Convergence of adaptive FEM for general elliptic PDE

Khamron Mekchay
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA

kxm@math.umd.edu

Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA

rhn@math.umd.edu

Abstract

We prove convergence of adaptive finite element methods (AFEM) for general (non-symmetric) second order linear elliptic PDE, thereby extending the result of Morin et al \cite{MNS00,MNS02}. The proof relies on quasi-orthogonality, which accounts for the bilinear form not being a scalar product, together with novel error and oscillation reduction estimates, which now do not decouple. We show that AFEM is a contraction for the sum of energy error plus oscillation. Numerical experiments, including oscillatory coefficients and convection-diffusion PDE, illustrate the theory and yield optimal meshes.