Finite Elements Methods for Interface Problems

Interface problems arise in several applications including heart models, cochlea models, aquatic animal locomotion, blood cell motion, front-tracking in porous media flows and material science, to name a few. One of the difficulties in these problems is that solutions are normally not smooth across interfaces, and therefore standard numerical methods will lose accuracy near the interface unless the meshes align to it. However, it is advantageous to have meshes that do not align with the interface, especially for time dependent problems where the interface moves with time. Remeshing at every time step can be prohibitively costly, can destroy the structure of the mesh, can deteriorate the well-conditioning of the stiffness matrix, and affect the stability of the problem. In this talk we present finite element methods for solving interface problems where the finite element triangulation does not fit the interface. We discuss methods for the Poisson and Stokes interface problems, transmission problem with high-contrast coefficients and fluid-structure interactions and solvers.