Chapter 5 A priori error estimates for nonconforming finite element approximations 5.1 Strang s first lemma

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1 Chapter 5 A priori error estimates for nonconforming finite element approximations 51 Strang s first lemma We consider the variational equation (51 a(u, v = l(v, v V H 1 (Ω, and assume that the conditions of the Lax-Milgram lemma are satisfied so that (51 admits a unique solution u V We assume that H is a null sequence of positive real numbers and (V h h H an associated family of conforming finite element spaces V h V, h H We approximate the bilinear form a(, : V V lr and the functional l( : V lr in (51 by bounded bilinear forms and bounded linear functionals (52 a h (, : V h V h lr, l h ( : V h lr, h H and consider the finite dimensional variational equations (53 a h (u h, v h = l h (v h, v h V h, h H Definition 51 Uniform V h -ellipticity The sequence (a h (, h H of approximate bilinear forms a h (, : V h V h lr is called uniformly V h -elliptic, if there exists a positive constant α such that uniformly in h H (54 a h (u h, u h α u h 2 V, u h V h Under the assumption of uniform V h -ellipticity, the variational equations (53 admit unique solutions u h V h The following result, known as Strang s first lemma, can be viewed as a generalization of Céa s lemma Theorem 51 Strang s first lemma Assume that (a h (, h H is a uniformly V h -elliptic family of bilinear forms and denote by u V and u h V h, h H the unique solutions of (51 and (53, respectively then, there exists a constant C lr +, independent of h H such that (55 u u h V ( ( a(v h, w h a h (v h, w h C inf u v h V + sup + v h V h w h V h w h V l(w h l h (w h + sup, h H w h V h w h V 1

2 2 Ronald HW Hoppe Proof Taking advantage of the uniform V h -ellipticity, for v h V h we obtain α u h v h 2 V a h (u h v h, u h v h ± a(u v h, u h v h = ( = a(u v h, u h v h + a(v h, u h v h a h (v h, u h v h + ( + l h (u h v h l(u h v h Using the boundedness of the bilinear form, ie, a(u, v M u V v V, and dividing the previous inequality by u h v h V, it follows that α u h v h 2 V M u h v h V + a(v h, u h v h a h (v h, u h v h u h v h V + Using the triangle inequality + l h(u h v h l(u h v h u h v h V M u h v h V + sup w h V h a(v h, w h a h (v h, w h w h V + + sup w h V h l h (w h l(w h w h V u u h V u v h V + u h v h V and the previous inequality, we deduce (55 Strang s first lemma shows that the upper bound for the global discretization error consists of two parts: the approximation error (56 inf v h V h u v h V and the consistency errors (57 (58 inf v h V h a(v h, w h a h (v h, w h sup, w h V h w h V l(w h l h (w h sup w h V h w h V As an application of Strang s first lemma, in the subsequent section we will study the effect of numerical integration in the finite element approximation of second order elliptic boundary value problems

3 Finite Element Methods 3 52 Numerical integration 521 Construction of quadrature formulas We consider the model variational equation (59 a(u, v = l(v, v V := H 1 0(Ω, where Ω lr d is supposed to be a polygonal resp polyhedral domain and the bilinear form a(, : V V lr and the functional l( : V lr are given according to (510 a(u, v := and (511 l(v := Ω i,j=1 Ω a ij u v dx, x i x j fv dx Here, f L 2 (Ω and a ij C(Ω, 1 i, j d, satisfying (512 a ij (x ξ i ξ j i,j=1 α ξi 2, x Ω, α > 0 We approximate (59 using a finite element space V h generated by a triangulation T h and finite elements (K, P K, Σ K, K T h, satisfying the assumptions (A1,(A2,(A3 of section 43 and ˆP ˆK H 1 ( ˆK which implies conformity, ie, V h V We approximate the bilinear form (510 and the functional (511 by evaluating the integrals by means of appropriate quadrature formulas with respect to the individual elements K T h of the triangulation Taking into account the affine equivalence of the finite elements with respect to a reference element ˆK F K : ˆK K ˆx F K (ˆx = B K ˆx + b K, K T h, due to the transformation of variables formula (513 ϕ(x dx = det(b K ˆϕ(ˆx dˆx, ˆϕ := ϕ F K, K ˆK a quadrature formula for an integral over the reference element ˆK induces a suitable quadrature formula for the associated integral over the

