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 L2- and H1-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 H1(Ω)-norm is only achieved under stringent assumptions on δ, namely, δ=O(h2p). Under weaker conditions on δ, optimal a priori estimates can be established in the L2- and in the H1(Ωδ)-norm, where Ωδ 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 L2(Ω)- and the H1(Ωδ)-norm can be expected but not order p for the approximation in the H1(Ω)-norm. Numerical examples in 2D and 3D illustrate and confirm our theoretical results.