Immersed finite element methods for elliptic interface problems
Elliptic interface problems appear in scientific computing and industrial applications. The interface is often caused by the problem involved two or more materials with different properties, such as the conductivity, densities,
or permeability. In this talk, we will introduce immersed finite element (IFE) methods for solving elliptic interface problems. The advantage of IFE methods is to use the unfitted mesh which is independent of the interface.
We first introduce a symmetric and consistent IFE method which is obtained by the classical IFE method by adding two correction terms to preserve the symmetry and consistency. Numerical examples are presented to show the method is stable and has optimal convergence.
Next, we introduce an augmented IFE method which is a fast iterative method. By introducing an augmented variable in the jump of the solution along the interface, we can utilize the FFT based fast Poisson solver to solve the interface problem. The augmented variable is updated using the GMRES iteration. It has been shown that the number of iterations is nearly independent of the jump in the coefficient and the mesh size. In addition, the method is extended to irregular boundary problems. Some numerical examples are presented to show the performance of the fast IFE method.