A posteriori dual-mixed adaptive finite element error control for Lamé and Stokes equations

Abstract

A unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the Herrmann formulation within the Hellinger-Reissner principle. This quasi-optimally converging extension of PEERS (Plane Elasticity Element with Reduced Symmetry) is called Dual-Mixed Hybrid formulation (DMH). Explicit residual-based a posteriori error estimates for DMH are introduced and are mathematically shown to be locking-free, reliable, and efficient. The estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation. Numerical evidence supports that the adaptive scheme leads to optimal convergence for Lamé and Stokes benchmark problems with singularities.

A posteriori error estimator and error control for contact problems

“…Recently, error control strategies, commonly known as AFEM refinement strategies, have been developed, e.g., in [5,21,25,42,45,46]. Here, special refinement rules in combination with a control over the data oscillation terms lead to a guaranteed error decay.…”

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

“…2×2;S is the space of symmetric tensors of which the components are in 16) where E loc T is a suitable set of edges being in the neighborhood of T . Such an operator can be quite easily constructed, e.g., we can set…”

“…Plate with hole under traction. In the last example, we consider a plate with a circular hole, subject to a shearing load on the right side, see [16].…”

Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

“…On the other hand, the use of adaptive algorithms based on a posteriori error estimates guarantees good convergence behavior of the finite element solution of a boundary value problem. Several a posteriori error estimators are already available in the literature for the usual mixed finite element method in linear elasticity (see [5,8,17,9,15,7]). Concerning the a posteriori error analysis of the augmented scheme presented in [11] in the case of pure homogeneous Dirichlet boundary conditions, an a posteriori error estimator of residual type was introduced in [3].…”