Local energy estimates for the finite element method on sharply varying grids
aa r X i v : . [ m a t h . NA ] A ug LOCAL ENERGY ESTIMATES FOR THE FINITE ELEMENTMETHOD ON SHARPLY VARYING GRIDS
ALAN DEMLOW, JOHNNY GUZM ´AN, AND ALFRED H. SCHATZ
Abstract.
Local energy error estimates for the finite element method for el-liptic problems were originally proved in 1974 by Nitsche and Schatz. Theseestimates show that the local energy error may be bounded by a local ap-proximation term, plus a global “pollution” term that measures the influenceof solution quality from outside the domain of interest and is heuristically ofhigher order. However, the original analysis of Nitsche and Schatz is restrictedto quasi-uniform grids. We present local a priori energy estimates that arevalid on shape regular grids, an assumption which allows for highly gradedmeshes and which much more closely matches the typical practical situation.Our chief technical innovation is an improved superapproximation result. Introduction
In this note we prove local energy error estimates for the finite element methodfor second-order linear elliptic problems on highly refined triangulations. Most apriori error analyses for the finite element method in norms other than the globalenergy norm place severe restrictions on the mesh. In particular, such error analysesare most often carried out under the assumption that the grid is quasi uniform , thatis, all simplices in the mesh are required to have diameter equivalent to some fixedparameter h . The typical practical situation is rather different. Many (especiallyadaptive) finite element codes enforce only shape regularity of elements, meaningthat all elements in the mesh must have bounded aspect ratio. Though it places aweak restriction upon the rate with which the diameters of elements in the meshmay change, shape regularity allows for the locally refined meshes that are neededto resolve the singularities and other sharp local variations of the solution thatoccur in the majority of practical applications.In the work [NS74] of Nitsche and Schatz, local energy error estimates wereestablished for interior subdomains under the assumption that the finite elementgrid is quasi-uniform. Such local energy estimates are helpful in understandingbasic error behavior, especially “pollution effects” of global solution properties onlocal approximation quality, and they also provide an important technical tool inmany proofs of pointwise bounds for the finite element method (cf. [SW95]). Inaddition, the most relevant error notion in applications is often related to some local norm or functional instead of to the global energy error, as evidenced by therecent surge of interest in ensuring control of the error in calculating “quantitiesof interest” in adaptive finite element calculations instead of merely controlling the Mathematics Subject Classification.
Primary 65N30, 65N15.The first author was partially supported by NSF grant DMS-0713770.The second author was partially supported by NSF grant DMS-0503050.The third author was partially supported by NSF grant DMS-0612599. default global energy error (cf. [BR01]). As a final example of the applicabilityof local energy estimates, we mention that the estimates of [NS74] have been usedto justify certain approaches to parallelization and adaptive meshing (cf. [BH00]).Thus local energy estimates are of broad and fundamental importance in finiteelement theory.Here we prove local energy error estimates under the assumption that the fi-nite element triangulation is shape regular instead of under the more restrictiveassumption of quasi uniformity required in [NS74]. In other words, we essentiallyprove that the results of Nitsche and Schatz hold under the restrictions typicallyplaced upon meshes in practical codes, which in particular allow for highly gradedgrids. Our main innovation is a novel “superapproximation” result which we stateand prove in §
2. In § locally elliptic, so that the PDE under consideration may bedegenerate or change type outside of the domain of interest. In contrast to [NS74],the results we present here are valid up to the domain boundary, allow for nonho-mogeneous Neumann, Dirichlet, and mixed boundary conditions, and also requireonly L ∞ regularity of the coefficients of the differential operator.2. An improved superapproximation result
An essential feature of the proofs of local error estimates given in [NS74], andalso of essentially all published proofs of local and maximum-norm a priori errorestimates for finite element methods, is the use of superapproximation properties.In essence, superapproximation bounds establish that a function in the finite ele-ment space multiplied by any smooth function can be approximated exceptionallywell by the finite element space.In order to fix thoughts, we shall in this section assume for simplicity that Ω ⊂ R n is a polyhedral domain; a more general situation is considered in § T h be a simplicial decomposition of Ω. Denote by h T the diameter of the element T ∈ T h . We assume throughout that the elements in T h are shape-regular, that is,each simplex T ∈ T h contains a ball of diameter c h T and is contained in a ballof radius C h T , where c and C are fixed. Let also S rh be a standard Lagrangefinite element space consisting of continuous piecewise polynomials of degree r − k u k H (Ω) = ( R Ω ( u + |∇ u | ) d x ) / , | u | W kp (Ω) = ( P | α | = k k D α u k pL p (Ω) ) /p , etc.A standard superapproximation result is as follows. Let ω ∈ C ∞ (Ω) with | ω | W j ∞ (Ω) ≤ Cd − j , 0 ≤ j ≤ r . Then for each χ ∈ S rh , there exists η ∈ S rh suchthat for each T ∈ T h satisfying d ≥ h T ,(2.1) k ωχ − η k H ( T ) ≤ C ( h T d k∇ χ k L ( T ) + h T d k χ k L ( T ) ) . Our modified result follows (cf. [Guz06]).
