Medius analysis and comparison results for first-order finite element methods in linear elasticity

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1 IMA Journal of Numerical Analysis Advance Access published November 7, 2014 IMA Journal of Numerical Analysis (2014) Page 1 of 31 doi: /imanum/dru048 Medius analysis and comparison results for first-order finite element methods in linear elasticity C. Carstensen Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D Berlin, Germany and Department of Computational Science and Engineering, Yonsei University, Seoul, South Korea Corresponding author: cc@math.hu-berlin.de and M. Schedensack Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D Berlin, Germany [Received on 23 September 2013; revised on 28 April 2014] his paper enfolds a medius analysis for first-order nonconforming finite element methods (FEMs) in linear elasticity named after Crouzeix Raviart and Kouhia Stenberg, which are robust with respect to the incompressible limit as the Lamé parameter λ tends to infinity. he new result is a best-approximation error estimate for the stress error in L 2 up to data-oscillation terms. Even for very coarse shape-regular triangulations, two comparison results assert that the errors of the nonconforming FEM are equivalent to those of the conforming first-order FEM. he explicit role of the parameter λ in those equivalence constants leads to an advertisement of the robust and quasi-optimal Kouhia Stenberg FEM, in particular for nonconvex polygons. he proofs are based on conforming companions, a new discrete Helmholtz decomposition and a new discrete-plus-continuous Korn inequality for Kouhia Stenberg finite element functions. Numerical evidence strongly supports the robustness of the nonconforming FEMs with respect to incompressibility locking and with respect to singularities, and underlines that the dependence of the equivalence constants on λ in the comparison of conforming and nonconforming FEMs cannot be improved. his work therefore advertises the Kouhia Stenberg FEM as a first-order robust discretization in linear elasticity in the presence of Neumann boundary conditions. Keywords: linear elasticity; nonconforming finite elements; Kouhia Stenberg; comparison; locking; discrete Korn inequality; Stokes equations. 1. Introduction he textbook apriorierror analysis of nonconforming finite element methods (FEMs) considers an inconsistency term with the normal derivative of the exact solution along edges and so requires H 3/2+ε regularity of the exact solution for some positive ε. his regularity request fails to hold for certain mixed boundary value problems in linear elasticity and leaves the impression that nonconforming FEMs may be more sensitive for near singularities than conforming FEM (Braess, 2007, p.111 and the web supplement). he medius analysis of Gudi (2010) and Carstensen et al. (2012b) does not rely on elliptic regularity at all and proves quasi-optimality for the linear elastic model problem of this paper in the sense that the total error is dominated by the approximation error. he medius analysis extends to nonconstant coefficients λ and μ and higher space dimensions, while the more involved precise analysis of c he authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 2of31 C. CARSENSEN AND M. SCHEDENSACK singular functions in the case of nonconvex polygons appears to be limited to the simple linear elastic model problem at hand. For a polygonal, bounded Lipschitz domain R 2 with closed Dirichlet boundary Γ D of positive length and (relatively open) Neumann boundary Γ N := \ Γ D with outer unit normal ν, the strong formulation of the Navier Lamé equations for volume forces f L 2 (; R 2 ) and applied tractions g L 2 (Γ N ; R 2 ) and homogeneous boundary conditions reads (in compact notation) div Cε(u) = f in, u = 0 on Γ D, Cε(u)ν = g on Γ N. he fourth-order elasticity tensor acts as CA := 2μA + λ tr(a)1 2 2 for positive Lamé parameters μ and λ, and for any general input variable A R 2 2, and the linear Green strain is ε(u) := (Du + Du )/2. he conforming first-order FEM of Fig. 1(a) (named after Courant (CFEM)) converges, but suffers from locking in the incompressible limit as λ (Falk, 2006; Braess, 2007; Brenner & Scott, 2008). his means for large values of λ that the L 2 error of the stresses shows the expected convergence rate for a very large number of degrees of freedom only. o overcome this phenomenon, finite element spaces should have good approximation properties for nearly incompressible materials. One possibility is the choice of a higher polynomial degree of the ansatz space ( 4) or the use of mixed methods. However, the lowest-order conforming mixed method of Arnold & Winther (2002) still has 30 degrees of freedom per triangle. Alternative approaches are the first-order nonconforming methods of Crouzeix and Raviart (Brenner & Sung, 1992) or ofkouhia & Stenberg (1995), which do not show such a locking phenomenon and are therefore of great interest. his paper enfolds a medius error analysis for the nonconforming FEM of Kouhia and Stenberg (KS-NCFEM) of Fig. 1(b) in the sense that mathematical arguments from an a posteriori error analysis lead to a priori error estimates. he notion of medius analysis was introduced in Gudi (2010) and leads to results which rely on no extra regularity of the weak solution and hold for arbitrarily coarse meshes with certain minimal conditions (a) (d) of Section 2.3. In this respect, the error analysis of this paper is a refinement of the error analysis in Kouhia & Stenberg (1995). he main result of this analysis is the best-approximation property of the discrete stress σ KS := Cε NC (u KS ) (ε NC and D NC are the piecewise analogues of ε and D, respectively) with respect to the exact stress σ := Cε(u) for the exact and discrete solutions u H 1 (; R 2 ) and u KS KS( ); i.e., σ σ KS L 2 () min σ Cε NC(v KS ) L v KS KS( ) 2 () + osc(f 2, ) + osc(g 2, E(Γ N )). he definitions of the data oscillations osc(f 2, ) and osc(g 2, E(Γ N )) and the precise definition of the Kouhia Stenberg FEM space KS( ) follow in Section 2. he notation A B abbreviates an inequality A CB with some mesh-size- and λ-independent generic constant 0 < C <. he constant may (a) (b) (c) Fig. 1. hree first-order FEMs for linear elasticity: (a) CFEM; (b) KS-NCFEM; (c) CR-NCFEM.

3 MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 3of31 depend on the constant α>0 in conditions (a) (d) of Section 2.3 and on μ. Since the multiplicative constant (hidden behind ) does not depend on λ, the aforementioned error estimate also holds in the incompressible limit λ. In other words, quasi-optimal convergence follows for the KS-NCFEM in the Stokes problem as well. he proof relies on a new discrete Helmholtz decomposition (heorem 3.1), a new discrete-pluscontinuous Korn inequality (heorem 4.1) and the conforming cubic companion of the nonconforming discrete solution from Lemma 3.3. his conforming companion J 3 v CR fulfils for all Crouzeix Raviart functions v CR the integral mean properties (v CR J 3 v CR ) dx = 0 and D NC (v CR J 3 v CR ) dx = 0 for all, and some stability and approximation properties. he nonconforming FEM of Crouzeix and Raviart (CR-NCFEM) (Brenner & Sung, 1992) of Fig. 1(c) only allows a discretization of the pure Dirichlet problem Γ D =, in which the (nonphysical) stress σ := CDu := μdu + (λ + μ) div(u)1 2 2 appears with its approximation σ CR := CD NC u CR in the Crouzeix Raviart FEM. he best-approximation result of this paper reads σ Cε NC (u CR ) L 2 () σ σ CR L 2 () σ Π 0 σ L 2 () + osc(f, ). Recent comparison results (Braess, 2009; Carstensen et al., 2012b) lead to equivalences of approximation classes for the Poisson model problem. he best-approximation results and further analysis of this paper lead to comparison results between the three considered FEMs of Fig. 1 with explicit dependence on the Lamé parameter λ in the equivalence constants. For the conforming discrete solution u C and the discrete stress σ C := Cε(u C ), the comparison between KS-NCFEM and CFEM reads λ 1 σ σ C L 2 () σ σ KS L 2 () σ σ C L 2 () + osc(f 2, ) + osc(g 2, E(Γ N )). (1.1) A detailed investigation of the gap in the dependence on λ, which is in fact sharp, is included in Section 6. For the pure Dirichlet problem Γ D =, the solutions of CR-NCFEM and KS-NCFEM (with σ KS := CD NC u KS ) exist and can be compared by and σ σ CR L 2 () σ σ KS L 2 () + osc(f, ) σ σ KS L 2 () σ σ CR L 2 () + osc(f 2, ). he paper focuses on the two-dimensional case; the generalization to higher dimensions is straightforward for CR-NCFEM and CFEM. he generalization of KS-NCFEM to three dimensions applies two nonconforming and one conforming FEM to the three components or two conforming and one nonconforming; the mathematical justification will be established in the near future (Hu & Schedensack). Within the scope of low-order methods, despite the equivalence results of this paper, the explicit dependence on the Lamé parameter λ strongly suggests the use of nonconforming discretizations for nearly incompressible materials. If Neumann boundary conditions are present, this advertises the use of KS-NCFEM which, therefore, is apparently far too underrated in the engineering community despite striking numerical examples in Kouhia & Stenberg (1995) and Carstensen & Funken (2001a). It may appear strange to employ some scheme which depends on the choice of the coordinate system, but (in the

