Improving the Rate of Convergence of High-Order Finite Elements on Polyhedra I: A Priori Estimates

ABSTRACT Let 𝒯 k be a sequence of triangulations of a polyhedron Ω ⊂ ℝ n and let S k be the associated finite element space of continuous, piecewise polynomials of degree m. Let u k ∈ 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 S k 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.

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