Error Reduction And Convergence For An Adaptive Mixed Finite Element Method

Carsten Carstensen , Ronald H.W. Hoppe

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 13

MSC 2000

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

Abstract
An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: After each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart-Thomas finite element method with a reduction factor $ \rho<1$ uniformly for the $L^2$ norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm does surprisingly not require any particular mesh design unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasi-orthogonality estimate. The proof does neither rely on duality nor on regularity.