Symmetry-Free, p-Robust Equilibrated Error Indication for the hp-Version of the FEM in Nearly Incompressible Linear Elasticity

Abstract

Abstract. We consider the extension of the p-robust equilibrated error estimator due to Braess, Pillwein and Schöberl to linear elasticity. We derive a formulation where the local mixed auxiliary problems do not require symmetry of the stresses. The resulting error estimator is p-robust, and the reliability estimate is also robust in the incompressible limit if quadratics are included in the approximation space. Extensions to other systems of linear second-order partial differential equations are discussed. Numerical simulations show only moderate deterioration of the effectivity index for a Poisson ratio close to .

Equilibrated Stress Reconstructions for Linear Elasticity Problems with Application to a Posteriori Error Analysis

“…Using the definitions (12) and (13), the formulation (14) is equivalent to (9). For interior vertices, the source term in (14a) has to verify the Neumann compatibility condition…”

“…First, error upper bounds are obtained with fully computable constants. Second, polynomial-degree robustness can be achieved for the Poisson problem in [4,11], for linear elasticity in [9], and for the related Stokes problem in [7]. Third, they allow one to distinguish among various error components, e.g., discretization, linearization, and algebraic solver error components, and to equilibrate adaptively these components in the iterative solution of nonlinear problems [10].…”

Equilibrated Stress Tensor Reconstruction and A Posteriori Error Estimation for Nonlinear Elasticity

“…The patch-wise equilibration technique was introduced in [8,13] for the Poisson problem using the Raviart-Thomas finite element spaces. In [14] it is extended to linear elasticity without any symmetry constraint by using linewise Raviart-Thomas reconstructions. Elementwise reconstructions from local Neumann problems requiring some pre-computations to determine the normal fluxes to obtain an equilibrated stress tensor can be found in [2,12,26,31], whereas in [30] the direct prescription of the degrees of freedom in the Arnold-Winther finite element space is considered.…”

“…Therefore, its implementation is independent and directly applicable to these laws, which makes the method convenient for FEM softwares in solid mechanics, which often provide a large choice of behavior laws. In addition, equilibrated error estimates were proven to be polynomial-degree robust for several linear problems in 2D, as the Poisson problem in [7,18], linear elasticity in [14] and the related Stokes problem in [10] and recently in 3D in [19]. This paper is organized as follows.…”

Interpolation in Jacobi-weighted spaces and its application to a posteriori error estimations of the p-version of the finite element method

“…In [AS97,AS98] the authors proposed an hp refinement strategy in which in every step and for every element they decided to do h adaptivity or p adaptivity based in the local regularity of the solutions. On the other hand, in [BPS09,DM13] p-robust equilibrated residual error estimates are obtained for the Poisson problem and Elasticity problem respectively.…”