4 4of31 C. CARSENSEN AND M. SCHEDENSACK presence of Neumann boundary conditions) the KS-NCFEM is the only known robustly quasi-optimal first-order scheme. he outline of this paper is as follows. Section 2 introduces the precise notation and states the main results, which imply the error estimates of this introduction. Section 3 presents some preliminary results which include the definition of the conforming companion and the new discrete Helmholtz decomposition. Sections 4 and 5 prove the main results including the new discrete-plus-continuous Korn inequality. Section 6 concludes the paper with numerical illustrations and provides striking numerical evidence for the equivalence of the three first-order methods and the claimed dependence on the equivalence constant as λ. hroughout this paper, standard notation on Lebesgue and Sobolev spaces and their norms is employed and further notation can be found in the following table for convenient reading. A B A CB with a mesh-size-independent constant C v k, v(k) he kth component of v R 2 A(k, j) he component kj of A R 2 2 A(k) he kth row (A(k,1), A(k,2)) of A R 2 2 a b = 2 j=1 a(j)b(j) for a, b R2 A : B = j,k=1,2 A(j, k)b(j, k) for A, B R Unit matrix in R 2 2 S Set of symmetric matrices; S :={A R 2 2 A = A } ε(u) Green strain (Du + (Du) )/2 C Elasticity tensor; CA = 2μA + λ tr(a)1 2 2 for A R 2 2 C Modified elasticity tensor; CA = μa + (μ + λ) tr(a)1 2 2 for A R 2 2 C D () (respectively, C N ()) Space of continuous functions with homogeneous boundary conditions on Γ D (respectively, Γ N ) V V :={v H 1 (; R 2 ) v ΓD = 0} Dv, w, divv Derivative (of a vector-valued function v V), gradient of a scalar-valued function w H 1 (), divergence of v Curl Curl v = ( v/ x 2, v/ x 1 ) L 2 (; R 2 ) for v H 1 (), Curl w = (Curl w(1);curlw(2)) L 2 (; R 2 2 ) for w V, N, E Shape-regular triangulation with the set of vertices N and set of edges E; Section 2.2 N (ω) Set of vertices in ω, N (ω) := N ω E(ω) ={E E E ω, E ω}, E Area of a triangle, length of an edge E mid(e) Midpoint of an edge E E P k ( ; R m ) Set of piecewise polynomials of degree k; Section 2.2 Π 0 Π 0 : L 2 (; R m ) P 0 ( ; R m ), L 2 projection on piecewise constants; Section 2.2 Π E L 2 projection onto E-piecewise constants; Section 2.2 h Piecewise constant mesh size, h := diam() for all [v] E Jump along an edge E; Section 2.2 osc(f, ), osc(f, ) Oscillations of f,osc(f, ) := h (f Π 0 f ) L 2 (), osc(f, ) := h (f Π 0 f ) L 2 () osc(g, E(Γ N )) Edge oscillations; Section 2.2

5 MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 5of31 D NC, NC,div NC,Curl NC Piecewise versions of D,, div,curl V C ( ) Conforming finite element space; Section 2.3 CR 1 D ( ) Nonconforming Crouzeix Raviart space; Section 2.3 V CR ( ) V CR ( ) := CR 1 D ( ) CR1 D ( ) KS( ) Finite element space of KS-NCFEM; KS( ) = (P 1 ( ) C D ()) CR 1 D ( ); Section 2.3 KS ( ) KS ( ) = CR 1 N ( ) (P 1( ) C N ()); Section 3 I NC : V V CR ( ) Nonconforming interpolation operator with (I NC v)(mid(e)) = v ds for all E E \ E(Γ E D) (, ) C 1 (σ, τ) C 1 := σ : C 1 τ dx for σ, τ L 2 (; S) 2. Notation and main results his section defines the linear elastic model problem and all the considered FEMs, and states the main results. 2.1 Linear elasticity Recall that the elastic body occupies the bounded Lipschitz domain with boundary = Γ D Γ N. We assume that Γ D consists of finitely many parts which lie on the outer boundary of (on the unbounded connectivity component of R 2 \ ). he weak formulation based on the Green strain seeks u V :={v H 1 (; R 2 ) v ΓD = 0} such that a(v, u) := ε(v) : Cε(u) dx = f v dx + g v ds for all v V. (2.1) Γ N For the pure Dirichlet problem, i.e., Γ D =, an integration by parts and the commutation of the derivatives for C0 (; R2 ) functions show that ε( ) : Cε( ) dx = D : CD dx on V C0 (; R2 ). he denseness of C0 (; R2 ) in V implies that the two bilinear forms are identical on V V. hus, for the pure Dirichlet problem, the equivalent weak formulation based on the full gradient seeks u H0 1(; R2 ) with Dv : CDu dx = f v dx for all v H0 1 (; R2 ). (2.2) Define the energy norm := a(, ) in V and the scalar product (σ, τ) C 1 := σ : C 1 τ dx for all σ, τ L 2 (; S). 2.2 riangulations Let denote some shape-regular triangulation of a polygonal, bounded Lipschitz domain into triangles, i.e., = and any two distinct triangles are either disjoint or share exactly one common

6 6of31 C. CARSENSEN AND M. SCHEDENSACK edge or one vertex. Let E denote the set of edges of and N denote the set of vertices. Define for ω R 2 the sets N (ω) := N ω and E(ω) :={E E int(e) ω} for the relative interior int(e) of an edge E E. We assume that the boundary edges E( ) match the boundary conditions in the sense that the boundary conditions change only at nodes Γ D Γ N N.Let P k (; R m ) :={v k : R m j = 1,..., m, the component v k (j) of v k is a polynomial of total degree k}, P k ( ; R m ) :={v k : R m, v k P k (; R m )} denote the set of polynomials and the set of piecewise polynomials, respectively; Π 0 : L 2 (; R m ) P 0 ( ; R m ) denotes the L 2 projection onto -piecewise constant functions or vectors, i.e., (Π 0 f ) = f dx := f dx/ for all with area and all f L2 (; R m ). he operator Π E denotes the L 2 projection onto E-piecewise constant functions or vectors, i.e., Π E g E = g ds := E E g ds/ E for all edges E E of length E. he volume oscillations read osc(f, ) := h (f Π 0 f ) L 2 () and osc(f, ) := h (f Π 0 f ) L 2 (), while the edge oscillations read osc(g, E(Γ N )) := E E(Γ N ) h 1/2 (g Π Eg) 2 L 2 (E) with piecewise constant mesh size h P 0 ( ) with h := diam() for all. he jump along an interior edge E E() with adjacent triangles + and, i.e., E = +, is defined by [v] E := v + v. Given the homogeneous Dirichlet boundary conditions, the jump along boundary edges E E(Γ D ) reads [v] E := v + for the triangle + with E +. For piecewise affine functions v NC P 1 ( ; R 2 ) the -piecewise gradient D NC v NC with (D NC v NC ) = D(v NC ) for all and, accordingly, ε NC (v NC ) and div NC (v NC ), exists and D NC v NC P 0 ( ; R 2 2 ) and ε NC (v NC ) P 0 ( ; S) and div NC (v NC ) P 0 ( ). 2.3 Discrete spaces CFEM. he Courant finite element space reads V C ( ) := (P 1 ( ) C D ()) (P 1 ( ) C D ()). he corresponding (unique) Galerkin approximation u C V C ( ) satisfies ε(v C ) : Cε(u C ) dx = f v C dx + g v C ds for all v C V C ( ). Γ N CR-NCFEM. Define the P 1 nonconforming space as CR 1 ( ) :={v CR P 1 ( ) v CR is continuous at midpoints of interior edges}. he nonconforming Crouzeix Raviart space reads CR 1 D ( ) :={v CR CR 1 ( ) v CR vanishes at midpoints of edges E E(Γ D )}.

