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

Abstract

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 a parameter δ that quantifies how well the interface is resolved by the finite element mesh. The optimal p-th order convergence in the H 1 ( Ω ) -norm is only achieved under stringent assumptions on δ, namely, δ = O ( h 2 p ) . Under weaker conditions on δ, optimal a priori estimates can be established in the L 2 - and in the H 1 ( Ω δ ) -norm, where Ω δ is a subdomain that excludes a tubular neighborhood of the interface of width O ( δ ) . 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 ( Ω ) - and the H 1 ( Ω δ ) -norm can be expected but not order p for the approximation in the H 1 ( Ω ) -norm. Numerical examples in 2D and 3D illustrate and confirm our theoretical results.