Local enrichment of finite elements for interface problems

• We consider the finite element approximation of interface problems on unfitted grids.
• We introduce a local enrichment of the original finite element space.
• We prove error estimates with optimal rate of convergence.
• Numerical experiments confirm the theoretical results.
• We investigate numerically the condition number of the involved matrices.

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 H1H1 and in L2L2 depending on the local regularity of the solution. Several numerical experiments confirm the theoretical results.