7 MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 7of31 For the discretization of the pure Dirichlet problem Γ D = of linear elasticity, define the space V CR ( ) := CR 1 D ( ) CR1 D ( ). Since the kernel of ε NC : V CR ( ) P 0 ( ; R 2 2 ) is in general not trivial, the weak formulation based on the full gradient is in use for the discretization and seeks u CR V CR ( ) with D NC v CR : CD NC u CR dx = f v CR dx for all v CR V CR ( ). (2.3) Here, the piecewise gradient D NC replaces the weak differential operator. KS-NCFEM. he Kouhia Stenberg approximation u KS KS( ) := (P 1 ( ) C D ()) CR 1 D ( ) satisfies ε NC (v KS ) : Cε NC (u KS ) dx = f v KS dx + g v KS ds for all v KS KS( ). (2.4) Γ N he following conditions (a) (d) are given for completeness and replace the assumptions on the sufficiently small mesh size of and the assumptions (AD) and (AN) of Kouhia & Stenberg (1995). hese conditions are, for example, fulfilled if at least one vertex of each triangle lies in the interior of the domain. he existence of α>0 which fulfils conditions (a) (d) ensures that the inf sup condition and a discrete Korn inequality hold. (a) For all E E() with N (E) N ( ), it holds that ν E (2) α. (b) If Γ N =, it holds that N Γ N = or there exists at least one E E(Γ N ) with ν E (2) α. (c) In the case that the two vertices of an interior edge E E() belong to the boundary, i.e., space N (E) N ( ), and ν E (1) <α, consider z N (E). For the nodal patch ω z := int( { z }) let ω 1, ω 2 ω z denote the two connected sets, which decompose the nodal patch in the upper and lower part (i.e., ω 1 ω 2 = E and ω 1 ω 2 int(e) = ω z ). hen there exist edges E k E( ω k ) E(Γ D ) for k = 1, 2 with ν Ek (1) >α(see Fig. 2(c)). (d) If the entire Dirichlet boundary is nearly horizontal, i.e., for all E E(Γ D ) it holds that ν E (1) < α, then there exist two adjacent edges on the Dirichlet boundary, i.e., there exist E, F E(Γ D ) with E = F and E F =. he generic multiplicative constants hidden in the notation are allowed to depend on α. Figure 2 illustrates conditions (a) (d). 2.4 Main results he main results below imply the statements of the introduction in Section 1. heorem 2.1 (Best approximation of KS-NCFEM) he exact and the discrete stress σ = Cε(u) and σ KS = Cε NC (u KS ) for the exact and discrete solutions u V and u KS KS( ), respectively, satisfy σ σ KS L 2 () min σ Cε NC(v KS ) L v KS KS( ) 2 () + osc(f 2, ) + osc(g 2, E(Γ N )).

8 8of31 C. CARSENSEN AND M. SCHEDENSACK (a) (b) (c) (d) Fig. 2. Illustrations of conditions (a) (d): (a) a situation excluded by condition (a); (b) triangulation which is excluded by condition (b); (c) a possible patch, which fulfils condition (c); (d) a situation excluded by condition (d). Remark 2.2 As one key ingredient for the proof of heorem 2.1, the error estimate of heorem 4.4 estimates the stress error of KS-NCFEM by some best-approximation error of the stress and the derivative in the piecewise constant functions plus some data approximation terms. heorem 2.3 (Best approximation of CR-NCFEM) Let σ := CDu for the exact solution u H0 1(; R2 ). For the pure Dirichlet problem Γ D =, the discrete stress σ CR := CD NC u CR for the discrete solution u CR V CR ( ) of (2.3) satisfies σ σ CR L 2 () σ Π 0 σ L 2 () + osc(f, ). heorem 2.4 (Comparison of CFEM, CR-NCFEM and KS-NCFEM) he exact stress σ = Cε(u) and the discrete stresses σ C = Cε(u C ) and σ KS = Cε NC (u KS ) satisfy λ 1 σ σ C L 2 () σ σ KS L 2 () σ σ C L 2 () + osc(f 2, ) + osc(g 2, E(Γ N )). For the pure Dirichlet problem Γ D =, the discrete stress σ CR = Cε NC (u CR ) satisfies σ σ CR L 2 () σ σ KS L 2 () + osc(f, ). In addition, the stress error of KS-NCFEM is comparable with the error of the nonsymmetric approximation σ := CDu from (2.2) through σ σ KS L 2 () σ σ CR L 2 () + osc(f 2, ).

9 MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 9of31 3. Preliminary results he following discrete Helmholtz decomposition and some properties of a conforming companion are required below; cf. Falk & Morley (1990) for a first decomposition of this type. o this end, define CR ( ) :={v CR CR( ) v CR (mid(e 1 )) = v CR (mid(e 2 )) for E 1, E 2 edges of the same connectivity component of Γ N }, V C ( ) :={v C P 1 ( ) H 1 () v C is constant along each connectivity component of Γ N }. Recall that the boundary conditions match the triangulation of the possibly multiply connected planar Lipschitz domain with Γ D Γ 0 for the outer boundary Γ 0 of (Γ 0 is defined as the boundary of the unbounded connectivity component of R 2 \ ). heorem 3.1 (Discrete Helmholtz decomposition) Let KS ( ) := CR ( ) VC ( ). hen it holds that P 0 ( ; S) = Cε NC (KS( )) (Curl NC (KS ( )) P 0 ( ; S)) and the sum is orthogonal with respect to the scalar product (, ) C 1 := : C 1 dx. Remark 3.2 For any v KS KS ( ) the assertion Curl NC v KS P 0( ; S) is equivalent to div NC v KS = 0. Proof of heorem 3.1. For α KS KS( ) and β KS KS ( ) with Curl NC (β KS ) P 0 ( ; S) let α C P 1 ( ) C D (), α CR CR 1 D ( ), β CR CR ( ) and β C VC ( ) with α KS = (α C, α CR ) and β KS = (β CR, β C ). hen α KS and β KS satisfy (Cε(α KS ),Curl NC β KS ) C 1 = ε NC (α KS ) :Curl NC β KS dx his and the L 2 orthogonalities = = D NC (α KS ) :Curl NC β KS dx ( α C Curl NC β CR + NC α CR Curl β C ) dx. (P 1 ( ) C D ()) L 2 Curl NC (CR ( )), NC CR 1 D ( ) L 2 Curl(V C ( )) (3.1) imply the orthogonality (with respect to the scalar product (, ) C 1) Given σ h P 0 ( ; S), letα KS KS( ) solve ε NC (v KS ) : Cε NC (α KS ) dx = Cε NC (KS( )) (Curl NC (KS ( )) P 0 ( ; S)). ε NC (v KS ) : σ h dx for all v KS KS( ). he jth row τ h (j) := (τ h (j,1), τ h (j,2)) P 0 ( ; R 2 ) of τ h := σ h Cε NC (α KS ) P 0 ( ; S) is piecewise constant for j = 1, 2. he discrete Helmholtz decomposition for Crouzeix Raviart and conforming P 1

