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

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 optimal convergence rate. For 2D and 3D, we show convergence if the nodal interpolation operator is replaced by the L2-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.

An Abstract Analysis of Optimal Goal-Oriented Adaptivity

“…The latter fails for general inhomogeneous Dirichlet conditions. We believe that the rigorous analysis of this problem is beyond the current work and requires further ideas beyond those of standard AFEM [3,15,25].…”

“…Although not used explicitly, we note that for newest vertex bisection, the triangulation T ⊕ T ′ is, in fact, the overlay of T and T ′ . For 1D bisection (e.g., for 2D BEM computations in Section 6), the algorithm from [2] satisfies (23)- (25) and guarantees that the local mesh-ratio is uniformly bounded. For meshes with first-order hanging nodes, (23)- (25) are analyzed in [12], while T-spline meshes for isogeometric analysis are considered in [38].…”

“…Theorem 17. Suppose that the mesh-refinement satisfies (23) as well as the mesh-closure estimate (24) and the overlay estimate (25). Let 0 < θ < θ ⋆ := (1 + C 2 stb C 2 rel ) −1 be sufficiently small.…”

“…Finally and for the ease of presentation, we focussed on (homogeneous) Dirichlet conditions throughout our experiments. We note that the extension to mixed Dirichlet-Neumann-Robin boundary conditions is easily possible; see [3,15,25] in the frame of standard AFEM. However, we note that our analysis currently requires that the Dirichlet data belongs to the coarsest trace space S 1 (T 0 | Γ ).…”

Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems

“…We note that in either case (A1)-(A4) are already known with R ⋆,• = T ⋆ \T • , and the corresponding constants depend only on uniform shape regularity of the triangulations T ⋆ ∈ T and the well-posedness of the continuous problem (2); see [CKNS08, CN12, FFP14]. The error estimator can be extended to mixed Dirichlet-Neumann-Robin boundary conditions, where inhomogeneous Dirichlet conditions are discretized by nodal interpolation for d = 2 and p = 1, see [FPP14], or by Scott-Zhang interpolation for d ≥ 2 and p ≥ 1,…”