Convergence and Quasi-Optimality of Adaptive FEM with Inhomogeneous Dirichlet Data

TL;DR

This paper proves convergence and quasi-optimality of an adaptive finite element method for elliptic PDEs with inhomogeneous Dirichlet data, using edge-based residual error estimators and various interpolation operators.

Contribution

It establishes convergence and quasi-optimality results for adaptive FEM with inhomogeneous Dirichlet data, including new analysis for the Scott-Zhang operator and practical error estimators.

Findings

Proves convergence of adaptive FEM with inhomogeneous Dirichlet data in 2D.

Shows convergence in 3D when using L^2-projection or Scott-Zhang interpolation.

Numerical experiments validate theoretical results.

Abstract

We consider the solution of a second order elliptic PDE with inhomogeneous Dirichlet data by means of adaptive lowest-order FEM. As is usually done in practice, the given Dirichlet data are discretized by nodal interpolation. As model example serves the Poisson equation with mixed Dirichlet-Neumann boundary conditions. For error estimation, we use an edge-based residual error estimator which replaces the volume residual contributions by edge oscillations. For 2D, we prove convergence of the adaptive algorithm even with quasi-optimal convergence rate. For 2D and 3D, we show convergence if the nodal interpolation operator is replaced by the L^2-projection or the Scott-Zhang quasi-interpolation operator. As a byproduct of the proof, we show that the Scott-Zhang operator converges pointwise to a limiting operator as the mesh is locally refined. This property might be of independent interest besides the current application. Finally, numerical experiments conclude the work.