4 4 Ronald HW Hoppe actual element K = F K ( ˆK of T h In particular, we consider the reference quadrature formula L (514 ˆQ ˆK( ˆϕ := ˆω l, ˆK ˆϕ(ˆb l, ˆK, with weights ˆω l, ˆK, 1 l L, and nodes ˆb l, ˆK ˆK, 1 l L We refer to (515 Ê ˆK( ˆϕ := ˆϕ(ˆx dˆx ˆQ ˆK( ˆϕ ˆK as the associated quadrature error functional Lemma 51 Quadrature formulas in the FEM Assume that ˆQ ˆK is a quadrature formula with respect to the reference element ˆK as given by (514 Then, for the affine equivalent element K := F K ( ˆK we obtain a quadrature formula according to (516 Q K (ϕ := L ω i,k ϕ(b i,k, ω l,k := det(b K ˆω l, ˆK, b l,k := F K (ˆb l, ˆK, 1 l L Moreover, referring to (517 E K (ϕ := K ϕ(x dx Q K (ϕ as the related quadrature error functional, we have (518 E K (ϕ = det(b K Ê ˆK( ˆϕ Proof The proof is a direct consequence of (513 Based on the quadrature formula (516, we define the approximate bilinear forms a h (, : V h V h lr by means of (519 a h (u h, v h := K T h L ω l,k u h (a ij x i u,j=1 v h x j (b l,k, and the approximate linear functionals l h ( : V h lr according to (520 l h (v h := L ω l,k (fv h (b l,k K T h

5 Finite Element Methods 5 Then, the finite element approximation of the model problem (59 amounts to the computation of u h V h, h H, as the solution of the approximate variational equation (521 a h (u h, v h = l h (v h, v h V h We note that according to the Lax-Milgram lemma, (521 admits a unique solution, if the bilinear form (a h (, h H is bounded and V h - elliptic We first derive sufficient conditions for the uniform V h -ellipticity of the family (a h (, h H of approximate bilinear forms Theorem 52 Sufficient conditions for uniform V h -ellipticity Let ˆQ ˆK be a quadrature formula with respect to the reference element ˆK with positive weights ˆω l, ˆK, 1 l L, and the property that there exists q ln such that ˆP ˆK P q ( ˆK, (i the quadrature formula is exact for polynomials ˆp P 2q 2 ( ˆK, or (ii the union of nodes L {ˆb l, ˆK} contains a P q 1 ( ˆK-unisolvent subset Then, there exists a positive constant α, independent of h H, such that (522 a h (v h, v h α v h 2 1,Ω, v h V h, h H Proof Observing v h K = p K P K and using the ellipticity condition (512, we find (523 Now, observing = L L α ω l,k ω l,k L v h (a ij x i i,j=1 p K (a ij x i i,j=1 ω l,k ( p K (b l,k 2 x i }{{} = Dp K (b l,k 2 v h x j (b l,k = p K x j (b l,k Dˆp ˆK(ˆb l, ˆKξ = Dp(b l,k (B K ξ, 1 l L,

6 6 Ronald HW Hoppe we have Dˆp ˆK(ˆb l, ˆK B K Dp(b l,k, 1 l L, and hence, using Theorem 43, we get L (524 ω l,k ( p K (b l,k 2 B K 2 x i }{{} = Dp K (b l,k 2 L ω l,k = det(b K B K 2 L ˆω l, ˆK ( ˆp ˆK ˆx i (ˆb l, ˆK 2 ( ˆp ˆK ˆx i (ˆb l, ˆK 2 (i Let us first assume that the quadrature formula ˆQ ˆK is exact for polynomials ˆp P 2q 2 ( ˆK Since ( ˆp ˆK 2 P 2q 2 ( ˆx ˆK, i we then have (525 ˆp ˆK 2 1, ˆK = ˆK ( ˆp ˆK ˆx i 2 dˆx = ( ˆp ˆK ˆx i (ˆb l, ˆK 2 Inserting (525 into (524 and using again Theorem 43, it follows that L ω l,k ( p K (526 (b l,k 2 x i det(b K B K 2 L ˆω l, ˆK = det(b K B K 2 ˆp ˆK 2 1, ˆK ( 2 B K B 1 K pk 2 1,K Due to the shape regularity of T h, h H, B K B 1 K ĥˆk ˆρ ˆK ( ˆp ˆK ˆx i (ˆb l, ˆK 2 = h k ρ K C Combining (523,(524 and (526, we deduce the existence of a positive constant α, independent of h H, such that L v h v h (527 ω l,k (a ij (b l,k α v h 2 1,K, v h V h x i x j i,j=1

