿÀ~ >$ z
we have by Galerkin orthogonality
8 < ?N [ > < ?N [ > < >@?t Ü Ü z Ü z ï
for arbitrary ï > dà~ > . Integrating cell-wise by parts and reordering terms, we conclude by elementary but tedious calculation that, for ( z ï > ,
8 < ?§ [ > < * Ü Ü z )F>@? l f Ü ( ? g Y < £" º X ( j ? g
YÅE Ä g < ¹ > º ¹ (¡º ? g v *
@?t ( ? g Y Â_ < $- < >¸?tF ( j ? Â_ < - < >¸?tX¡#" ( YÅE Ä g ¹ (¡º ? g v 17
(47)
with the following notation of cell and edge residuals:
+À< -
$-
>¸?X g Y3 >$ n¹ ./ ¾ ¿ > º if · , 3Á , < >¸?X » ¯¼ = >$ if · 0 , if · ) n ¹ #"$ > ./ º if · , Á ¾¿ * < >¸?X » ¯¼ if · , 0 ) +#" >$ if · , )
From the error representation (47), we obtain the following error estimate: Theorem 2. For the error Ü the a posteriori error estimate
>
in the interior penalty scheme (7), there holds
RonpW RonpW m R W R W m RsrtW RruW 8 < ?N f lKm ge q g Y geq g Y geq gv (48) Ü ghji m RywxW RywoW with the cell residuals g and weight factors q g being defined by m RIn W RonpW g J +õ< >@?ZJ g q g J z ï > J g m R W EVÐ n`Î R W npÎ g g JX$- < >@?ZJ g q g E g J z ï > J g m RruW EVÐ rMÎ RsrtW rMÎ n g g J1- < >¸?ZJ g q g E g JF¡#" < z ï >@?u Ä E gÐ ¹ z ï > º J g for arbitrary ï > d~ > . Theorem 2 provides a posteriori estimates for arbitrary functionals of the error. This also includes the -error estimates. To see this, we take the special functional
8 < È? < tÈ?ZJ J Ð n * Ü Ü The corresponding dual solution z d~ satisfies d 6 < ? and the a priori bound JXL z J A C V (49) C where the stability constant only depends on the domain . From (48) and the interpolation estimates (13), (14), we infer that
l E m RonpW E m R W E m RruW `n Î J Ü J AC32C f g g Y g g Y g g v * gh¡i
(50)
This a posteriori error estimate is asymptotically optimal, too. Next, we state an a posteriori error estimate for the locally averaged error as considered in our a priori error analysis. In this case the dual solution is just the regularized Green 9 function, z ö ; introduced in the proof of Theorem 1. We have the estimate
7 8 9 ; < ? 7@A Ü
l m oR npW IR n W m R W R W m sR ruW sR rtW n < >¸? f geq g Y g(q g Y g(q g v gh¡i 18
(51)
where the residual terms mated as follows:
m yR wxW g
are as defined above and the weights
RonpW J ö ;9 ï >J g R W npÎ q g E g J ö ;9 ï RsrtW rtÎ q g E g JZ¡#" < ö
RswxW
can be esti-
q g
7
q g
> J g 9; >@u? E Ä ¹ ;9 > J g * ï g ö ï º
Numerical results 2
The a posteriori error estimate in Theorem 2 is tested at the same configuration as before; see Figure 3. The error is evaluated at the point [ < *54@u*64#? , which is the point with maximum error in Figure 2 (the value is taken from the solution in the upper right cell adjacent to this point) . We compare three types of “error estimators”. The first one, ¤ < >¸? , is obtained directly from the error representation (47), avoiding the use of triangle and H¨older inequalities:
8 < ?N Ü
¤ < >¸? f
g
R¤ W g *
(52)
7 gR ¤ W 7 : 7 gR ¤ W 7 *
The second estimator uses the local refinement indicators
n < >¸? f
The third one,
< >@?
g
g
RonpW
f g
, is given by Theorem 2 as described above:
< >¸? f g
R W g f
g
m m m ® goR npW q g oR npW Y gR W q g R W Y gR ruW q g sR ruW ´ * Ó
Ó
For practical evaluation of the error estimators, we solve the dual problem (45) on the current mesh with bi-quadraticÓ polynomials obtaining z d ~ > . We decided for exact computation in the higher order space to avoid additional error contributions. For a more efficient computation, z can be obtained by post-processing; see [8] for such strategies and their influence on the estimator. Ó The quality of the resulting approximate error estimators sured by the “effectivity index”: Ó
7 w < >¸? 7 7 8 ;9 < ? 7 * Þ98;: Ü RonpW Mesh adaptation is based on “error indicators” g
w < >¸?t$7 ã _ Â
, is mea-
R W
and g , respectively. For mesh refinement, a fixed fraction (here 20%) of the grid cells with largest indicator are refined.
19
Ó
Ü ¸? 4.288e-3 1.652e-3 6.528e-4 2.917e-4 1.401e-4 6.751e-5 3.373e-5 1.742e-5 9.068e-6 Ó
Ó
eff
Ü ¸? 6.127e-3 2.296e-3 9.431e-4 4.182e-4 2.000e-4 9.678e-5 4.875e-5 2.522e-5 1.324e-5
Þ
eff
1.42 1.39 1.44 1.43 1.43 1.43 1.44 1.45 1.46
Ó Ó Table 2: Point error Ü \< [ ? and Ü \< [ ? Ü \< [ ?KY ¤ < >¸? and the corresponding a poste >@? and n < >@? ( is maximum refinement level). riori estimators ¤ < Ó ¤ < >¸? Ü ¸? is asymptotically optimal while estima tor n < >¸? appears to be off by Ó a factor of aboutÓ < a . The error representation (52) suggests to consider 8 < >@? 8 < >¸?ÈY ¤ < >¸? as new approximation. This post-processing step can improve accuracy dramatically, as shown in Table 2. The mesh obtained in the eighth step of this iteration is shown in Figure 3. Table 2 presents the results obtained by the adaptive refinement process with a refined mesh shown in Figure 3. The error estimator ¤ < >¸? is asymptotically optimal while the estimator n < >¸? appears to be only suboptimal. In Table 3, we show results for estimator < >@? . Since this estimator involves H¨older and triangle inequalities on each cell, there is no cancellation between the different terms of the estimator. Therefore, the error is over-estimated by a factor between 4 and 5. The errors resulting from adaptive refinement —based on the estimator < >¸? here— are comparable to those of Table 2. Therefore, we conclude that both estimators are suited as refinement criteria. Finally, in Table 4, we compare the “error/mesh-ratio” 20
Ü
for uniform and
Figure 3: Initial coarse mesh and an adapted mesh (level computation. uniform refinement Ü
