A Multigrid Algorithm for Immersed Interface Problems

Many physical problems involve interior interfaces across which the coefficients in the problem, the solution, its derivatives, the flux, or the source term may have jumps. These interior interfaces may or may not align with a underlying Cartesian grid. Zhilin Li, in his dissertation, showed how to discretize such elliptic problems using only a Cartesian grid and the known jump conditions to second order accuracy. In this paper, we describe how to apply the full multigrid algorithm in this context. In particular, the restriction, interpolation, and coarse grid problem will be described. Numerical results for several model problems are given to demonstrate that good rates can be obtained even when jumps in the coefficients are large and do not align with the grid.