10 10 of 31 C. CARSENSEN AND M. SCHEDENSACK functions (Arnold & Falk, 1989) remains true for mixed boundary conditions and interchanged discrete spaces as P 0 ( ; R 2 ) = NC CR 1 D ( ) Curl V C ( ), P 0 ( ; R 2 ) = (P 1 ( ) C D ()) Curl NC CR ( ). (his can be proved, for example, by the orthogonalities (3.1) and a dimension argument.) his guarantees the existence of p C P 1 ( ) C D (), p CR CR ( ), q CR CR 1 D ( ) and q C VC ( ) with τ h (1) = p C + Curl NC p CR and τ h (2) = NC q CR + Curl q C. (Here, p C,Curl NC p CR, NC q CR and Curl q C are understood as row vectors.) Since τ h is orthogonal to Cε NC (KS( )) with respect to (, ) C 1 and since τ h P 0 ( ; S), the functions v C P 1 ( ) C D () and v CR CR 1 D ( ) satisfy ( v C ; NC v CR ) : (τ h (1); τ h (2)) dx = ε NC (v C, v CR ) : τ h dx = 0. his implies the L 2 orthogonalities his and the orthogonalities (3.1) lead to τ h (1) L 2 (P 1 ( ) C D ()) and τ h (2) L 2 NC CR 1 D ( ). p C 2 L 2 () = Analogous arguments prove NC q CR 2 L 2 () = 0. Hence, p C ( p C τ h (1)) dx = 0. τ h = Curl NC (p CR, q C ) Curl NC (KS ( )). Lemma 3.3 here exists an operator J 3 :CR 1 D ( ) (P 3( ) C D ()) with the conservation properties (v CR J 3 v CR ) dx = 0 for all, (3.2a) and the approximation and stability properties E (v CR J 3 v CR ) dx = 0 for all E E, (3.2b) h 1 (v CR J 3 v CR ) L 2 () NC (v CR J 3 v CR ) L 2 () min ϕ H 1 () C D () NC (v CR ϕ) L 2 () NC v CR L 2 (). (3.3)

11 MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 11 of 31 Remark 3.4 he conservation property along edges (3.2b) and an integration by parts reveal the conservation property of the gradients Π 0 J 3 = NC in the sense that J 3 v CR dx = NC v CR dx for all and all v CR CR 1 D ( ). Proof of Lemma 3.3. he design is based on three successive steps. Step 1. he operator J 1 :CR 1 D ( ) P 1( ) C D () acts on any function v CR CR 1 D ( ) by averaging the function values at each node z N ( Γ N ): J 1 v CR (z) = (z) 1 (z) v CR (z) for all z N ( Γ N ) (3.4) with (z) :={ z }. (his operator is also known as the enriching operator in the context of fast solvers; see Brenner, 1996.) he arguments of Carstensen et al. (2012a, heorem 5.1) prove the approximation property h 1 (v CR J 1 v CR ) L 2 () E E( Γ D ) E E( Γ D ) min v H 1 () C D () his and an inverse estimate imply the stability property NC (v CR J 1 v CR ) L 2 () min v H 1 () C D () E 1 [v CR ] E 2 L 2 (E) E [ NC v CR τ E ] E 2 L 2 (E) NC (v CR v) L 2 (). (3.5) NC (v CR v) L 2 (). (3.6) Step 2. Given any edge E = conv{a, b} E( Γ N ) with nodal P 1 conforming basis functions ϕ a, ϕ b P 1 ( ) C() (defined by ϕ a (a) = ϕ b (b) = 1,ϕ a (z) = 0forz N \{a} and ϕ b (z) = 0forz N \{b}), the quadratic edge-bubble function E := 6 ϕ a ϕ b has the support ω E and satisfies E E ds = 1. For any function v CR CR 1 D ( ) the operator J 2 : CR 1 D ( ) P 2( ) C D () acts as J 2 v CR := J 1 v CR + An immediate consequence of this choice is E E E( Γ N ) ( ) (v CR J 1 v CR ) ds E. E J 2 v CR ds = v CR ds for all E E. E

12 12 of 31 C. CARSENSEN AND M. SCHEDENSACK An integration by parts shows, for the vertex P E N () \ E opposite to E E() in the triangle, the trace identity (v CR J 1 v CR ) ds = CR J 1 v CR ) dx + E (v 1 (x P E ) NC (v CR J 1 v CR ) dx. 2 he scaling E L 2 () h shows ( ) (v CR J 1 v CR ) ds E h 1 E E() his together with (3.5) and (3.6) yield h 1 E L 2 () (v CR J 2 v CR ) L 2 () (v CR J 1 v CR ) ds E E() E h 1 v CR J 1 v CR L 2 () + NC (v CR J 1 v CR ) L 2 (). min v H 1 () C D () he stability property of J 2 follows with an inverse estimate: NC (v CR J 2 v CR ) L 2 () h 1 (v CR J 2 v CR ) L 2 () min v H 1 () C D () NC (v CR v) L 2 (). (3.7) NC (v CR v) L 2 (). (3.8) Step 3. On any triangle = conv{a, b, c} with nodal basis functions ϕ a, ϕ b, ϕ c, the cubic volume bubble function reads := 60ϕ a ϕ b ϕ c H 1 0 () and enjoys the scaling L 2 () 1. Define J 3 v CR := J 2 v CR + ( ) (v CR J 2 v CR ) dx. hen J 3 fulfils the conservation properties (3.2) and ( ) 2 (v CR J 2 v CR ) dx ( ) 2 (v CR J 2 v CR ) dx L 2 () h 1 (v CR J 2 v CR ) 2 L 2 (). his together with (3.7) and (3.8) imply ( ) D NC (v CR J 3 v CR ) L 2 () D NC (v CR J 2 v CR ) L 2 () + (v CR J 2 v CR ) dx L 2 () min v H 1 () C D () NC (v CR v) L 2 (). his and a Poincaré inequality lead to (3.3).

13 MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 13 of 31 Lemma 3.5 Any v KS KS( ) satisfies (Π 0 σ σ KS ) : ε NC (v KS ) dx ( (σ Π 0 σ)(2) L 2 () + osc(f 2, ) + osc(g 2, E(Γ N ))) NC v KS (2) L 2 (). Proof. Given any v KS KS( ),letv C P 1 ( ) C D () and v CR CR 1 D ( ) be the components of v KS, i.e., v KS = (v C, v CR ). Lemma 3.3 guarantees the existence of J 3 v CR P 3 ( ) H 1 () C D () with (v CR J 3 v CR ) dx = 0 = NC (v CR J 3 v CR ) dx for all and NC (v CR J 3 v CR ) L 2 () NC v CR L 2 (). (3.9) Since Π 0 σ is piecewise constant, the integral mean property (3.9) implies Π 0 σ : ε NC (0, v CR ) dx = Π 0 σ : ε NC (0, J 3 v CR ) dx = (Π 0 σ σ): D(0, J 3 v CR ) dx + σ : ε(0, J 3 v CR ) dx. Since σ is the stress of the exact solution and J 3 v CR H 1 () C D (), the Cauchy Schwarz inequality implies (Π 0 σ σ): D(0, J 3 v CR ) dx + σ : ε(0, J 3 v CR ) dx (σ Π 0 σ)(2) L 2 () J 3 v CR L 2 () + f 2 J 3 v CR dx + g 2 J 3 v CR ds. Γ N he triangle inequality and the stability property (3.9) show J 3 v CR L 2 () J 3 v CR NC v CR L 2 () + NC v CR L 2 () NC v CR L 2 (). he combination of the above inequalities yields Π 0 σ : ε NC (0, v CR ) dx (σ Π 0 σ)(2) L 2 () NC v CR L 2 () + f 2 J 3 v CR dx + g 2 J 3 v CR ds. Γ N Since σ and σ KS are the stresses of the exact and the discrete solution, respectively, it follows that (Π 0 σ σ KS ) : ε NC (v KS ) dx = (Π 0 σ σ KS ) : ε NC (0, v CR ) dx = Π 0 σ : ε NC (0, v CR ) dx f 2 v CR dx g 2 v CR ds. Γ N

