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 (h^{2 p}). Under weaker conditions on δ, optimal a priori estimates can be established in the L²- and in the H¹ (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 L² (Ω)- and the H¹ (Ω_δ)-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.