Local enrichment of finite elements for interface problems

We consider interface problems for second order elliptic partial differential equations with Dirichlet boundary conditions. It is well known that the finite element discretization may fail to produce solutions converging with optimal rates unless the mesh fits with the discontinuity interface. We introduce a method based on piecewise linear finite elements on a non-fitting grid enriched with a local correction on a sub-grid constructed along the interface. We prove that our method recovers the optimal convergence rates both in H1 and in L2 depending on the local regularity of the solution. Several numerical experiments confirm the theoretical results.