Convergence and Complexity of Adaptive Finite Element Methods for Elliptic Partial Differential Equations

Abstract

In this paper, we study adaptive finite element approximations in a perturbation framework, which makes use of the existing adaptive finite element analysis of a linear symmetric elliptic problem. We analyze the convergence and complexity of adaptive finite element methods for a class of elliptic partial differential equations when the initial finite element mesh is sufficiently fine. For illustration, we apply the general approach to obtain the convergence and complexity of adaptive finite element methods for a nonsymmetric problem, a nonlinear problem as well as an unbounded coefficient eigenvalue problem.