# Residual-based a posteriori error analysis for symmetric mixed Arnold–Winther FEM

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## Abstract

This paper introduces an explicit residual-based a posteriori error analysis for the symmetric mixed finite element method in linear elasticity after Arnold–Winther with pointwise symmetric and \(H({\text {div}})\)-conforming stress approximation. The residual-based a posteriori error estimator of this paper is reliable and efficient and truly explicit in that it solely depends on the symmetric stress and does neither need any additional information of some skew symmetric part of the gradient nor any efficient approximation thereof. Hence, it is straightforward to implement an adaptive mesh-refining algorithm. Numerical experiments verify the proven reliability and efficiency of the new a posteriori error estimator and illustrate the improved convergence rate in comparison to uniform mesh-refining. A higher convergence rate for piecewise affine data is observed in the \(L^2\) stress error and reproduced in non-smooth situations by the adaptive mesh-refining strategy.

## Mathematics Subject Classification

65N15 65N30## 1 Introduction

### 1.1 Overview

The design of a pointwise symmetric stress approximation \(\sigma _h\in L^2(\varOmega ;{\mathbb {S}})\) with divergence in \(L^2(\varOmega ;{\mathbb {R}}^d)\), written \(\sigma _h \in H({\text {div}},\varOmega ; \mathbb S)\), has been a long-standing challenge [2] and the first positive examples in [5] initiated what nowadays is called the finite element exterior calculus [4]. The a posteriori error analysis of mixed finite element methods in elasticity started with [11] on PEERS [3], where the asymmetric stress approximation \(\gamma _h\) arises in the discretization as a Lagrange multiplier to enforce weakly the stress symmetry. This allows the treatment of the term \({\mathbb {C}}^{-1} \sigma _h + \gamma _h\) as an approximation of the (nonsymmetric) functional matrix *Du* for the displacement field [11] with the arguments of [1, 9] developed for mixed finite element schemes for a Poisson model problem. Here and throughout, \({\mathbb {C}}\) denotes a fourth-order elasticity tensor with two Lamé constants \(\lambda \) and \(\mu \) and \({\mathbb {C}}^{-1}\) is its inverse. Mixed finite element methods appear attractive in the incompressible limit for they typically avoid the locking phenomenon [12] as \(\lambda \rightarrow \infty \).

*v*with Green strain \(\varepsilon (v):={\text {sym}}D v\) or of some skew-symmetric approximation \(\gamma _h\) motivated from the first results in [11] on PEERS. In fact,

*any*choice of a piecewise smooth and pointwise skew-symmetric \(\gamma _h\) allows for an a posteriori error control of the symmetric stress error \(\sigma -\sigma _h\) in [15]. Its efficiency, however, depends on the (unknown and uncontrolled) efficiency of the choice of \(\gamma _h\) as an approximation to the skew-symmetric part \(\gamma \) of

*Du*.

*explicit*residual-based a posteriori error estimator of the nonconforming residual with the typical contributions to \(\eta ({\mathscr {T}},\sigma _h)\) computed from the (known) Green strain approximation \(\varvec{\varepsilon }_h:= {\mathbb {C}}^{-1} \sigma _h\). Besides oscillations of the applied forces in the volume and along the Neumann boundary, there is a volume contribution \(h_T^2\Vert {\text {rot}}{\text {rot}}\varvec{\varepsilon }_h\Vert _{L^2(T)}\) for each triangle \(T\in {\mathscr {T}}\) and an edge contribution with the jump \([\varvec{\varepsilon }_h]_E\) across an interior edge

*E*with unit normal \(\nu _E\), tangential unit vector \(\tau _E\), and length \(h_E\), namely

The analysis is restricted to the two dimensional case, since it involves explicit calculations in two dimensions without any reference to the exterior calculus but with inhomogeneous Dirichlet and Neumann boundary data. The main result is reliability and efficiency to control the stress error robustly in the sense that the multiplicative generic constants hidden in the notation \(\lesssim \) do neither depend on the (local or global) mesh-size nor on the parameter \(\lambda >0\) but may depend on \(\mu >0\) and on the shape regularity of the underlying triangulation \({\mathscr {T}}\) of the domain \(\varOmega \) into triangles through a lower bound of the minimal angle therein.

