Improving the rate of convergence of high-order finite elements on polyhedra I:: A priori estimates

Let T-k be a sequence of triangulations of a polyhedron Omega subset of R-n and let S-k be the associated finite element space of continuous, piecewise polynomials of degree m. Let u(k) epsilon S-k 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 u(k). 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.