14 14 of 31 C. CARSENSEN AND M. SCHEDENSACK he combination of the previous displayed formulas proves (Π 0 σ σ KS ) : ε NC (v KS ) dx (σ Π 0 σ)(2) L 2 () NC v CR L 2 () + f 2 (J 3 v CR v CR ) dx + g 2 (J 3 v CR v CR ) ds. Γ N (3.10) Since the integral mean of J 3 v CR v CR vanishes on triangles, the trace inequality (Brenner & Scott, 2008, p.282) followed by a Poincaré inequality yields, for E E(Γ N ) and with E E(), h 1/2 (J 3 v CR v CR ) L 2 (E) h 1 (J 3v CR v CR ) L 2 () + NC (J 3 v CR v CR ) L 2 () NC (J 3 v CR v CR ) L 2 (). Since the integral mean of J 3 v CR v CR vanishes on edges, this leads to E g 2 (J 3 v CR v CR ) ds h 1/2 (g 2 Π E g 2 ) L 2 (E) NC (J 3 v CR v CR ) L 2 (). Since the integral mean of J 3 v CR v CR vanishes on triangles, (3.10) implies (Π 0 σ σ KS ) : ε NC (v KS ) dx ( (σ Π 0 σ)(2) L 2 () + osc(f 2, ) + osc(g 2, E(Γ N ))) NC v CR L 2 (). his concludes the proof. he nonconforming interpolation operator I NC : V V CR ( ) is defined by (I NC v)(mid(e)) = E v ds for all E E \ E(Γ D ) and fulfils the integral mean property D NC I NC = Π 0 D in the sense that D NC I NC v = Dv dx for all and all v V. (3.11)

15 MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 15 of 31 Lemma 3.6 Any β KS KS ( ) with Curl NC β KS P 0 ( ; S) satisfies ε NC (I NC u u KS ) :Curl NC β KS dx (Du Π 0 Du)(1) L 2 () Curl NC β KS (1) L 2 (). Proof. According to the definition, β KS (1) CR ( ) and β KS (2) VC ( ). he orthogonalities (3.1) and Curl NC β KS S show, for any φ C P 1 ( ) C D (), that ε NC (I NC u u KS ) :Curl NC β KS dx = ( NC I NC u(1) φ C ) Curl NC β KS (1) dx. Since φ C P 1 ( ) C D () is arbitrary, this implies ε NC (I NC u u KS ) :Curl NC β KS dx min NCI NC u(1) φ C L φ C P 1 ( ) C D () 2 () Curl NC β KS (1) L 2 (). (3.12) he integral mean property (3.11) ofi NC and Carstensen et al. (2012a, heorem 5.1) show min φ C P 1 ( ) C D () NCI NC u(1) φ C L 2 () u(1) NC I NC u(1) L 2 () = u(1) Π 0 u(1) L 2 (). (3.13) he combination of (3.12) and (3.13) concludes the proof. 4. Proof of heorem 2.1 he main step in the proof of heorem 2.1 is the error estimate σ σ KS L 2 () σ Π 0 σ L 2 () + (Du Π 0 Du)(1) L 2 () + osc(f 2, ) + osc(g 2, E(Γ N )) from heorem 4.4. he discrete-plus-continuous Korn inequality from heorem 4.1 allows control of the nonsymmetric term Du Π 0 Du L 2 () in terms of the symmetric stress error σ Cε NC (v KS ) L 2 (). his proves heorem 2.1. he remaining parts of this section prove first heorem 4.1 and then heorem 4.4. heorem 4.1 generalizes the discrete Korn inequality from Kouhia & Stenberg (1995) in that the underlying function space is V + KS( ) and not just KS( ). hen Carstensen & Funken (2001b, Remark 4.1.v) gives the general warning that the Korn inequality in the form of heorem 4.1 is only stated but not proved completely in Bao & Barrett (1998). heorem 4.1 (Discrete-plus-continuous Korn inequality) For a triangulation which fulfils conditions (c),(d) of Section 2.3, any v NC V + KS( ) satisfies D NC v NC L 2 () ε NC (v NC ) L 2 (). Remark 4.2 he discrete-plus-continuous Korn inequality could be proved for slightly weaker conditions than conditions (c),(d) from Section 2.3 as in the situation of Fig. 2(d). In those situations, the proof of heorem 4.1 considers some larger neighbourhoods of the patches. In the situation of Fig. 2(d), it is not guaranteed that those patches do not become arbitrarily large under some refinement strategies

16 16 of 31 C. CARSENSEN AND M. SCHEDENSACK and so the constant from the discrete Korn inequality is not uniformly bounded. For the ease of this presentation and the sake of clarity, the slightly stronger versions (c),(d) are assumed. he proof of heorem 4.1 considers a set of vertices Z N defined by z Z if and only if at least one of the following conditions (i) (iii) is fulfilled with α>0 from conditions (c), (d) of Section 2.3: (i) he node z N () is an interior node. (ii) he node z N ( ) is a boundary node with ν E (1) >α for all E E(ω z ) for the nodal patch ω z := int( { z }), and if {E E(Γ D ω z ) ν E (1) <α} = 1, then {E E(Γ D ω z ) ν E (1) >α} > 0. (iii) he node z N ( ) is a boundary node and there exists an edge E E(ω z ) with N (E) N ( ) and ν E (1) <α, which decomposes the patch ω z in the two domains ω 1, ω 2 (i.e., ω 1, ω 2 connected with ω 1 ω 2 = E and ω 1 ω 2 int(e) = ω z ). For each of the two domains ω 1 and ω 2 there exist E 1 E( ω 1 ) E(Γ D ) and E 2 E( ω 2 ) E(Γ D ) on the Dirichlet boundary with ν E1 (1) >αand ν E2 (1) >α, as depicted in Fig. 2(c). Recall that the generic multiplicative constants hidden in the notation may depend on α. he set Z contains all interior nodes and some nodes on the boundary, for which some local discrete Korn inequality holds on the nodal patches. he proof of heorem 4.1 uses the fact that, under conditions (c), (d) of Section 2.3, the set Z is large enough to prove the theorem even if that set is empty and the mesh is very coarse (without any interior nodes). he first step of this proof is the subsequent lemma. Lemma 4.3 (Characterization of rigid body motions) Let be a triangulation which fulfils conditions (c),(d) of Section 2.3 and define Z as above. hen any v KS KS( ) with ε NC (v KS ωz ) = 0on the nodal patch ω z for z Z is continuous on ω z. For E E() with ν E (1) α, any v KS KS( ) with ε NC (v KS ωe ) = 0 on the edge patch ω E := int( { E E()}) is continuous on ω E. Proof. he critical situation concerns horizontal edges as depicted in Fig. 3(a). For interior nodes the rigid body motions are fixed through two midpoints of those horizontal edges (see Fig. 3(c)). For nodes on the boundary, condition (iii) guarantees that the rigid body motions are fixed by the boundary conditions. In the case of the edge patches, such critical situations are excluded. Proof of heorem 4.1. Define Γ D := {E E(Γ D ) ν E (1) >α or F E(Γ D ) with F = E and F E = }. he point of departure is the discrete Korn inequality for piecewise H 1 functions (Brenner, 2004, Equation (1.19)), D NC v NC L 2 () ε NC (v NC ) L 2 () + v NC L 2 ( Γ D ) + E E() E 1 [v NC ] E 2 L 2 (E). For any vertex z Z set (z) :={ z }, the set of all triangles with vertex z, and define E z := {E E( Γ D ) z E and if E E( Γ D ) and ν E (1) <α, then E( Γ D ) > 1} and let ω z := int( (z)) be the nodal patch. On KS( (z)) :={v KS P 1 ( (z), R 2 ) w KS KS( ) s.t. v KS = w KS ωz } the maps ρ 1 (v KS ) := E 1 [v KS ] E 2 L 2 (E) E E z (4.1) and ρ 2 (v KS ) := inf v V(ω z ) ε NC(v KS v) L 2 (ω z )

