Improving the rate of convergence of high-order finite elements on polyhedra II: Mesh refinements and interpolation

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 ∈ H¹ (Ω) be the solution of the Poisson problem- Δu = f ∈ H^m-1(Ω), m ≥ 2, u = 0 on . 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∥_H1(Ω) ≤ C dim(S_k) ^-m/3∥f∥_Hm-1(ω), with C independent of k and f. Our method relies on the a priori estimate ∥u∥_D ≤ C∥f∥_Hm-1(Ω) in certain anisotropic weighted Sobolev spaces [image omitted], 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 Tk' 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| = k|AB| if A is more singular than B. We can choose 2-m/a. This allows us to use a uniform refinement of the tetrahedra that are away from the edges to construct T′_k.