Georgia Institute of Technology College of Sciences

In this talk, I will introduce the Immersed Finite Element Methods (IFEM) for one and two dimensional elliptic interface problems based on Cartesian triangulations. The key is to modify the basis functions so that the homogeneous jump conditions are satisfied in the presence of discontinuity in the coefficients. Both non-conforming and conforming finite element spaces are considered. Corresponding interpolation functions are proved to be second order accurate in the maximum norm. For non-homogeneous jump conditions, we have developed a new strategy to transform the original interface problem to a new one with homogeneous jump conditions using the level set function. If time permits, I will also explain some recent progress in this direction including the augmented IFEM for piecewise constant coefficient, and a SVD free version of the method.