A Posteriori Dual-Mixed Adaptive Finite Element Error Control for Lamé and Stokes Equations

Carsten Carstensen , Paola Causin , Riccardo Sacco

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 28 pp.

MSC 2000

65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

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 Lam\'e and Stokes benchmark problems with singularities.