Convergence of Anisotropic Mesh Adaptation via Metric Optimization

Abstract

Adaptive finite element methods (AFEMs) are an increasingly common means of automatically controlling error in numerical simulations. Proofs of convergence and rate of convergence exist for AFEMs; however, these proofs typically rely upon a nested structure for the sequence of meshes. A metric adaptive finite element method (MAFEM) utilizes the continuous mesh model and instead seeks to optimize a Riemannian metric field for a given cost, from which a mesh is generated. This meshing process results in a sequence of nonnested meshes. In this paper we introduce a proof of convergence for a class of MAFEM, utilizing an optimization statement to relate the error on the sequence of meshes. In addition, we prove that such a sequence of meshes will demonstrate the optimal asymptotic rate of convergence for a given polynomial order. Finally some numerical results demonstrate the performance of the algorithm for a singularly perturbed linear advection diffusion problem.