7 Finite Element Methods 7 Forming the sum over all K T h in (527, gives the assertion (ii What remains to be shown is that the assertion also holds true, if we assume that the union L {ˆb l, ˆK} contains a P q 1 ( ˆK-unisolvent subset We claim that in this case (525 has to be replaced by (528 ( ˆp ˆK ˆx i (ˆb l, ˆK 2 Ĉ ˆp ˆK 2 1, ˆK, where Ĉ is a positive constant The proof then proceeds in the same way as before In order to verify (528, it suffices to show that ( L ˆω l, ˆK ( ˆp ˆK ˆx i (ˆb l, ˆK 2 1/2 provides a norm on the quotient space ˆP ˆK/P 0 ( ˆK, since so does 1, ˆK, and we may conclude taking advantage of the equivalence of norms on finite dimensional spaces For that purpose, we assume L ˆω l, ˆK ( ˆp ˆK ˆx i (ˆb l, ˆK 2 = 0 Then, the positivity of the weights ˆω l, ˆK, 1 l L, yields But for each i {1,, d} ˆp ˆK ˆx i (ˆb l, ˆK = 0, 1 i d, 1 l L and hence, ˆp ˆK ˆx i P q 1 ( ˆK, ˆp ˆK ˆx i 0, since it vanishes on a P q 1 ( ˆK-unisolvent subset With Theorem 52 at hand, we may apply Strang s first lemma If the solution u V of (59 satisfies u V H k+1 (Ω, k ln,, for the approximation error we get (529 inf v h V h u v h 1,Ω u Π h u 1,Ω C h k u k+1,ω Therefore, we have to provide sufficient conditions ensuring that the consistency error does not deteriorate this order of convergence, ie,

8 8 Ronald HW Hoppe we are looking for consistency error estimates of the form (530 sup w h V h a(v h, w h a h (v h, w h w h V C(a ij, u h k, (531 sup w h V h l(w h l h (w h w h V C(l h k, where C(a ij, u and C(l are positive constants, independent of h H The following auxiliary result will be used to establish such consistency error estimates Lemma 52 Generalized Leibniz formula Assume that Ω lr d is a Lipschitz domain and u H m (Ω and v W m, (Ω for some m ln 0 Then, there exists a positive constant C(m, d such that m (532 uv m,ω C(m, d u m j,ω v j,,ω Proof The proof is left as an exercise Theorem 53 j=0 Consistency error estimate I Let ˆQ ˆK be a quadrature formula with respect to the reference element ˆK with positive weights ˆω l, ˆK, 1 l L, and the property that there exists k ln such that (533 ˆP ˆK = P k ( ˆK, (534 Ê ˆK( ˆϕ = 0, ˆϕ P 2k 2 ( ˆK Then, there exists a positive constant C, independent of h H, such that for all a W k, (Ω and all p, q P k (K (535 E K (a q x i p x j C h k K a k,,k p 1,K q k,k Proof Since q x i, p x j P k 1 (K, it suffices to prove (535 for E K (avw, a W k, (K, v, w P k 1 (K In view of (518 in Lemma 51, we have where E K (avw = det(b K Ê ˆK(âˆvŵ â W k, ( ˆK, ˆv, ŵ P k 1 ( ˆK

9 Finite Element Methods 9 (i Since âˆv W k, ( ˆK, we first provide an estimate for Ê ˆK( ˆϕŵ, ˆϕ W k, ( ˆK, ŵ P k 1 ( ˆK By direct estimation and using the equivalence of norms on P k 1 ( ˆK, we obtain L Ê ˆK( ˆϕŵ = ˆϕŵ dˆx ˆω l, ˆK ( ˆϕŵ(ˆb ˆK l, ˆK Ĉ ˆϕŵ 0,, ˆK Ĉ ˆϕ 0,, ˆK ŵ 0,, ˆK Ĉ ˆϕ k,, ˆK ŵ 0, ˆK, which proves the continuity of the functional Ê ˆK(ŵ : W k, ( ˆK lr given by (Ê ˆK(ŵ( ˆϕ := Ê ˆK( ˆϕŵ with (536 Ê ˆK(ŵ Ĉ ŵ 0, ˆK Since P k 1 ( ˆK Ker(Ê ˆK(ŵ, using (536 in the Bramble-Hilbert lemma, we get (537 Ê ˆK( ˆϕŵ Ĉ ˆϕ k,, ˆK ŵ 0, ˆK (ii We now consider the case ˆϕ = âˆv where â W k, ( ˆK and ˆv P k 1 ( ˆK In view of the generalized Leibniz formula (cf Lemma 52 and taking ˆv k,, ˆK = 0 into account, we obtain (538 Ê ˆK(âˆvŵ Ĉ Now, Theorem 43 tells us ( k 1 â k j,, ˆK ˆv j, ˆK ŵ 0, ˆK (539 â k j,, ˆK Ĉ hk j K a k j,,k, 0 j k 1, j=1 (540 ˆv j, ˆK Ĉ det(b K 1/2 h j K v j,k, 0 j k 1, (541 ŵ 0, ˆK Ĉ det(b K 1/2 w 0,K Using (538-(541, we finally obtain E K (avw C h k K ( k 1 a k j,,k v j,k j=0 w 0,K C h k K a k,,k v k 1,K w 0,K Theorem 54 Consistency error estimate II

