Each \(H^{1/2}\)-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in \(\mathbb R^d\) (Q2842460)

The authors consider the solution of second-order elliptic partial differential equations in \(\mathbb R^d\) with inhomogeneous Dirichlet data. The numerical approximation is performed using an \(h\)-adaptive finite element method (FEM) with fixed polynomial order \(p \in \mathbb N\). 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 \(p \geq 1\). For error estimation, the authors use a residual error estimator which includes the Dirichlet data oscillations. It is proved that each \(H^{1/2}\)-stable projection yields convergence of the adaptive algorithm even with quasi-optimal convergence rate. Numerical experiments with the Scott-Zhang projection are presented.