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.
Highlights
We prove that ρl andl are locally equivalent (Lemma 4) and obtain reliability and efficiency ofl (Proposition 5)
The proof relies on the analytical observation that, under adaptive mesh-refinement, the Scott-Zhang projection converges pointwise to a limiting operator (Lemma 19), which might be of independent interest
We show that the edgebased estimatorl from (7)–(8) is locally equivalent to the element-based error estimator ρl from the previous section
Summary
Adaptive finite element methods, convergence analysis, quasi-optimality, inhomogeneous Dirichlet data. Given some initial mesh T0, the algorithm generates successively locally refined meshes Tl with corresponding discrete solutions Ul ∈ S1(Tl) of (4). The second main result is Theorem 18 which states that the outcome of the adaptive algorithm is quasi-optimal in the sense of Stevenson [29]: Provided the given data (f, g, φ) ∈. The proof relies on the analytical observation that, under adaptive mesh-refinement, the Scott-Zhang projection converges pointwise to a limiting operator (Lemma 19), which might be of independent interest. Newest vertex bisection guarantees that any sequence Tl of generated meshes with Tl+1 = refine(Tl) is uniformly shape regular in the sense of (21).
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