10 10 Ronald HW Hoppe Let ˆQ ˆK be a quadrature formula with respect to the reference element ˆK with positive weights ˆω l, ˆK, 1 l L, and the property that there exists k ln such that (542 ˆP ˆK = P k ( ˆK, (543 Ê ˆK( ˆϕ = 0, ˆϕ P 2k 2 ( ˆK Assume that g W k,p (Ω and p P k (K and suppose that k ln satisfies (544 k > d p Then, there exists a constant C lr +, independent of h H, such that for all K T h (545 E K (gp C (meas(k ( p h k K g k,p,k p 1,K Proof where In view of (518 in Lemma 51, we have E K (gp = det(b K Ê ˆK(ĝˆp ĝ W k,p ( ˆK, ˆp P k ( ˆK Denoting by ˆQ : L 2 ( ˆK P 1 ( ˆK the orthogonal projection of L 2 ( ˆK onto P k ( ˆK, we split Ê ˆK(ĝˆp according to (546 Ê ˆK(ĝˆp = Ê ˆK(ĝ ˆQˆp + Ê ˆK(ĝ(ˆp ˆQˆp and estimate both parts separately (Note that a direct estimation (without such a splitting would result in a non optimal estimate (i Estimation of the first term in (546 In view of (544 and the Sobolev imbedding theorem, the space W k,p ( ˆK is continuously imbedded in C 0 ( ˆK Hence, for ˆψ W k,p ( ˆK Since Ê ˆK( ˆψ Ĉ ˆψ 0,, ˆK Ĉ ˆψ k,p, ˆK Ê ˆK( ˆψ = 0, ˆψ Pk 1 ( ˆK, the Bramble-Hilbert lemma infers (547 Ê ˆK( ˆψ Ĉ ˆψ k,p, ˆK Now, for ˆψ = ĝ ˆQˆp, observing that ˆQˆp l,, ˆK = 0, l 2,

11 Finite Element Methods 11 the generalized Leibniz formula implies ĝ ˆQˆp ( k,p, ˆK Ĉ ĝ k,p, ˆK ˆQˆp 0,, ˆK + ĝ k 1,p, ˆK ˆQˆp 1,, ˆK Taking advantage of the equivalence of norms on P 1 ( ˆK, we get (548 ĝ ˆQˆp ( k,p, ˆK Ĉ ĝ k,p, ˆK ˆQˆp 0, ˆK + ĝ k 1,p, ˆK ˆQˆp 1, ˆK Since ˆQ is an orthogonal projection, we have (549 ˆQˆp 0, ˆK ˆp 0, ˆK Moreover, since ˆQˆp = ˆp, ˆp P 0 ( ˆK, the Bramble-Hilbert lemma gives ˆp ˆQˆp 1, ˆK Ĉ ˆp 1, ˆK, whence (550 ˆQˆp 1, ˆK ˆp ˆQˆp 1, ˆK + ˆp 1, ˆK (1 + Ĉ ˆp 1, ˆK Combining (547-(550, we obtain (551 Ê ˆK(ĝ ˆQˆp ( Ĉ ĝ k,p, ˆK ˆp 0, ˆK + ĝ k 1,p, ˆK ˆp 1, ˆK (ii Estimation of the second term in (546 Since ˆp ˆQˆp = 0 for ˆp P 1 ( ˆK, we may assume k 2 We further choose r [1, large enough so that the mappings W k,p ( ˆK W k 1,r ( ˆK, W k 1,r ( ˆK C 0 ( ˆK represent continuous imbeddings Case 1 1 p < d We may choose 1 = 1 1 so that by the Sobolev imbedding theorem the mapping W 1,p ( ˆK L r ( ˆK and hence, also the mapping r p d W k,p ( ˆK W k 1,r ( ˆK is a continuous imbedding Moreover, in view of (544, we have k 1 d r = k d p > 0 Then, the Sobolev imbedding theorem also guarantees that the mapping W k 1,r ( ˆK C 0 ( ˆK is a continuous imbedding Case 2 d < p In this case, due to the Sobolev imbedding theorem the mapping W 1,p ( ˆK L r ( ˆK is a continuous imbedding for all r [1, ], if d < p, and for all r [1,, if d = p It follows that the same holds