17 MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 17 of 31 (a) (b) (c) Fig. 3. Illustration of critical situations for Lemma 4.3: (a) an excluded infinitesimal rigid body motion; (b) situation of Fig. 3(a) embedded in a triangulation, excluded by condition (c); (c) interior patch. define two seminorms, where V(ω z ) :={w L 2 (ω z ; R 2 ) v V with w = v ωz }. he triangle inequality implies that infimizing sequences v n V(ω z ) in (4.1) are bounded in H 1 (ω z ; R 2 ). Since V(ω z ) is a closed subspace of the reflexive space H 1 (ω z ; R 2 ), there exist a subsequence v nk and a function v V(ω z ) with v nk v. his and the weak lower semicontinuity of the norm ε( ) L 2 (ω z ) on V(ω z ) imply that the infimum is in fact a minimum. If ρ 2 (v KS ) = 0forv KS KS( (z)), then there exists some v V(ω z ) with ε NC (v KS ) = ε(v). herefore, w KS := v v KS P 1 ( (z); R 2 ) is a piecewise rigid body motion. his implies v P 1 ( (z); R 2 ) C(ω z ; R 2 ) KS( (z)) and therefore w KS KS( (z)). Lemma 4.3 implies that w KS C(ω z ; R 2 ) is continuous. Hence, v KS = v w KS C(ω z ; R 2 ) and v KS E 0forE E z Γ D and therefore ρ 1 (v KS ) = 0. Since ρ 1 and ρ 2 are seminorms on the finite-dimensional space KS( (z)), there exists a constant C( (z)), such that ρ 1 C( (z))ρ 2. A scaling argument shows that the constant C( (z)) is independent of the mesh size and depends on the minimal angle in (z) and on α>0 from conditions (c),(d) of Section 2.3 only. For E E( Γ D ) with ν E (1) α, a similar argument shows the inequality ρ 1 ρ 2 for the two seminorms (of v KS KS( (z))) and ρ 1 (v KS ) := E 1/2 [v KS ] E L 2 (E) ρ 2 (v KS ) := inf ε NC(v KS v) L v V(ω E ) 2 (ω E ). Note that for all E E( Γ D ) with ν E (1) <α, conditions (c),(d) guarantee the existence of a node z Z with E E z. Since the length of edges E E(Γ D ) on the Dirichlet boundary is bounded

18 18 of 31 C. CARSENSEN AND M. SCHEDENSACK ( E 1), the sum over all vertices z Z and the bounded overlap of the patches show v KS L 2 ( Γ D ) + E 1 [v KS ] E 2 L 2 (E) E E() E 1 [v KS ] E 2 L 2 (E) E E( Γ D ) E E( Γ D ) ν E (1) α E 1 [v KS ] E 2 L 2 (E) + E 1 [v KS ] E 2 L 2 (E) z Z E E z inf v V ε NC(v KS v) L 2 (). (4.2) For v NC V + KS( ) and v V and v KS KS( ) with v NC = v + v KS, it holds that [v NC ] E = [v KS ] E and v NC ΓD = v KS ΓD. Inequality (4.2) implies v NC L 2 ( Γ D ) + E 1 [v NC ] E 2 L 2 (E) inf ε NC(v KS w) L w V 2 () E E() he remaining part of this section proves heorem 4.4. heorem 4.4 It holds that ε NC (v NC ) L 2 (). σ σ KS L 2 () σ Π 0 σ L 2 () + (Du Π 0 Du)(1) L 2 () + osc(f 2, ) + osc(g 2, E(Γ N )). Remark 4.5 It remains an open question whether one can neglect the term (Du Π 0 Du)(1) L 2 () in the upper bound in heorem 4.4; it is not clear how to control this term by the stress error. he inf sup condition from heorem 4.6 plays an important role for the independence from λ in the proof of heorem 4.4. heorem 4.6 (Inf sup condition, Kouhia & Stenberg, 1995) Let satisfy conditions (a),(b) from Section 2.3. hen it holds that p 0 L 2 () sup p 0 div NC v KS dx (4.3) v KS KS( )\{0} D NC v KS L 2 () for all p 0 P 0 ( ) if Γ N = and for all p 0 P 0 ( ) with p 0 dx = 0ifΓ N =. Proof. he first paper (Kouhia & Stenberg, 1995) on this nonconforming FEM aims at an asymptotic result for sufficiently fine mesh sizes and therefore reasonably ignores the possibly pathological cases on coarse meshes. Following the arguments of Kouhia & Stenberg (1995, pp ), one can verify that condition (a) is stronger than Kouhia & Stenberg (1995, condition (AD), p. 198) but avoids the

19 MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 19 of 31 modification of the domain necessary in Kouhia & Stenberg (1995, Step 4 of the proof). In fact, the proof in Kouhia & Stenberg (1995) reduces the discrete stability to that on the continuous level but changing the mesh results in changing the domain. One possible criticism is that the change of the continuous inf sup constant with respect to the change of the domain is neglected without a detailed discussion in Kouhia & Stenberg (1995). Conditions (a),(b) of Section 2.3 are sufficient to argue on the original domain in a way analogous to Kouhia & Stenberg (1995, pp ). Since there are no additional ideas in the proof, further details of this technicality are omitted. Proof of heorem 4.4. he triangle inequality implies that it suffices to consider the difference Π 0 σ σ KS L 2 (). hel 2 -orthogonal decomposition in the isochoric and deviatoric part reads Π 0 σ σ KS 2 L 2 () = dev(π 0σ σ KS ) 2 L 2 () tr(π 0σ σ KS ) L 2 (). For Γ N = the homogeneous boundary conditions of u and u KS allow an integration by parts for the second term. he continuity condition E [u KS] E ds = 0 for all E E() leads to tr(π 0 σ)dx = 0 = tr(σ KS ) dx, (4.4) i.e., tr(π 0 σ σ KS ) P 0 ( )/R. heorem 4.6 guarantees, for Γ N = and Γ N =, the existence of v KS KS( ) with D NC v KS L 2 () = 1 and tr(π 0 σ σ KS ) L 2 () = tr(π 0 σ σ KS ) div NC v KS dx (Π 0 σ σ KS ) : D NC v KS dx dev(π 0 σ σ KS ) : D NC v KS dx. he application of Lemma 3.5 to the first term of the right-hand side yields tr(π 0 σ σ KS ) L 2 () (σ Π 0 σ)(2) L 2 () + osc(f 2, ) + osc(g 2, E(Γ N )) + dev(π 0 σ σ KS ) L 2 (). he analysis of dev(π 0 σ σ KS ) L 2 () remains. Algebraic manipulation shows dev CA : dev CA A : CA for all A R 2 2. Applied to the above situation this reads dev(π 0 σ σ KS ) 2 L 2 () (Π 0 σ σ KS ) : ε NC (I NC u u KS ) dx. (4.5) he point is that C dev A does not depend on λ. heorem 3.1 guarantees the existence of α KS KS( ) and β KS KS ( ) with the property that Curl NC β KS P 0 ( ; S) and Π 0 σ σ KS = Cε NC (α KS ) +