### 1.2 Linear elastic model problem

More essential will be a discussion on the precise conditions on the Neumann data *g* and its discrete approximation \(g_h\) below for they belong to the essential boundary conditions in the mixed finite element method based on the dual formulation.

*V*and the aforementioned stress space \(H({\text {div}},\varOmega ;{\mathbb {S}}) \), namely,

Throughout this paper, the model problem considers truly mixed boundary conditions with the hypothesis that both \(\varGamma _D\) and \(\varGamma _N\) have positive length. The remaining cases of a pure Neumann problem or a pure Dirichlet problem require standard modification and are immediately adopted. The presentation focuses on the case of isotropic linear elasticity with constant Lamé parameters \(\lambda \) and \(\mu \) for brevity and many results carry over to more general situations (cf. Remarks 1 and 2 for instance).

### 1.3 Mixed finite element discretization

Let \({\mathscr {T}}\) denote a shape-regular triangulation of \(\varOmega \) into triangles (in the sense of Ciarlet [8]) with set of nodes \({\mathscr {N}}\), set of interior edges \({\mathscr {E}}(\varOmega )\), set of Dirichlet edges \({\mathscr {E}}(\varGamma _D)\) and set of Neumann edges \({\mathscr {E}}(\varGamma _N)\). The triangulation is compatible with the boundary pieces \(\varGamma _D\) and \(\varGamma _N\) in that the boundary condition changes only at some vertex \({\mathscr {N}}\) and \(\varGamma _D\) (resp. \(\overline{\varGamma _N}\)) is partitioned in \({\mathscr {E}}(\varGamma _D)\) (resp. \({\mathscr {E}}(\varGamma _N)\)).

*g*in the normal trace space

*h*)

*f*under some extra condition (N) on the Neumann data approximation \(g_h\) implied by the first and zero moment orthogonality assumption \(g-g_h\perp P_1( {\mathscr {E}}(\varGamma _N);{\mathbb {R}}^2)\) (\(\perp \) indicates orthogonality in \(L^2(\varGamma _N)\)) met in all the numerical examples of this paper.

For simple benchmark examples with piecewise polynomial data *f* and *g*, there is even a superconvergence phenomenon visible in numerical examples. The arguments of this paper allow a proof of fourth-order convergence of the \(L^2\) stress error \(\Vert \sigma -\sigma _h\Vert ={\mathscr {O}}( h^4 )\) in the lowest-order Arnold–Winther method with \(k=1\) for a smooth stress \(\sigma \in H^4(\varOmega ;{\mathbb {S}})\) with \(f=f_h\in P_1({\mathscr {T}};{\mathbb {R}}^2)\) and \(g=g_h\in G({\mathscr {T}})\). (In fact, once the data are not piecewise affine, the arising oscillation terms are only of at most third order and the aforementioned convergence estimates are sharp.)

This is stated as Theorem 5 in the appendix, because the a priori error analysis lies outside of the main focus of this work. It is surprising though that adaptive mesh-refining suggested with this paper recovers this higher convergence rate even for the inconsistent Neumann data in the Cook membrane benchmark example below.

### 1.4 Explicit residual-based a posteriori error estimator

The novel explicit residual-based error estimator for the discrete solution \((\sigma _h, u_h)\) to (3) depends only on the Green strain approximation \({\mathbb {C}}^{-1}\sigma _h\) and its piecewise derivatives and jumps across edges.

*E*of length \(h_E\), let \(\nu _E\) denote the unit normal vector (chosen with a fixed orientation such that it points outside along the boundary \(\partial \varOmega \) of \(\varOmega \)) and let \(\tau _E\) denote its tangential unit vector; by convention \(\tau _E = (0,-1; 1,0) \nu _E\) with the indicated asymmetric \(2\times 2\) matrix. The tangential derivative \(\tau _E\cdot \nabla \bullet \) along an edge (or boundary) is identified with the one-dimensional derivative \(\partial \bullet /\partial s\) with respect to the arc-length parameter

