Finite Element Methods for Interface Problems with Locally Modified Triangulations

Abstract

Interface problems arise in many applications such as heat conduction in different materials. The partial differential equations (PDEs) that describe these applications have domains that consist of different subdomains. The different subdomains can have complicated shapes or can have different properties. For instance, different subdomains can represent different phases of the same material, such as water and ice. The coefficients of the PDEs can be discontinuous across the interfaces of the subdomains, and the source terms can be singular. Due to these irregularities, the solutions to the PDEs can be nonsmooth or even discontinuous. Here we restrict ourselves to interface problems that do not depend on time and can be expressed in terms of elliptic or elasticity PDEs.

We present finite element methods (FEMs) for elliptic and elasticity problems with interfaces. The FEMs are based on body-fitted meshes with a locally modified triangulation. A FEM based on a body-fitted mesh uses a triangulation that is aligned with the interfaces. However, for complicated interfaces it can be difficult and expensive to generate such triangulations. That is why we use a locally modified triangulation based on Cartesian meshes. We first form a Cartesian mesh, then move the grid points near the interfaces to the interfaces. This leads to a locally modified triangulation. We use the standard FEM with the locally modified triangulation to solve the elliptic and elasticity problems with interfaces. By FEM theory, the method is second order accurate in the infinity norm for piecewise smooth solutions. We present some numerical examples to show the second order accuracy of the method.

We also present a new second order finite difference method that does not require to compute the curvature. At points away from the interface we can approximate the PDE by using the standard 5-point scheme. At points where the interface crosses the 5-point scheme, we still use the 5-point scheme by introducing some ghost values for the grid points on the other side of interface. The price is that we need to find an equation for each ghost value. We will use the interface conditions, either the jump in Dirichlet or Neumann boundary conditions, to form the equations for the ghost values to complete the linear system. We also present some numerical examples to show the second order accuracy of the method.