An a posteriori error estimator for the Lame equation based on equilibrated fluxes

Abstract

Abstract. We derive a new a posteriori error estimator for the Lamé system based on H(div)-conforming elements and equilibrated fluxes. It is shown that the estimator gives rise to an upper bound where the constant is one up to higher order terms. The lower bound is also established using Argyris elements. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests.

Realistic computable error bounds for three dimensional finite element analyses in linear elasticity

“…The computation of the estimator involves computations over patches of elements sharing a vertex, with the computation of the three dimensional version of the field ≈ σ K constructed using a three dimensional analog of the Arnold, Douglas and Gupta element. We show that, if one wishes to avoid additional post-processing steps as in [18], then one is obliged to make use of so-called "macro-element" techniques and we believe the one presented here is the simplest one available.…”

“…Introduction. Error estimators for linear elasticity problems go back at least as far as [16] with many more estimators being subsequently obtained for two dimensional linear elasticity [7,8,11,12,18,19,24] and three dimensional linear elasticity [10-13, 17, 20, 24]. However, the majority of estimators obtained are not actually computable since they involve either (a) generic unknown constants [7,[10][11][12]24], or (b) the solution of (local) infinite dimensional problems (which cannot be solved exactly) [8,13,19,20].…”

“…Upper bounds can be obtained at the expense of the solution of a global finite element problem [17]. However, the only locally computable guaranteed error estimators for two dimensional linear elasticity problems, of which we are aware, are those in [4,18].…”

“…A vital part of the analysis in both [4,18] is the construction of a suitable field ≈ σ K whose normal components and divergence have certain properties. In [18] this field was constructed using the Arnold and Winther finite element [6] whilst in [4] the Arnold, Douglas and Gupta finite element [5] was used.…”

“…In [18] this field was constructed using the Arnold and Winther finite element [6] whilst in [4] the Arnold, Douglas and Gupta finite element [5] was used. In the current work we extend this result to three dimensional domains in the context of approximation on meshes comprised of tetrahedra.…”