12 12 Ronald HW Hoppe true for the mapping W k,p ( ˆK W k 1,r ( ˆK Further, choosing r so large that k 1 d r > 0, the Sobolev imbedding theorem implies that W k 1,r ( ˆK C 0 ( ˆK is a continuous imbedding Now, in either case we obtain Ê ˆK(ĝ(ˆp ˆQˆp Ĉ ĝ(ˆp ˆQˆp 0,, ˆK Ĉ ĝ 0,, ˆK ˆp ˆQˆp 0,, ˆK Ĉ ĝ k 1,r, ˆK ˆp ˆQˆp 0,, ˆK Consequently, the linear functional Ê ˆK(ˆp : W k 1,r ( ˆK lr given by (Ê ˆK(ˆp(ĝ := Ê ˆK(ĝ(ˆp ˆQˆp is continuous with Ê ˆK(ˆp Ĉ ˆp ˆQˆp 0,, ˆK Since Ê ˆK(ˆp = 0, ˆp P k 2 ( ˆK, by the Bramble-Hilbert lemma (552 Ê ˆK(ĝ(ˆp ˆQˆp Ĉ ĝ k 1,r, ˆK ˆp ˆQˆp 0, ˆK Moreover, ˆQˆp = ˆp, ˆp P0 ( ˆK, and hence, (553 ˆp ˆQˆp o, ˆK Ĉ ˆp 1, ˆK In order to estimate ĝ k 1,r, ˆK in (552, we note that in view of the Sobolev imbedding theorem the mapping W 1,p (ˆk L r (ˆk is a continuous imbedding, whence ˆf ( 0,r, ˆK Ĉ ˆf 0,p, ˆK + ˆf 1,p, ˆK, and hence, (554 ĝ k,r, ˆK Ĉ ( ĝ k 1,p, ˆK + ĝ k,p, ˆK Using (553 and (554 in (552, we get (555 Ê ˆK(ĝ(ˆp ˆQˆp ( Ĉ ĝ k 1,p, ˆK + ĝ k,p, ˆK ˆp 1, ˆK Finally, combining (546,(551,(555 and observing that Theorem 43 infers ĝ k j,p, ˆK Ĉ (det(b K 1/p h k j K g k j,p,k, 0 j 1,

13 Finite Element Methods 13 ˆp j, ˆK Ĉ (det(b K 1/2 h j K p j,k, 0 j 1, we arrive at the assertion After having estimated both the approximation and the consistency errors, we are now in a position to establish an a priori estimate of the global discretization error in the H 1 (Ω-norm For this purpose, we provide the following auxiliary result Lemma 53 Generalized Schwarz inequality for sums Assume a i, b i, c i lr, 1 i m, and r 1, r 2, r 3 ln satisfying 1 r r 2 + = 1 Then, there holds 1 r 3 (556 m a i b i c i ( m a i r 1/r1 ( m 1 b i r 1/r2 ( m 2 c i 3 r 1/r3 Proof The proof is left as an exercise Theorem 55 A priori estimate in the H 1 (Ω-norm In addition to the assumptions (A1,(A2,(A3, suppose that for some integer k ln we have (557 ˆP = Pk ( ˆK, and that (558 H k+1 ( ˆK C s ( ˆK is a continuous imbedding, where s ln 0 is the maximal order of partial derivatives in the definition of the set ˆΣ ˆK of degrees of freedom Moreover, let ˆQ ˆK be a quadrature formula with respect to the reference element ˆK with positive weights ˆω l, ˆK, 1 l L, satisfying (559 Ê ˆK( ˆϕ = 0, ˆϕ P 2k 2 ( ˆK As far as the data of the model problem (59 are concerned, we require the existence of some p 2 with k > d such that the following p regularity assumptions are met (560 a ij W k, (Ω, 1 i, j d, f W k,p (Ω We denote by u V := H0(Ω 1 the solution of (59 and by u h V h, h H the solution of its finite element approximation (521, and we assume (561 u V H k+1 (Ω