Theorem 2.1.
Let ω ∈ C ∞ (Ω) with | ω | W j ∞ (Ω) ≤ Cd − j for ≤ j ≤ r . Then foreach χ ∈ S rh , there exists η ∈ S rh such that for each T ∈ T h satisfying d ≥ h T , (2.2) k ω χ − η k H ( T ) ≤ C ( h T d k∇ ( ωχ ) k L ( T ) + h T d k χ k L ( T ) ) . Remark . There are two differences between (2.1) and (2.2). First, in (2.1) weconsider approximation of ωχ , whereas in (2.2) we consider approximation of ω χ . OCAL ENERGY ESTIMATES ON SHARPLY VARYING GRIDS 3
Secondly, in (2.1) the norms on the right hand side involve only χ , whereas in (2.2)the H seminorm involves ωχ . If we think of ω as a cutoff function, this distinctionbecomes vitally important: ωχ has the same support as ω χ , whereas the supportof χ is generally larger than that of ωχ . This seemingly minor difference will allowus to establish local energy estimates on grids that are only assumed to be shaperegular. Proof.
Let I h : C (Ω) → S rh be the standard Lagrange interpolant. We shall choose η = I h ( ω χ ) in (2.2). For T ∈ T h , we may use standard approximation theory (cf.[BS02]) to calculate k ω χ − I h ( ω χ ) k H ( T ) ≤ Ch n/ T k ω χ − I h ( ω χ ) k W ∞ ( T ) ≤ Ch n/ r − T | ω χ | W r ∞ ( T ) . (2.3)Noting that D α χ = 0 for all multiindices α with | α | = r , recalling that h T d ≤ Ch n/ r − T | ω χ | W r ∞ ( T ) ≤ C ( r X i =2 h i − T | ω | W i ∞ ( T ) ) k χ k L ( T ) + Ch n/ r − T X | α | =1 , | β | = r − k D α ω D β χ k L ∞ ( T ) ≤ C h T d k χ k L ( T ) + Ch n/ r − T X | α | =1 , | β | = r − k D α ω D β χ k L ∞ ( T ) . (2.4)We next consider the terms k D α ω D β χ k L ∞ ( T ) above. Since | α | = 1, we have D α ω = 2 ωD α ω . Let ˆ ω = | T | R T ω d x so that k ω − ˆ ω k L ∞ ( T ) ≤ Ch T | ω | W ∞ ( T ) ≤ C h T d . Employing inverse estimates, we thus have Ch n/ r − T X | α | =1 , | β | = r − k D α ω D β χ k L ∞ ( T ) ≤ Cd − h n/ r − T X | β | = r − k ωD β χ k L ∞ ( T ) ≤ Cd − h n/ r − T X | β | = r − ( k ( ω − ˆ ω ) D β χ k L ∞ ( T ) + k ˆ ωD β χ k L ∞ ( T ) ) ≤ C ( h T d k χ k L ( T ) + h T d | ˆ ωχ | H ( T ) ) ≤ C ( h T d k χ k L ( T ) + h T d | (ˆ ω − ω ) χ | H ( T ) + h T d | ωχ | H ( T ) ) . (2.5)Using an inverse inequality, we find that h T d | (ˆ ω − ω ) χ | H ( T ) ≤ h T d ( | ω | W ∞ ( T ) k χ k L ( T ) + k ˆ ω − ω k L ∞ ( T ) | χ | H ( T ) ) ≤ C h T d ( 1 d k χ k L ( T ) + h T d | χ | H ( T ) ) ≤ C h T d k χ k L ( T ) . (2.6)Inserting (2.6) into (2.5) and the result into (2.4) and (2.3) completes the proof of(2.2). (cid:3) A. DEMLOW, J. G´UZMAN, AND A. SCHATZ Local H estimates In this section we state and prove a local H estimate that is valid on highlygraded grids. We now let Ω be a domain in R n , and let Ω be a bounded subdomainof Ω. We decompose ∂ Ω ∩ ∂ Ω (if it is nonempty) into a Dirichlet portion Γ D and aNeumann portion Γ N . For the sake of simplicity, we assume that Γ D is polyhedraland that Γ N is either polyhedral or