Each H^{1/2}-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in R^d

TL;DR

This paper proves that using $H^{1/2}$-stable projections in adaptive finite element methods ensures convergence and near-optimality for solving elliptic PDEs with inhomogeneous Dirichlet data, supported by numerical experiments.

Contribution

It demonstrates that $H^{1/2}$-stable projections guarantee convergence and quasi-optimality in adaptive FEM for elliptic PDEs with inhomogeneous Dirichlet data, extending previous results.

Findings

$H^{1/2}$-stable projections ensure convergence of adaptive FEM.

The adaptive algorithm achieves quasi-optimal convergence rates.

Numerical experiments confirm theoretical results.

Abstract

We consider the solution of second order elliptic PDEs in $\R^d$ with inhomogeneous Dirichlet data by means of an $h$-adaptive FEM with fixed polynomial order $p\in\N$. As model example serves the Poisson equation with mixed Dirichlet-Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of an $H^{1/2}$-stable projection, for instance, the $L^2$-projection for $p=1$ or the Scott-Zhang projection for general $p\ge1$. For error estimation, we use a residual error estimator which includes the Dirichlet data oscillations. We prove that each $H^{1/2}$-stable projection yields convergence of the adaptive algorithm even with quasi-optimal convergence rate. Numerical experiments with the $L^2$- and Scott-Zhang projection conclude the work.