20 20 of 31 C. CARSENSEN AND M. SCHEDENSACK Curl NC β KS. Lemmas 3.5 and 3.6 yield (Π 0 σ σ KS ) : ε NC (I NC u u KS ) dx = ε NC (α KS ) : (Π 0 σ σ KS ) dx + Curl NC β KS : ε NC (I NC u u KS ) dx ( (Π 0 σ σ)(2) L 2 () + osc(f 2, ) + osc(g 2, E(Γ N ))) NC α KS (2) L 2 () + (Du Π 0 Du)(1) L 2 () Curl NC β KS (1) L 2 (). (4.6) A similar argument as for the decomposition of Π 0 σ σ KS in the isochoric and the deviatoric part at the beginning of the proof bounds the term Curl NC β KS (1) 2 L 2 () by (Curl NC β KS, C 1 Curl NC β KS ) C 1. For this purpose Curl NC β KS is L 2 -orthogonally decomposed in the isochoric and the deviatoric part, i.e., Curl NC β KS 2 L 2 () = dev(curl NC β KS ) 2 L 2 () tr(curl NC β KS ) L 2 (). (4.7) For Γ N = the function α KS satisfies tr(cε NC (α KS )) dx = (2μ + 2λ) div NC (α KS ) dx = (2μ + 2λ) [α KS ] E ν E ds = 0. E E E With (4.4), it follows that tr(curl NC β KS ) dx = 0. he inf sup condition for Kouhia Stenberg functions, heorem 4.6, guarantees, for Γ N = and Γ N =, the existence of v KS KS( ) with D NC v KS L 2 () = 1 and tr(curl NC β KS ) L 2 () tr(curl NC β KS ) div NC v KS dx. For β KS = (β CR, β C ) with β CR CR 1 N ( ) and β C P 1 ( ) C N () and v KS = (v C, v CR ) with v C P 1 ( ) C D () and v CR CR 1 D ( ),itfollowsthat tr(curl NC β KS ) L 2 () (Curl NC β KS dev Curl NC β KS ) : D NC v KS dx Since v C τ E vanishes on Γ D, an integration by parts leads to Curl NC β CR v C dx = dev Curl NC β L 2 () D NC v KS L 2 () + Curl NC β CR v C dx + Curl β C NC v CR dx. (4.8) E E() E [β CR ] E ds v C τ E + E E(Γ N ) E β CR ds v C τ E = 0.

21 MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 21 of 31 Since β C τ E vanishes on E E(Γ N ), Curl β C NC v CR dx = v CR ds( β C τ E ) + [v CR ] E ds( β C τ E ) = 0. E E(Γ D ) E E E() E ogether with (4.7) and (4.8), it follows that tr(curl NC β KS ) L 2 () + Curl NC β KS L 2 () dev Curl NC β KS L 2 (). Since dev CA : dev CA A : CA for all A R 2 2, it follows, as above, that Curl NC β KS 2 L 2 () (Curl NC β KS,Curl NC β KS ) C 1. heorem 4.1 implies NC α KS (2) 2 L 2 () ε NC(α KS ) 2 L 2 () (Cε NC(α KS ), Cε NC (α KS )) C 1. he orthogonality of the decomposition Π 0 σ σ KS = Cε NC (α KS ) + Curl NC β KS (, ) C 1, together with the above estimate, implies with respect to ( 1/2 NC α KS (2) L 2 () + Curl NC β KS L 2 () (Π 0 σ σ KS ) : ε NC (I NC u u KS ) dx). Inequality (4.6) proves ( 1/2 (Π 0 σ σ KS ) : ε NC (I NC u u KS ) dx) (Π 0 σ σ)(2) L 2 () + osc(f 2, ) + osc(g 2, E(Γ N )) + (Du Π 0 Du)(1) L 2 (). his and (4.5) conclude the proof of heorem Proof of heorems 2.3 and 2.4 he first part of this section proves heorem 2.3, while the second proves heorem 2.4. he proof of heorem 2.3 is based on the following lemma. It corresponds to Lemma 3.5 for Crouzeix Raviart functions. Lemma 5.1 Any v CR V CR ( ) satisfies ( σ σ CR ) : D NC v CR dx ( σ Π 0 σ L 2 () + osc(f, )) D NC v CR L 2 ().

22 22 of 31 C. CARSENSEN AND M. SCHEDENSACK Proof. Lemma 3.3 implies, with a piecewise Poincaré inequality for J 3 (applied componentwise), ( σ σ CR ) : D NC v CR dx = = f (J 3 v CR v CR ) dx + ( C(D NC I NC u) σ CR ) : D NC v CR dx ( C(D NC I NC u) σ): DJ 3 v CR dx h (f Π 0 f ) L 2 () (J 3 v CR v CR )/h L 2 () (5.1) + Π 0 σ σ L 2 () DJ 3 v CR L 2 () osc(f, ) D NC v CR L 2 () + Π 0 σ σ L 2 () D NC v CR L 2 (). Proof of heorem 2.3. he point of departure is an inequality of Carstensen & Rabus (2012, Lemma 3.8), σ σ CR 2 L 2 () ( σ σ CR ) : (Du D NC u CR ) dx + h f 2 L 2 (). (5.2) Define the bubble function b := (ϕ, ϕ ) P 3 ( ; R 2 ) with ϕ as in the proof of Lemma 3.5. he property b dx implies h f L 2 () osc(f, ) + h Π 0 f L ( ) osc(f, ) + b Π 0 f dx. he scaling b L 2 () h and an integration by parts show b Π 0 f dx b L 2 () f Π 0 f L 2 () + b f dx osc(f, ) + Db : (σ Π 0 σ)dx. Since Db L 2 () 1 and (A + A ) : (A + A ) 4A : A, it follows that Db : (σ Π 0 σ)dx σ Π 0σ L 2 () σ Π 0 σ L 2 (). Altogether, h f L 2 () osc(f, ) + σ Π 0 σ L 2 ().

23 his, (5.2) and Lemma 5.1 imply σ σ CR 2 L 2 () MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 23 of 31 ( σ Π 0 σ): (Du Π 0 Du) dx + ( σ σ CR ) : D NC (I NC u u CR ) dx + osc 2 (f, ) + σ Π 0 σ 2 L 2 () σ Π 0 σ 2 L 2 () + osc2 (f, ) + ( σ Π 0 σ L 2 () + osc(f, )) D NC (I NC u u CR ) L 2 (), where the last inequality follows from Du Π 0 Du L 2 () σ Π 0 σ L 2 (). he Young inequality 2 ab α a 2 + α 1 b 2 for α>0 implies ( σ Π 0 σ L 2 () + osc(f, )) D NC (I NC u u CR ) L 2 () 1/(4α)( σ Π 0 σ L 2 () + osc(f, )) 2 + α D NC (I NC u u CR ) 2 L 2 (). For sufficiently small α the last term is absorbed. It follows that σ σ CR L 2 () σ Π 0 σ L 2 () + osc(f, ). he remaining parts of this section are devoted to the proof of heorem 2.4, which is based on the following proposition. Proposition 5.2 For u KS KS( ) and 1 λ it holds that min u KS v C NC λ 1/2 min u KS v NC. v C V C ( ) v V Proof. he arguments of Carstensen et al. (2012a, heorem 5.1) prove the crucial point, namely min D NC(v KS v C ) L v C V C ( ) 2 () min D NC(v KS v) L v V 2 (). (his is proved for scalar functions and the pure Dirichlet problem in Carstensen et al., 2012a, but the local arguments in the proof are still valid for the weaker boundary conditions and for two components.) he estimate and v KS NC λ 1/2 ε NC (v KS ) L 2 () λ 1/2 D NC v KS L 2 () D NC (v KS v) L 2 () v KS v NC conclude the proof of the proposition. Proof of heorem 2.4. he proof follows in three steps.