*s*. The jump \([v]_E\) of any piecewise continuous scalar, vector, or matrix

*v*across an interior edge \(E = \partial T_+\cap \partial T_-\) shared by the two triangles \(T_+\) and \(T_-\) such that \(\nu _E\) points outside \(T_+\) along \(E\subset \partial T_+\) reads

### Theorem 1

*E*and a degree \(m\ge k+2\), let \(\varPi _{m,E}:L^2(E)\rightarrow P_{m}(E)\) denote the \(L^2\) projection onto polynomials of degree at most

*m*. For any \(E\in {\mathscr {E}}(\varGamma _D)\) define the two Dirichlet data oscillation terms

### Theorem 2

### 1.5 Outline of the paper

The remaining parts of this paper provide a mathematical proof of Theorems 1 and 2 and numerical evidence in computational experiments on the novel a posteriori error estimation and its robustness as well as on associated mesh-refining algorithms.

The proof of the reliability of Theorem 1 in Sect. 2 adopts arguments of [11, 15] and carries out two integration by parts on each triangle plus one-dimensional integration by parts along all edges. The resulting terms are in fact locally efficient in Sect. 3 with little generalizations of the bubble-function methodology due to Verfürth [24]. The five lemmas of Sect. 3 give slightly sharper results and in total imply Theorem 2.

The point in Theorems 1 and 2 is that the universal constants \(C_{\text {rel}}\) and \(C_{\text {eff}}\) may depend on the Lamé parameter \(\mu \) but are independent of the critical Lamé parameter \(\lambda \) as supported by the benchmark examples of the concluding Sect. 4. Adaptive mesh-refining proves to be highly effective with the novel a posteriori error estimator even for incompatible Neumann data. Four benchmark examples with the Poisson ratio \(\nu =0.3\) or 0.4999 provide numerical evidence of the robustness of the reliable and efficient a posteriori error estimation and for the fourth-order convergence of Theorem 5.

Three appendices highlight some improvements in the numerical benchmarks: Appendix A explains the improved convergence order for piecewise affine data and B and C explain how to treat incompatible Neumann data successfully.

### 1.6 Comments on general notation

Standard notation on Lebesgue and Sobolev spaces and norms is adopted throughout this paper and, for brevity, \(\Vert \cdot \Vert :=\Vert \cdot \Vert _{L^2(\varOmega )}\) denotes the \(L^2\) norm. The piecewise action of a differential operator is denoted with a subindex *NC*, e.g., \(\nabla _{NC}\) denotes the piecewise gradient \((\nabla _{NC} \bullet )|_T := \nabla (\bullet |_T)\) for all \(T\in {\mathscr {T}}\). Sobolev functions are usually defined on open sets and the notation \(W^{m,p}(T)\) (resp. \(W^{m,p}({\mathscr {T}})\)) substitutes \(W^{m,p}({\text {int}}(T))\) for a (compact) triangle *T* and its interior \({\text {int}}(T)\) (resp. \(W^{m,p}({\text {int}}({\mathscr {T}}))\)) and their vector and matrix versions.

The colon denotes the scalar product \(A:B:=\sum _{\alpha ,\beta =1,2} A_{\alpha ,\beta } B_{\alpha ,\beta }\) of \(2\times 2\) matrices *A*, *B*. The inequality \(A\lesssim B\) between two terms *A* and *B* abbreviates \(A\le C\, B\) with some multiplicative generic constant *C*, which is independent of the mesh-size and independent of the one Lamé parameter \(\lambda \ge 0\) but may depend on the other \(\mu >0\) and may depend on the shape-regularity of the underlying triangulation \({\mathscr {T}}\) and the parameter *k* related to the polynomial degree of the scheme.

## 2 Proof of reliability

This section is devoted to the proof of Theorem 1 based on a Helmholtz decomposition of [11] with two parts as decomposed in Theorem 3 below. The critical part is the \(L^2\) product of \({\mathbb {C}}^{-1} (\sigma -\sigma _h)\) times the \({\text {Curl}}\) of an unknown function \({\text {Curl}}\beta \). The observation from [15] is that one can find an Argyris finite element approximation \(\beta _h\) to \(\beta \in H^2(\varOmega )\) such that the continuous function \(\phi :=\beta -\beta _h\in H^2(\varOmega )\) vanishes at all vertices \({\mathscr {N}}\) of the triangulation. Two integration by parts on each triangle plus one-dimensional integration by parts along the edges \({\mathscr {E}}\) of the triangulation eventually lead to a key identity.

