A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction

Abstract

The nonconforming triangular piecewise quadratic finite element space by Fortin and Soulie can be used for the displacement approximation and its combination with discontinuous piecewise linear pressure elements is known to constitute a stable combination for incompressible linear elasticity computations. In this contribution, we extend the stress reconstruction procedure and resulting guaranteed a posteriori error estimator developed by Ainsworth, Allendes, Barrenechea and Rankin [2] and by Kim [18] to linear elasticity. In order to get a guaranteed reliability bound with respect to the energy norm involving only known constants, two modifications are carried out: (i) the stress reconstruction in next-to-lowest order Raviart-Thomas spaces is modified in such way that its anti-symmetric part vanishes in average on each element; (ii) the auxiliary conforming approximation is constructed under the constraint that its divergence coincides with the one for the nonconforming approximation. An important aspect of our construction is that all results hold uniformly in the incompressible limit. Global efficiency is also shown and the effectiveness is illustrated by adaptive computations involving different Lamé parameters including the incompressible limit case.

Equilibrated Stress Reconstruction and a Posteriori Error Estimation for Linear Elasticity

“…This additional term measures the nonconformity of u h by its jumps and can be estimated by its distance to an appropriate conforming approximation as it is done in [1] and [19]. Moreover, the quadratic nonconforming elements allow to improve the localization to an element-wise computation, see again [1,19] and, for the linear elasticity case, [6]. Finally, it should be mentioned that our stress equilibration procedure and its analysis does, of course, also apply to the standard finite element approximation without the pressure variable whenever the finite size of λ admits this.…”

“…Remarks on the generalization to other finite element pairs, including nonconforming ones, will be given at the end of this manuscript. Specifically, the case of quadratic nonconforming finite elements is studied in detail in [6].…”

“…h denotes the element-wise L 2 projection onto the space of polynomials of degree k. Moreover, on sides located on the Neumann boundary Γ N , (5) and (6)…”

A posteriori error analysis for the mixed Laplace eigenvalue problem: investigations for the BDM‐element

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A posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem can be derived using a reconstruction in the standard H 1 0 -conforming space for the primal variable of the mixed problem. A unified framework for a posteriori error estimation based on stress reconstruction for the Stokes system is carried out in [5] and an extension to the linear elasticity problem is presented in [6]. This paper shows that the extension for the BDM-element is not straightforward.

Let Ω ⊂ R 2 be a polygon.

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“…In the case of Raviart-Thomas finite elements of arbitrary polynomial degree the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient. A unified framework for a posteriori error estimation based on stress reconstruction for the Stokes system is carried out in [5] and an extension to the linear elasticity problem is presented in [6]. Let Ω ⊂ R 2 be a polygon.…”