Improving the Rate of Convergence of ‘high Order Finite Elements’ on Polyhedra Ii: Mesh Refinements and Interpolation

Given a bounded polyhedral domain Ω ⊂ R3, we construct a sequence of tetrahedralizations (i.e., meshes) T ′ k that provides quasi-optimal rates of convergence with respect to the dimension of the aproximation space for the Poisson problem with data f ∈ Hm−1(Ω), m ≥ 2. More precisely, let Sk be the Finite Element space of continuous, piecewise polynomials of degree m ≥ 2 on T ′ k and let uk ∈ Sk be the finite element approximation of the solution u of the Poisson problem −∆u = f , u = 0 on the boundary, then ‖u − uk‖H1(Ω) ≤ C dim(Sk)‖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 D = D a+1 (Ω), with a > 0 small and determined 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 chose 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 .