### Lemma 1

### Proof

*E*for the last term shows \( ([\varvec{\varepsilon }_h]_E\tau _E, (\partial \phi /\partial s )\nu _E)_{L^2(E)} = - (\partial [\varvec{\varepsilon }_h]_E\tau _E/\partial s , \phi \nu _E)_{L^2(E)}\). This proves

*E*when \([\varvec{\varepsilon }_h]_E\) is replaced by \(\varvec{\varepsilon }_h\). The combination of the latter identities with the first displayed formula of this proof verifies the asserted representation formula. \(\square \)

The contribution of \(\varepsilon (u)={\mathbb {C}}^{-1} \sigma \) times the \({\text {Curl}}^2\phi \in L^2(\varOmega ;{\mathbb {S}})\) exclusively leads to boundary terms. Throughout this paper, suppose that the Dirichlet data \(u_D\) satisfies \(u_D\in C(\varGamma _D)\cap H^2({\mathscr {E}}(\varGamma _D))\) in the sense that \(u_D\) is globally continuous with \(u_D|_E\in H^2(E;{\mathbb {R}}^2)\) for all \(E\in {\mathscr {E}}(\varGamma _D)\).

### Lemma 2

*z*of \(\varGamma _D\) in its relative interior satisfy

### Proof

*v*and \(\phi \), when integration by parts arguments show that the left-hand side is equal to

*z*in \(\varGamma _D\) with a jump of the normal unit vector. The substitution of the boundary conditions concludes the proof. \(\square \)

The consequence of the previous two lemmas is a representation formula for the error times a typical function \({\text {Curl}}^2 \phi \). To understand why the contributions on the Neumann boundary of \(\phi \) and \(\nabla \phi \) disappear along \(\varGamma _N\), some details on the Helmholtz decomposition are recalled from the literature. For this, let \(\varGamma _0,\ldots , \varGamma _J\) denote the compact connectivity components of \(\overline{\varGamma _N}\).

### Theorem 3

The second ingredient is an approximation \(\beta _h\) of \(\beta \) from the Helmholtz decomposition in Theorem 3 based on the Argyris finite element functions \( A({\mathscr {T}}) \subset C^1(\varOmega )\cap P_5({\mathscr {T}})\) [7, 8, 20]. The local mesh-size \(h_{\mathscr {T}}\in P_0({\mathscr {T}})\) in the triangulation \({\mathscr {T}}\) is defined as its diameter \(h_{\mathscr {T}}|_T:=h_T\) on each triangle \(T\in {\mathscr {T}}\).

### Lemma 3

The combination of all aforementioned arguments leads to the following estimate as an answer to the question of Sect. 1.1 in terms of directional derivatives of \(\varvec{\varepsilon }_h:= {\mathbb {C}}^{-1}\sigma _h\). Recall the definition of \(\eta ({\mathscr {T}},\sigma _h)\) from (4).

### Theorem 4

### Proof

Before the proof of Theorem 1 concludes this section, three remarks and one lemma are in order.

### Remark 1

*(nonconstant coefficients)* The main parts of the reliability analysis of this section hold for rather general material tensors \({\mathbb {C}}\) as long as \(\varvec{\varepsilon }_h:= {\mathbb {C}}^{-1}\sigma _h\) allows for the existence of the traces and the derivatives in the error estimator (4) in the respective \(L^2\) spaces. For instance, if \(\lambda \) and \(\mu \) are piecewise smooth with respect to the underlying triangulation \({\mathscr {T}}\).

