Convergence and quasi‐optimality of adaptive FEM with inhomogeneous Dirichlet data

Abstract

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.

EachH1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd

“…Moreover, in the 2D case, where the Dirichlet data are discretized by means of nodal interpolation, linear convergence and even quasi-optimality can be shown for the usual Dörfler marking due to some additional orthogonality relation of 1D nodal interpolation, cf. [14].…”

“…[4,24]. Moreover, besides the 2D works [14,22], no convergence result for AFEM with inhomogeneous Dirichlet data is found in the literature, yet.…”

“…The earlier works [14,22] considered lowest-order finite elements p = 1 in 2D and nodal interpolation to discretize g. However, this situation is very special in the sense that the entire analysis in [14,22] is strictly bound to the lowest-order case and cannot be generalized to R d , since nodal interpolation of the Dirichlet data is well-defined if and only if d = 2. [22] used an error estimator which is obtained by solving local problems on stars, and proved convergence of the related adaptive algorithm.…”

“…This convergence result, however, relies on an artificial marking criterion which first consists of Dörfler marking [12] for the estimator, and afterwards some possible enrichment of the set of marked elements to guarantee linear convergence of volume and Dirichlet oscillations. In [14], the common residual-based error estimator is analyzed, and combined marking for estimator plus Dirichlet oscillations is shown to lead to convergence and even quasi-optimality of AFEM.…”

Efficiency and Optimality of Some Weighted-Residual Error Estimator for Adaptive 2D Boundary Element Methods

“…Following the pioneering works [9,15,30] which analyzed quasi-optimality of AFEM for homogeneous Dirichlet problems, the successors included non-symmetric problems [16,21], inhomogeneous Dirichlet/Neumann conditions [4,23], and even nonlinearities [7,21] into the AFEM analysis. However, many of the ingredients which appear in their proofs were mathematically open for adaptive BEM (ABEM).…”