Optimal a priori estimates for higher order finite elements for elliptic interface problems

We analyze higher order finite elements applied to second order elliptic interface problems. Our a priori error estimates in the L-2- and H-1-norm are expressed in terms of the approximation order p and delta parameter A that quantifies how well the interface is resolved by the finite element mesh. The optimal p-th order convergence in the H-1(Omega)-norm is only achieved under stringent assumptions on delta, namely, delta = O(h(2p)). Under weaker conditions on delta, optimal a priori estimates can be established in the L2- and in the H-1 (Omega(delta))-norm, where Omega(delta) is a subdomain that excludes a tubular neighborhood of the interface of width O(delta). In particular, if the interface is approximated by an interpolation spline of order p and if full regularity is assumed, then optimal convergence orders p + 1 and p for the approximation in the L-2(Omega)- and the H-1 (Omega(delta))-norm can be expected but not order p for the approximation in the H-1(Omega)-norm. Numerical examples in 2D and 3D illustrate and confirm our theoretical results. (C) 2009 IMACS. Published by Elsevier B.V. All rights reserved.