### Remark 2

*(constant coefficients)*The overall assumption of constant Lamé parameters \(\lambda \) and \(\mu \) allows a simplification in the error estimator (4). It suffices to have \(\mu \) globally continuous and \(\mu \) and \(\lambda \) piecewise smooth to guarantee

### Remark 3

*(related work)* Although the work [22] concerns a different problem (bending of a plate of fourth order) with a different discretization (even nonconforming in \(H({\text {div}})\)), some technical parts of that paper are related to those of this by a rotation of the underlying coordinate system and the substitution of \({\text {div}}{\text {div}}\) by \({\text {rot}}{\text {rot}}\) etc. Another Helmholtz decomposition also allows for a discrete version and thereby enables a proof of optimal convergence of an adaptive algorithm with arguments from [13, 19].

*V*, namely

### Lemma 4

### Proof

There are several variants of the tr-dev-div lemma known in the literature [6, Proposition 9.1.1]. The version in [11, Theorem 4.1] explicitly displays a version with \(\Vert {\text {div}}\tau \Vert \) replacing \(\Vert {\text {div}}\tau \Vert _{-1}\). Since its proof is immediately adopted to prove the asserted version, further details are omitted. \(\square \)

## 3 Local efficiency analysis

The local efficiency follows with the bubble-function technique for \(C^1\) finite elements [24, Sec 3.7]. This section focuses on a constant \({\mathbb {C}}\) for linear isotropic elasticity with constant Lamé parameters \(\lambda \) and \(\mu \) such that \(\varvec{\varepsilon }_h:={\mathbb {C}}^{-1}\sigma _h\in P_{k+2}({\mathscr {T}})\) for some \(\sigma _h\in AW _k({\mathscr {T}})\) is a polynomial of degree at most \(k+2\). Apart from this, the Lamé parameters do not further arise in this section.

The moderate point of departure is the volume term for each triangle \(T\in {\mathscr {T}}\) with barycentric coordinates \(\lambda _1,\lambda _2,\lambda _3\in P_1(T)\) and their product, the cubic volume bubble function, \(b_T:=27\,\lambda _1\lambda _2\lambda _3 \in W^{1,\infty }_0(T)\) plus its square \(b_T^2\in W^{2,\infty }_0(T)\) with \(0\le b_T^2\le 1\), \( \Vert b_T\Vert _{L^2(T)} \lesssim 1\), and \(|b_T|_{H^2(T)} \lesssim h_T^{-2}\) etc.

### Lemma 5

### Proof

The localization of the first edge residual involves the piecewise quadratic edge-bubble function \(b_E\) with support \(T_+\cup T_-\) for an interior edge \(E=\partial T_+\cap \partial T_-\) shared by the two triangles \(T_+\) and \(T_-\) with edge-patch \(\omega _E := {\text {(}}T_+\cup T_-)\). With an appropriate scaling \(b_E|_T=4\lambda _1\lambda _2\) for the two barycentric coordinates \(\lambda _1,\lambda _2\) on \(T\in \{T_+,T_-\}\) associated with the two vertices of *E*. Then \(b_E\in W^{1,\infty }(\omega _E)\) and \(b_E^2\in W^{2,\infty }(\omega _E)\) satisfy \(0\le b_E^2\le b_E\le 1\) and \(|b_E|_{H^1(E)} \lesssim h_E^{-1}\) etc.

*E*to \(\omega _E\). Throughout this section those functions are polynomials and given \(\rho _E\in P_m(E)\), their coefficients (in some fixed basis) already define an algebraic object that is a natural extension \(\rho \in P_m({\hat{E}})\) along the straight line \({\hat{E}}:={\text {mid}}(E)+{\mathbb {R}}\, \tau _E\) that extends

*E*with midpoint \({\text {mid}}(E)\) and tangential unit vector \(\tau _E\). This and

*E*for any \(\rho _E\in P_m(E)\), which is constant in the normal direction, \(\nabla P_E(\rho _E)\cdot \nu _E\equiv 0\). This design is different from that in [24].