Lipschitz. Let u satisfy − div( A ∇ u ) + b · ∇ u + cu = f in Ω ,u = g D on Γ D ,∂u∂n A = g N on Γ N . (3.1)Here A is an n × n coefficient matrix that is uniformly bounded and positive definitein Ω, b ∈ L ∞ (Ω ) n , c ∈ L ∞ (Ω ), and ∂∂n A is the conormal derivative with respectto A . We also assume that Ω ⊂ R n . Note that we make no assumptions about thedifferential equation solved by u outside of Ω .Let H D, (Ω ) = { u ∈ H (Ω ) : u | Γ D = 0 } , and let H D (Ω ) = u ∈ H (Ω ) : u | Γ D = g D } . Also let H < ( B ) = { u ∈ H (Ω ) : u | Ω \ B = 0 } for subsets B of Ω .Thus functions in H < ( B ) are zero on ∂B \ ∂ Ω, but may be nonzero on portions of ∂B coinciding with ∂ Ω, or put in other terms, functions in H < ( B ) are compactlysupported in B modulo ∂ Ω. Rewriting (3.1) in its weak form, we find that u ∈ H D (Ω ) satisfies L ( u, v ) := Z Ω ( A ∇ u ∇ v + b · ∇ uv + cuv ) d x = Z Ω f v d x − Z Γ N g N v d σ, v ∈ H D, (Ω) ∩ H < (Ω ) . (3.2)Following [NS74], we do not assume that L is coercive over H (Ω ), but ratherwe make a local coercivity assumption: R1: Local coercivity.
There exists a constant d > B is the intersectionof any open sphere of diameter d ≤ d with Ω , then L is coercive over H < ( B ),that is, for some constant C > C ) − k u k H ( B ) ≤ L ( u, u ) ≤ C k u k H ( B ) , u ∈ H < ( B ) . Remark . R1 may be satisfied in one of two ways. It may happen that L iscoercive over H (Ω ), in which case no further argument is needed. R1 so long asa Poincar´e inequality(3.4) k u k L ( B ) ≤ Cd k u k H ( B ) holds for balls B as in R1 having small enough diameter (cf. Remark 1.2 of [NS74]).Such Poincar´e inequalities always hold for interior balls. If B is the nontrivialintersection of an open ball with Ω, then (3.4) holds for d ≤ d small enough underthe restrictions we have placed on ∂ Ω ∩ ∂ Ω ; here d depends on the properties of ∂ Ω ∩ ∂ Ω .Next we make assumptions concerning the finite element approximation u h of u .Let T be a triangulation such that Ω ⊂ ∪ T ∈T T and T ∩ Ω = ∅ for all T ∈ T .Let h T = diam ( T ) for T ∈ T . We denote our trial finite element space by S D .We do not assume that S D ⊂ H D (Ω). In addition, we let S D, = S D ∩ H D, (Ω ) OCAL ENERGY ESTIMATES ON SHARPLY VARYING GRIDS 5 be our trial finite element space. We assume that u h is the local finite elementapproximation to u on Ω , that is, u h ∈ S D and(3.5) L ( u − u h , v h ) = 0 for all v h ∈ S D, ∩ H < (Ω ) . We do not explicitly fix u h on the Dirichlet portion of the boundary, but rather im-plicitly assume that u h | Γ D is set equal to some appropriate interpolant or projectionof g D .Next we state properties that S D and S D, must possess in order to prove thedesired local energy error estimate. Let ˜ d ≤ d be a fixed parameter, and let G and G be arbitrary subsets of Ω with G ⊂ G and dist ( G , ∂G \ ∂ Ω) = ˜ d >
A1: Local interpolant.
