Improving the Rate of Convergence of High-Order Finite Elements on Polyhedra II: Mesh Refinements and Interpolation

Abstract

We construct a sequence of meshes 𝒯 k ′ that provides quasi-optimal rates of convergence for the solution of the Poisson equation on a bounded polyhedral domain with right-hand side in H m−1, m ≥ 2. More precisely, let Ω ⊂ ℝ3 be a bounded polyhedral domain and let u ∈ H 1(Ω) be the solution of the Poisson problem − Δ u = f ∈ H m−1(Ω), m ≥ 2, u = 0 on ∂ Ω. Also, let S k be the finite element space of continuous, piecewise polynomials of degree m ≥ 2 on 𝒯 k ′ and let u k ∈ S k be the finite element approximation of u, then ‖u − u k ‖ H 1(Ω) ≤ C dim(S k )−m/3 ‖f‖ H m−1(Ω), with C independent of k and f. Our method relies on the a priori estimate ‖u‖𝒟 ≤ C ‖f‖ H m−1(Ω) in certain anisotropic weighted Sobolev spaces , with a > 0 small, determined only by Ω. The weight is the distance to the set of singular boundary points (i.e., edges). The main feature of our mesh refinement is that a segment AB in 𝒯 k ′ will be divided into two segments AC and CB in 𝒯 k+1′ as follows: |AC| = |CB| if A and B are equally singular and |AC| = κ |AB| if A is more singular than B. We can choose κ ≤ 2−m/a . This allows us to use a uniform refinement of the tetrahedra that are away from the edges to construct 𝒯 k ′.