14 14 Ronald HW Hoppe Then, there exists a positive constant C lr, independent of h H, such that the following a priori error estimate for the global discretization error holds true (562 u u h 1,Ω C h ( u k k+1,ω + a ij k,,ω u k+1,ω + f k,p,ω i,j=1 Proof The assumptions apply that the family (a h (, h H of approximate bilinear forms a h (, : V h V h lr, h H, is uniformly V h -elliptic We may thus apply Strang s first lemma and provide appropriate estimates of the approximation error and the consistency error (i Approximation error In view of the regularity assumption (561, it follows from Theorem 45 that (563 u Π h u 1,Ω C h k u k+1,ω (ii Consistency error, Part I By Theorem 53 and the Cauchy-Schwarz inequality, for w h V h we obtain (564 a(π h u, w h a h (Π h u, w h K T h C i,j=1 K T h h k K C h k ( Π K u w h E K (a ij x i x j ( a ij k,,k Π K u k,k w h 1,K i,j=1 ( a ij k,,k i,j=1 K T h Π K u 2 k,k 1/2 wh 1,Ω Applying Theorem 45 once more, we get ( 1/2 (565 Π K u 2 k,k K T h ( 1/2 u k,k + u Π K u 2 k,k K T h u k,k + C h u k+1,ω C u k+1,ω

15 Using (565 in (564 yields (566 (567 Finite Element Methods 15 a(π h u, w h a h (Π h u, w h sup w h V h w h 1,Ω C h k a ij k,,ω u k+1,ω i,j=1

16 16 Ronald HW Hoppe (ii Consistency error, Part II Using Theorem 54, for w h V h it follows that (568 l(w h l h (w h E K (fw h K T h C K T h (meas(k ( p h k K f k,p,k w h 1,K Applying the generalized Schwarz inequality (Lemma 53 to the last sum in (568 with results in (569 m := card(t h, l(w h l h (w h 1 r 1 := p, r 2 := p, r 3 := 2, K T h E K (fw h We thus get C h k (meas(ω ( p f k,p,ω w h 1,Ω l(w h l h (w h (570 sup C h k (meas(ω ( p f k,p,ω w h V h w h 1,Ω Finally, combining (563,(566 and (570, the assertion is a direct consequence of Strang s first lemma 522 Special quadrature formulas We first consider quadrature formulas for simplicial triangulations Lemma 54 Quadrature formulas for simplicial triangulations Let ˆK be a non degenerate 2-simplex with vertices â i, 1lei 3, and denote by â ij, 1 i < j 3, the midpoints of the edges and by â 123 the center of gravity then, there holds: (i The quadrature formula (571 ˆϕ(ˆx dˆx meas( ˆK ˆϕ(â 123 ˆK is exact for polynomials ˆp P 1 ( ˆK, ie, (572 Ê(ˆp = 0, ˆp P 1 ( ˆK

17 Finite Element Methods 17 (ii The quadrature formula (573 ˆϕ(ˆx dˆx 1 3 ˆK meas( ˆK 1 i<j 3 ˆϕ(â ij is exact for polynomials ˆp P 2 ( ˆK, ie, (574 Ê(ˆp = 0, ˆp P 2 ( ˆK (iii The quadrature formula (575 ˆϕ(ˆx dˆx 1 ( meas( ˆK 3 60 ˆK 3 ˆϕ(â i + 8 is exact for polynomials ˆp P 3 ( ˆK, ie, 1 i<j 3 (576 Ê(ˆp = 0, ˆp P 3 ( ˆK ˆϕ(â ij + 27 ˆϕ(â 123 Proof Let λ i (ˆx, 1 i 3, be the barycentric coordinates of ˆx ˆK Then, for α i ln 0, 1 i 3, there holds 3 3 d! α i! (577 λ α i i (ˆx dˆx = meas( ˆK ˆK ( 3 α i + d! The proof of (577 is left as an exercise We next consider quadrature formulas for rectangular triangulations Lemma 55 Quadrature formulas for rectangular triangulations Let ˆK := [0, 1] 2 with vertices â 1 := (0, 0, â 2 := (1, 0, â 3 := (1, 1, â 4 := (0, 1 and center â s := (1/2, 1/2 Then, there holds (i formula (578 ˆϕ(ˆx dˆx 1 4 ˆϕ(â i 4 is exact for polynomials ˆp Q 1 ( ˆK, ie, ˆK (579 Ê(ˆp = 0, ˆp Q 1 ( ˆK The quadrature