### Lemma 6

### Proof

*E*, the test function \(\phi \in H^2_0(\omega _E)\subset H_0^2(\varOmega )\) leads in Lemma 1 to

*E*and \(\varepsilon (v)\perp {\text {Curl}}^2\phi \), an inverse estimate shows

*b*and

For any edge \(E\in {\mathscr {E}}(\varGamma _D)\), the edge-bubble function \(b_E=4\lambda _1\lambda _2\in W^{1,\infty }(\omega _E)\) for the two barycentric coordinates \(\lambda _1,\lambda _2\) associated with the two vertices of *E* and \(b_E\) vanishes on the remaining sides \(\partial \omega _E{\setminus } E\) of the aligned triangle \(\overline{\omega _E}\). The Dirichlet data \(u_D\) allows for some polynomial approximation \(\varPi _{m,E} u_{D}\in P_{m}(E)\) of a maximal degree bounded by \(m\ge k+2\); recall the definition of \({\text {osc}}_I(u_D,E)\) from (5).

### Lemma 7

### Proof

*E*, the residual \(\tau _E\cdot ( \varvec{\varepsilon }_h\tau _E - \partial u_D/\partial s)\) is well approximated by its \(L^2\) projection \(\rho _E:= (\tau _E\cdot ( \varvec{\varepsilon }_h\tau _E - \varPi _{m,E} \partial u_D/\partial s))\) onto \(P_{m}(E)\). The Pythagoras theorem based on the \(L^2\) orthogonality reads

*b*from (10) lead to an admissible test function \( \phi := b P_E \rho _E \in W^{2,\infty }_0(\omega _E)\). Two successive integration by parts as in Lemma 1 show

*E*and \(\rho _E\) is the \(L^2\) projection of \(\tau _E\cdot ( \varvec{\varepsilon }_h\tau _E- \partial u_D/\partial s)\), the left-hand side equals \(\Vert b_E \rho _E\Vert _{L^2(E)}^2 - ( ( 1-\varPi _{m,E}) \partial u_D/\partial s, b_E^2 \rho _E )_{L^2(E)} \). The scaling argument which leads to (11) shows that the left-hand side of (11) is \(\lesssim \Vert \rho _E\Vert _{L^2(E)}\). The combination with the previously displayed identity leads to

The edge-bubble functions for the second edge residuals are defined slightly differently to ensure some vanishing normal derivative.

### Lemma 8

### Proof

*E*. The polynomial \(\rho _E:= \tau _E\cdot ( [{\text {rot}}_{NC}\varvec{\varepsilon }_h]_E - \partial [\varvec{\varepsilon }_h]_E/\partial s \, \nu _E)\) and its extension \(P_E \rho _E\) define the test function \(\phi := b P_E \rho _E\in C_0^\infty (\omega _E)\) in Lemma 1. The representation formula and \((\varepsilon (v),{\text {Curl}}^2\phi )_{L^2(\omega _E)}=0\) lead to

The efficiency of the last edge contribution involves the second Dirichlet data oscillation \({\text {osc}}_{II}(u_D,E)\) from (6).

### Lemma 9

### Proof

Select a maximal open ball \(B(x_E,2r_E)\cap \varOmega \subset \omega _E\) around a point \(x_E\in E\) with maximal radius \(2r_E\) such that \(B(x_E,2r_E)\cap \omega _E\) is a half ball. The regularization \(b:=\chi _{B(x_E,r_E)}*\eta _{r_E}\in C_c^\infty ({\mathbb {R}}^2)\) of the characteristic function \(\chi _{B(x_E,r_E)}\) attains values in [0, 1] and a positive integral mean \(h_E^{-1} \int _E b\, ds \approx 1\) along *E* (depending only on the shape regularity of \({\mathscr {T}}\)); *b* vanishes on \(\partial \omega _E{\setminus } E\) and its normal derivative \(\nabla b\cdot \nu =0\) vanishes along the entire boundary \(\partial \omega _E\).

## 4 Numerical examples

This section is devoted to numerical experiments for four different domains to demonstrate robustness in the reliability and efficiency of the a posteriori error estimator \(\eta ({\mathscr {T}}_\ell ,\sigma _\ell )\). The implementation follows [12, 15, 16] for \(k=1\) with Lamé parameters \(\lambda \) and \(\mu \) from \(\lambda = E\nu /( (1+\nu )(1-2\nu ))\) and \(\mu ={E}/(2(1+\nu ))\) for a Young’s modulus \(E=10^5\) and various Poisson ratios \( \nu =0.3\) and \( \nu = 0.4999\).

