Convergence of an Adaptive Finite Element Method for Controlling Local Energy Errors

A number of works concerning rigorous convergence theory for adaptive finite element methods (AFEM) for controlling global energy errors have appeared in recent years. However, many practical situations demand AFEM designed to efficiently compute quantities which depend on the unknown solution only on some subset of the overall computational domain. In this work we prove convergence of an adaptive finite element method for controlling local energy errors. The first step in our convergence proof is the construction of novel a posteriori error estimates for controlling a weighted local energy error. This weighted local energy notion admits versions of standard ingredients for proving convergence of AFEM such as quasi-orthogonality and error contraction, but modulo “pollution terms” which use weaker norms to measure effects of global solution properties on the local energy error. We then prove several convergence results for AFEM based on various marking strategies, including a contraction result in the case of convex polyhedral domains.