18 18 Ronald HW Hoppe (ii The quadrature formula (580 ˆϕ(ˆx dˆx ˆϕ(â s is exact for polynomials ˆp Q 1 ( ˆK, ie, (581 Ê(ˆp = 0, ˆp Q 1 ( ˆK ˆK (iii Let k ln 0 and denote by ˆb i (0, 1, 1 i k+1, the nodes and by ˆω i, 1 i k+1, the weights of the Gauss-Legendre quadrature formula 1 0 ˆϕ(ˆx dˆx k+1 ˆω i ˆϕ(ˆb i, which is exact for polynomials ˆp P 2k+1 (ˆk Then, the quadrature formula (582 ˆϕ(ˆx dˆx ˆω i1 ˆω i2 ˆϕ(ˆb i1, ˆb i2 ˆK i 1,i 2 {1,,k+1} is exact for polynomials ˆp Q 2k+1 ( ˆK, ie, (583 Ê(ˆp = 0, ˆp Q 2k+1 ( ˆK Proof The proof is left as an exercise We note that the quadrature formulas (571,(573,(575 and (578, (580,(582 can be easily generalized to 3D 53 Strang s second lemma In this section, we will derive an abstract error estimate, known as Strang s second lemma, that is applicable for finite element approximations of variational equations (584 a(u, v = l(v, v V, where the finite element spaces V h, h H, are no longer subspaces of the underlying infinite dimensional function space V We assume that (V h h H is a family of finite dimensional linear spaces and suppose that h, h T h, provides a mesh-dependent norm on V h + V We further suppose that the linear functional l is well defined on V h + V We consider an associated family of approximate bilinear forms (585 a h (, : (V h + V (V h + V lr

19 Finite Element Methods 19 We suppose that the family (a h (, h H is uniformly V h -elliptic in the sense that there exists a positive constant α, independent of h H, such that (586 a h (v h, v h α v h 2 V h, v h V h, h H We further suppose that the family (a h (, h H is uniformly bounded on V h + V in the sense that there exists a positive constant M, independent of h H, such that (587 a h (u, v M u h v h, u, v V h + V, h H We approximate (584 by the finite dimensional variational equations (588 a h (u h, v h = l(v h, v h V h, h H Strang s second lemma may be viewed as another generalization of Céa s lemma Theorem 56 Strang s second lemma Assume that (a h (, h H is a uniformly bounded and uniformly V h -elliptic family of approximate bilinear forms and suppose that u V and u h V h, h H, are the unique solutions of the variational equations (584 and (588, respectively Then, there exists a positive constant C lr, independent of h H, such that ( a h (u, w h l(w h (589 u u h h C inf u v h h + sup v h V h w h V h w h h Proof Taking advantage of the uniform V h -ellipticity and observing (588, for an arbitrary v h V h we obtain α u h v h 2 h a h (u h v h, u h v h = ( = a h (u v h, u h v h + l(u h v h a h (u, u h v h By the uniform boundedness (587 we get α u h v h h a h (u h v h, u h v h M u v h h + l(u h v h a h (u, u h v h u h v h h M l(w h a h (u, w h u v h h + sup w h V h w h h In conjunction with the triangle inequality u u h h u v h h + u h v h h, the previous inequality gives the assertion

20 20 Ronald HW Hoppe As in the case of Strang s first lemma, we see that the upper bound for the global discretization error consists of an approximation error (590 inf v h V h u v h h and the consistency error (591 sup w h V h l(w h a h (u, w h w h h, which can be estimated separately 54 A priori error estimate for nonconforming finite element approximations As an example for a nonconforming finite element in the discretization of second order elliptic boundary value problems with respect to a family of shape-regular simplicial triangulations T h, h H, of the computational domain Ω lr d we consider the lowest order Crouzeix- Raviart element CR 1 (K := (K, P K, Σ K, K T h, also known as the nonconforming P1-element: (592 P K := P 1 (K, Σ K := { p(m i (K, 1 i d + 1 }, p P K, where m i (K, 1 i d + 1, are the midpoints of the edges (d = 2 and the centers of gravity of the faces (d = 3, respectively (cf Figure 51 Fig 51: Nonconforming Crouzeix-Raviart element The associated Crouzeix-Raviart finite element space CR 1 (Ω, T h, composed by the Crouzeix-Raviart elements CR 1 (K, K T h, is then