There exists a local interpolant I such that for each u ∈ H < ( G ), Iu ∈ S D ∩ H < ( G ), and for each u ∈ H D, (Ω ), Iu ∈ S D, . A2: Inverse properties.
For each χ ∈ S D , T ∈ T h , 1 ≤ p ≤ q ≤ ∞ , and 0 ≤ ν ≤ s ≤ r with r sufficiently small,(3.6) k χ k W sq ( T ) ≤ Ch ν − s + np − nq T k χ k W νp ( T ) . A3: Superapproximation.
Let ω ∈ C ∞ (Ω ) ∩ H < ( G ) with | ω | W j ∞ (Ω ) ≤ Cd − j forintegers 0 ≤ j ≤ r with r sufficiently large. For each χ ∈ S D, and for each T ∈ T h satisfying d ≤ h T ,(3.7) k ω χ − I ( ω χ ) k H ( T ) ≤ C ( h T d k∇ ( ωχ ) k L ( T ) + h T d k χ k L ( T ) ) , where the interpolant I is as in A1 above. Remark . A1, A2, and A3 are satisfied by standard finite element spaces definedon shape-regular triangular grids. A1 also essentially requires that the finite elementmesh resolve G \ G , i.e., that ˜ d ≥ K max T ∩ G = ∅ h T with K large enough.We begin by proving a Caccioppoli-type estimate for “discrete harmonic” func-tions. Such a statement was also proved in [NS74] as a preliminary to local energyestimates, though the proof we give below more closely follows [SW77]. Lemma 3.3.
Let G ⊂ G ⊂ Ω be given, and let dist ( G , ∂G \ ∂ Ω) = d with d ≤ d where d is the parameter defined in the assumption R1. Let also A1, A2,and A3 hold with ˜ d = d , and assume that u h ∈ S D, satisfies (3.8) L ( u h , v h ) = 0 for all v h ∈ S D, ∩ H < (Ω ) . In addition let max T ∩ G = ∅ h T d ≤ . Then (3.9) k u h k H ( G ) ≤ C d k u h k L ( G ) . Here C depends only on the constants in (3.6) and (3.7) and the coefficients of L .Proof. We assume that G is the intersection of a ball B d of radius d with Ω ; thegeneral case may be proved using a covering argument. Let then G and G be theintersections with Ω of balls having the same center as G and having radii d and d , respectively, and without loss of generality let G be the corresponding ball ofradius d . Let then ω ∈ C ∞ ( G ) be a cutoff function which is 1 on G and which A. DEMLOW, J. G´UZMAN, AND A. SCHATZ satisfies k ω k W j ∞ ( G ) ≤ Cd − j , 0 ≤ j ≤ r . We may then apply the assumptions A1through A3 to the pairs G and G , and G and G .Using (3.3), we first compute that(3.10) k u h k H ( G ) ≤ k ωu h k H ( G ) ≤ CL ( ωu h , ωu h ) . Using the fact that k∇ ω k L ∞ (Ω) ≤ Cd , we compute that for any ǫ > L ( ωu h , ωu h ) = L ( u h , ω u h ) − Z Ω u h [ A ∇ ( ωu h ) ∇ ω + u h A ∇ ω ∇ ω + A ∇ ω ∇ ( ωu h ) + ωu h b ∇ ω ] d x ≤| L ( u h , ω u h ) | + C d ǫ k u h k L ( G ) + ǫ k ωu h k H ( G ) . (3.11)Next we use (3.8), (3.7), and the fact that k ω u h k H ( G ) ≤ k ωu h k H ( G ) + Cd k u h k L ( G ) to compute L ( u h , ω u h ) = L ( u h , ω u h − I ( ω u h )) ≤ C X T ∩ G = ∅ h T k u h k H ( T ) ( 1 d | ωu h | H ( T ) + 1 d k u h k L ( T ) ) . (3.12)Using (3.6) and the fact that h T d ≤
1, we have for ǫ as above that Ch T k u h k H ( T ) ( 1 d | ωu h | H ( T ) + 1 d k u h k L ( T ) ) ≤ Cǫd k u h k L ( T ) + ǫ | ωu h | H ( T ) . (3.13)Inserting (3.13) into (3.12), noting that T ∩ G = ∅ implies that T ⊂ G (sincemax T ∩ G = ∅ h T ≤ d ) and carrying out further elementary manipulations then yieldsthat for ǫ > L ( u h , ω u h ) ≤ Cǫd k u h k L ( G ) + ǫ k ωu h k H ( G ) . Inserting (3.14 into (3.11) and the result into (3.10) yields(3.15) k ωu h k H ( G ) ≤ Cǫd k u h k L ( G ) + 2 ǫ k ωu h k H ( G ) . Taking ǫ = so that we may kick back the last term above, employing the triangleinequality, and inserting the result into (3.10) then completes the proof of (3.9). (cid:3) We now prove a local energy error estimate. In our proof below we shall follow[NS74] by using a local finite element projection in order to split the finite elementerror into an approximation error and a “discrete harmonic” term which may bebounded using Lemma 3.3. We note, however, that the use of a local finite elementprojection is not necessary, and our final local error estimate may in fact be provedwith some simple modifications to the proof of Lemma 3.3 above. These two stylesof proof are essentially equivalent. Local finite element projections have been usedfor example in [NS74], [SW77], [SW95], and [AL95] in order to prove local a priorierror estimates. The methodology of Lemma 3.3 in which no local projections areused has been employed in for example [Dem04] and [Guz06] in order to provelocal a priori error estimates and in [LN03] and [Dem07] in order to prove local aposteriori error estimates.
