Improving the Rate of Convergence of ‘high Order Finite Elements’ on Polyhedra I: a Priori Estimates

Let Tk be a sequence of triangulations of a polyhedron Ω ⊂ Rn and let Sk be the associated finite element space of continuous, piecewise polynomials of degree m. Let uk ∈ Sk be the finite element approximation of the solution u of a second order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence Sk ensures optimal rates of convergence for the sequence uk. The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet–Neumann boundary conditions on a polygon.