24 24 of 31 C. CARSENSEN AND M. SCHEDENSACK Step 1. he inclusion V C ( ) KS( ) and Galerkin orthogonality show, together with Proposition 5.2, u KS u C NC min u KS v C NC v C V C ( ) his implies the following inequality for the energy norm: λ 1/2 min v V u KS v λ 1/2 u u KS. u u C u u KS NC + u C u KS NC (1 + λ 1/2 ) u u KS NC. Since CA 2 λ(a : CA), it follows that Step 2. he inequalities σ σ C L 2 () λ 1/2 u u C λ u u KS NC λ σ σ KS L 2 (). σ σ KS L 2 () σ σ C L 2 () + osc(f 2, ) + osc(g 2, E(Γ N )), σ σ KS L 2 () σ σ CR L 2 () + osc(f 2, ) are direct consequences of heorems 2.1 and 4.4. Step 3. he inequality (A + A ) : (A + A ) 4 A : A implies (σ σ CR ) : (σ σ CR ) ( σ σ CR ) : ( σ σ CR ). From heorem 4.1 it follows, for σ KS := CD NC u KS, that (if Γ D = ) σ σ KS 2 L 2 () = μ2 D NC (u u KS ) 2 L 2 () + (2μ(λ + μ) + (λ + μ) 2 ) div NC (u u KS ) 2 L 2 () 4μ 2 ε NC (u u KS ) 2 L 2 () + (4μλ + λ2 ) div NC (u u KS ) 2 L 2 () = σ σ KS 2 L 2 (). Altogether, σ σ CR L 2 () σ σ CR L 2 () σ σ KS L 2 () + osc(f, ) σ σ KS L 2 () + osc(f, ). his concludes the proof of heorem Numerical investigations his section provides numerical evidence that the claimed equivalence of σ CR and σ KS is independent of the parameter λ for the pure Dirichlet problem in linear elasticity and that the dependence of the equivalence constants in (1.1)onλ = k for k = 6, 7, 8, 9 cannot be improved.

25 6.1 Preliminaries MEDIUS ANALYSIS AND COMPARISON FOR FEM IN ELASICIY 25 of 31 hroughout this section, the elastic modulus is E = 10 5 and the Poisson ratio varies between ν = 0.4, 0.49, 0.499, with corresponding values of μ = E/(2(1 + ν)) and λ = Eν/((1 + ν)(1 2ν)) = k for k = 6, 7, 8, 9. he initial triangulations 0 of all four numerical examples are depicted in Figs 4 and 5. he discrete problems are solved on a sequence of triangulations l obtained by successive red-refinements; a red-refinement of a triangle subdivides each triangle into four congruent subtriangles via straight lines through the edges midpoints, as depicted in Fig. 4(a). Since the error is known only in the first example, the averaging error estimator defined in Carstensen & Funken (2001a, Equation (2.17)) serves as an error indicator. Although the proofs of efficiency and reliability from Carstensen & Funken (2001a) provide no information about the efficiency and reliability constants, there is numerical evidence that the averaging error estimator often yields results very close to the exact error (Carstensen & Funken, 2001a). he first example confirms this observation and so partly justifies the use of this error estimator for the further examples. Let denote the area of a triangle and τ E the tangent of an edge E E. he residual error estimators η C (u C ) := f 2 L 2 () + 1/2 η CR (u CR ) := f 2 L 2 () + 1/2 η KS (u KS ) := f 2 L 2 () + 1/2 + 1/2 E E()\E(Γ D ) E E()\E(Γ D ) E E()\E(Γ N ) E E()\E(Γ N ) [σ C ] E ν E 2 L 2 (E) (1, 0) ([σ KS ] E ν E ) 2 L 2 (E) 1/2 [D NC u CR ] E τ E 2 L 2 (E) [D NC u KS ] E τ E 2 L 2 (E) 1/2, 1/2, for CFEM, CR-NCFEM and KS-NCFEM, respectively, are reliable and efficient (Carstensen & Funken, 2001a; Carstensen & Rabus, 2012). In contrast to Carstensen & Funken (2001a), the normal jump of the second component of the stress is omitted for KS-NCFEM in the spirit of Dari et al. (1995). A close investigation of the dependency on the parameter λ for ν = 0.4, 0.49, and in the comparison result (1.1) considers the quotients q(ν, l) := σ ν σ l,ν C L 2 ()/ σ ν σ l,ν KS L 2 () for l = 1,..., 9. (6.1) Here and in Sections 6.2 and 6.4, σ ν denotes the exact stress for the Poisson ratio ν, and σ l,ν C and denote, respectively, the discrete stresses of CFEM and KS-NCFEM for the Poisson ratio ν and σ l,ν KS the lth red-refinement l := red (l) ( 0 ) of 0. (For the experiment from Section 6.4 the quotients are approximated by the corresponding values of the averaging error estimator.)

26 26 of 31 C. CARSENSEN AND M. SCHEDENSACK 6.2 Academic example Under homogeneous pure Dirichlet boundary conditions, the unit square = (0, 1) 2 is loaded with the applied force ( ) 2μπ f (x, y) = 3 cos(πy) sin(πy)(2 cos(2πx) 1) 2μπ 3 cos(πx) sin(πx)(2 cos(2πy) 1) (written as a function of the coordinates x and y) so that (2.1) leads to the exact smooth solution ( π cos(πy) sin 2 ) (πx) sin(πy) u(x, y) = π cos(πx) sin 2. (πy) sin(πx) Given the initial mesh 0 of Fig. 4(b) with one interior node and eight interior edges, the three FEMs with the number of degrees of freedom lead on each triangulation l to the discrete stresses σ C, σ CR, σ KS ; on level zero, for instance, ndof = 2 for CFEM, ndof = 16 for CR-NCFEM and ndof = 9forKS- NCFEM. he convergence history plot of Fig. 6 displays various errors and error estimators versus the number of degrees of freedom for the Poisson ratios ν = 0.4, 0.49, 0.499, (from dark to light; in the color picture in the online edition of this paper the values correspond to red, blue, green, cyan) for the three FEMs. he graphs of the averaging error estimators and the exact error of CR-NCFEM and KS-NCFEM for all values of ν lie on top of each other and the values of the residual error estimator for KS-NCFEM and also the values of the residual error estimator for CR-NCFEM behave in the same way. For the initial triangulation 0 of Fig. 4(b) with two degrees of freedom in CFEM, the averaging error estimator strongly underestimates and is omitted. Apart from that case, the values of the averaging error estimator are very close to the exact error. his example therefore serves as an empirical validation of the averaging error estimator in the following examples where it is expected to indicate the (unknown) errors to high accuracy. Equivalent convergence rates are observed for all three FEMs with a strong dependency on λ for CFEM, while the errors in KS-NCFEM and CR-NCFEM are of similar size. able 1 displays the quotients (6.1) and reveals a linear dependency on λ. his is clear numerical evidence that the dependence on λ in the first estimate of (1.1) and in heorem 2.4 is sharp. (a) (b) Fig. 4. (a) Red-refined triangle. (b) Initial triangulation 0 on the unit square from Section 6.2.