Each H1/2-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in ℝd

Abstract

We consider the solution of second order elliptic PDEs in ℝd with inhomogeneous Dirichlet data by means of an h-adaptive FEM with fixed polynomial order p ∈ 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 H1/2-stable projection, for instance, the L2[-projection for p = 1 or the Scott-Zhang projection for general p ≥ 1. For error estimation, we use a residual error estimator which includes the Dirichlet data oscillations. We prove that each H1/2-stable projection yields convergence of the adaptive algorithm even with quasi-optimal convergence rate. Numerical experiments with the Scott-Zhang projection conclude the work.