### 4.1 Academic example

*f*depends only on the Lamé parameter \(\mu \) and not on the critical Lamé parameter \(\lambda \). For uniform mesh refinement, Fig. 1 displays the robust third-order convergence of the a posteriori error estimator \(\eta ({\mathscr {T}}_\ell ,\sigma _\ell )\) as well as the Arnold–Winther finite element stress error. The convergence is robust in the Poisson ratio \(\nu \rightarrow 1/2\) and the a posteriori error estimator proves to be reliable and efficient. In this example, the oscillations \({\text {osc}}(f,{\mathscr {T}}_\ell )\) dominate the a posteriori error estimator.

### 4.2 Circular inclusion

### 4.3 L-shaped benchmark

Figure 5 shows suboptimal convergence \({\mathscr {O}}(N_\ell ^{-0.27})\), namely an expected rate \(\alpha \) in terms of the maximal mesh-size, for uniform and fourth-order \(L^2\) stress convergence for adaptive mesh-refinement.

### 4.4 Cook membrane problem

One of the more popular benchmarks in computational mechanics is the tapered panel \(\varOmega \) with the vertices *A*, *B*, *C*, *D* of Fig. 6 clamped on the left side \(\varGamma _D={\text {conv}}(D,A)\) (with \(u_D\equiv 0\)) under no volume force (\(f\equiv 0\)) but applied surface tractions \(g = (0,1)\) along \({\text {conv}}(B,C)\) and traction free on the remaining parts \({\text {conv}}(A,B)\) and \({\text {conv}}(C,D)\) along the Neumann boundary.

This example is a particular difficult one for the Arnold–Winther MFEM because of the incompatible Neumann boundary conditions on the right corners [12, 15, 16]. That means, although *g* is piecewise constant, *g* does not belong to \(G({\mathscr {T}})\) for any triangulation. In the two Neumann corner vertices *B* and *C* we therefore strongly impose the values \(\sigma _\ell (B) = (0.2491 , 0.7283; 0.7283 , 0.6676)\) and \(\sigma _\ell (C) = ( 3/20, 11/20 ; 11/20 , 11/60)\) for the design of \(g_\ell \in G({\mathscr {T}}_\ell )\).

Since the exact solution is unknown, the error approximation rests on a reference solution \({\tilde{\sigma }}\) computed as \(P_5({\mathscr {T}})\) displacement approximation on the uniform refinement of the finest adapted triangulation.

The large pre-asymptotic range of the convergence history plot in Fig. 7 illustrates the difficulties of the Arnold–Winther finite element method in case of incompatible Neumann boundary conditions according to its nodal degrees of freedom. Once the resulting and dominating boundary oscillations (caused by the necessary choice of discrete compatible Neumann conditions in \(G({\mathscr {T}}_\ell )\)) \({\text {osc}}(g-g_\ell ,{\mathscr {E}}_\ell (\varGamma _N))\) are resolved through adaptive mesh-refining, even the fourth-order \(L^2\) stress convergence is visible in a long asymptotic range in (the approximated error and) the equivalent error estimator.

This example underlines that adaptive mesh-refining is unavoidable in computational mechanics with optimal rates and a large saving in computational time and memory compared to naive uniform mesh-refining.

### 4.5 Comments

The generic constants in this paper are not worked out explicitly in detail and so a numerical comparison with the earlier paper [15] cannot be quantitatively. It is conjectured that the residual-based error estimation with the reliability constants (for a guaranteed upper error bound) overestimates the true error up to an order of magnitude.

The qualitative comparison in Fig. 5 (without the reliability constants for the estimators) provides numerical evidence that the error estimators of this paper converge with the same convergence rates as those from [15] and it also indicates global equivalence of the errors with the two error estimators. The theoretical evidence in [15] for efficiency depends on unrealistically high regularity assumptions – unlike the general efficiency results of this paper.

## Notes

### Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). The work has been written, while the three authors enjoyed the hospitality of the Hausdorff Research Institute of Mathematics in Bonn, Germany, during the Hausdorff Trimester Program *Multiscale Problems: Algorithms, Numerical Analysis and Computation*.

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