OCAL ENERGY ESTIMATES ON SHARPLY VARYING GRIDS 7
Theorem 3.4.
Let G ⊂ G ⊂ Ω be given, and let dist ( G , ∂G \ ∂ Ω) = d with d ≤ min { d , d } where d is the parameter defined in the assumption R1 and d is defined in Remark 3.1. Let also A1, A2, and A3 hold with ˜ d = d . In additionlet max T ∩ G = ∅ h T d ≤ . Then k u − u h k H ( G ) ≤ C min u h − χ ∈ S D, ( k u − χ k H ( G ) + 1 d k u − χ k L ( G ) )+ C d k u − u h k L ( G ) . (3.16) Here C depends only on the constant C in (2.2) and the coefficients of L .Proof. We assume that G is the intersection of a ball B d of radius d with Ω ; thegeneral case may be proved using a covering argument. Let G be the intersectionwith Ω of a ball having the same center as G and having radius d , and withoutloss of generality let G be the corresponding ball of radius d . Let then ω ∈ C ∞ ( G )be a cutoff function which is 1 on G and which satisfies k ω k W j ∞ ( G ) ≤ Cd − j ,0 ≤ j ≤ r . Note that we may apply Lemma 3.3 with G on the left hand side ofthe estimate (3.9) and G on the right hand side.Next we let P ( ωu ) be a local finite element projection of ωu . In particular, welet P ( ωu ) ∈ S D ∩ H < ( G ) with u h − P ( ωu ) = 0 on Γ D ∩ ∂G satisfy(3.17) L ( ωu − P ( ωu ) , v h ) = 0 , v h ∈ S D, ∩ H < ( G ) . The local coercivity condition (3.3) then implies the stability estimate(3.18) k P ( ωu ) k H ( G ) ≤ C k ωu k H ( G ) . Recalling that u h − P ( ωu ) = 0 on Γ D ∩ ∂G while employing (3.9) and using(3.4) while recalling that ω ≡ G , we compute that k u − u h k H ( G ) ≤ k ωu − P ( ωu ) k H ( G ) + k P ( ωu ) − u h k H ( G ) ≤k ωu − P ( ωu ) k H ( G ) + Cd k P ( ωu ) − u h k L ( G ) ≤k ωu − P ( ωu ) k H ( G ) + Cd ( k P ( ωu ) − ωu k L ( G ) + k u − u h k L ( G ) ) ≤ C k ωu − P ( ωu ) k H ( G ) + Cd k u − u h k L ( G ) . (3.19)Next we employing the triangle inequality along with (3.18) while recalling that k ω k W j ∞ ( G ) ≤ Cd − j in order to find that k ωu − P ( ωu ) k H ( G ) ≤ C k ωu k H ( G ) ≤ C ( k u k H ( G ) + 1 d k u k L ( G ) ) . (3.20)In order to complete the proof of (3.16), we first insert (3.20) into (3.19) andfinally write u − u h = ( u − χ ) + ( χ − u h ) with u h − χ ∈ S D, . (cid:3) References [AL95] Douglas N. Arnold and Xiao Bo Liu,
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