21 given by (593 CR 1 (Ω, T h := Finite Element Methods 21 { v h L 2 (Ω v h K P 1 (K, K T h, v h is continuous in m(f, f F h (Ω }, where m(f, f F h (Ω are the midpoints (centers of gravity of interior edges (faces Obviously, in general (594 CR 1 (Ω, T h H 1 (Ω The subspace CR 1,0 (Ω, T h is defined by means of CR 1,0 (Ω, T h := { v h CR 1 (Ω, T h v h (m(f = 0, f F h ( Ω } We consider the approximation of Poisson s equation under homogeneous Dirichlet boundary conditions (595 a(u, v := u vdx = l(v := fv dx, v H0(Ω 1 by Ω (596 a h (u h, v h = l(v h, v h CR 1,0 (Ω, T h, h H, where the mesh dependent bilinear form a h (, : H 1 0(Ω CR 1,0 (Ω, T h H 1 0(Ω CR 1,0 (Ω, T h lr is given by (597 a h (u, v := K T h (Ω K Ω u v dx, u, v H 1 0(Ω CR 1,0 (Ω, T h We associate with (597 the mesh dependent norm (598 v h := a h (v, v, v CR 1,0 (Ω, T h H 1 0(Ω We will derive a quasi-optimal a priori error estimate in the h - norm by an application of Strang s second lemma: Theorem 57 A priori error estimate in the h -norm Let u H0(Ω 1 H s (Ω, s = 2 (d = 2, s = 3 (d = 3, be the solution of (595 and u h CR 1,0 (Ω, T h, h H, its nonconforming P1-approximations Then, there exists a constant C > 0, depending only on the shape regularity of the triangulations T h (Ω, h H, such that (599 u u h h C h u s,ω

22 22 Ronald HW Hoppe Proof It is easily verified that the family (a h (, h H of bilinear forms is uniformly V h -elliptic (V h := CR 1,0 (Ω, T h According to Strang s second lemma we have to estimate the approximation error (5100 inf v h CR 1,0 (Ω,T h u v h h and the consistency error a h (u, w h l(w h (5101 sup w h CR 1,0 (Ω,T h w h h (i Estimation of the approximation error Since S 1,0 (Ω, T h CR 1,0 (Ω, T h, in view of Theorem 47 we have (5102 inf v h CR 1,0 (Ω,T h u v h h inf v h S 1,0 (Ω,T h u v h h C h u s,ω (ii Estimation of the consistency error Observing that u(x = f(x faa x Ω, by Green s formula we obtain for w h CR 1,0 (Ω, T h (5103 L u (w h := a h (u, w h l(w h = = ( u w h dx fw h dx K T h K = ( = ( K T h K = K T h f K f K u f w h dx + u n w h dσ K u n w h dσ Denoting by w h f the integral mean 1 w h f := w h dσ, f F h (Ω, meas(f it follows that (5104 L u (w h = K T h f f K f u n (w h w h f dσ =

23 Finite Element Methods 23 Moreover, since I hu P n 0(K ν, 1 ν 2, we have I h u n (w h w h f dσ = 0, f K ν and hence, (5104 gives rise to L u (w h = K T h f K f (u I h u n (w h w h f dσ The Cauchy-Schwarz inequality yields L u (w h ( (u I hu 1/2 2 dσ w h w h f 2 dσ n K T h f K f f }{{}}{{} := I 1 := I 2 We will estimate I 1 and I 2 separately (ii 1 Estimation of I 1 Taking advantage of the affine equivalence of the Crouzeix-Raviart elements, for the reference element ˆK we find by the trace theorem and the Bramble-Hilbert lemma (u I hu 2 dσ C u I h u 2 n 2, ˆK C u 2 s, ˆK ˆK By a standard scaling argument (5105 (u I hu 2 dσ C h u 2 s,k n K (ii 2 Estimation of I 2 Again, for the reference element ˆK we find by the Bramble-Hilbert lemma ŵ h ŵ h ˆf 2 dˆσ C ŵ h 2 1, ˆK, ŵ h P 1 ( ˆK, ˆf F( ˆK, ˆf and by a standard scaling argument, for w h CR 1,0 (Ω, T h and f F(K, K T h we obtain (5106 w h w h f 2 dσ C h w h 2 1,K f

24 24 Ronald HW Hoppe Using (5105 and (5106, we finally obtain L u (w h 3 C h K T h u s,k w h 1,K ( C h K T h u 2 s,k K T h w h 2 1,K 1/2 = C h u s,ω w h h

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