A comparison of algebraic multigrid and geometric immersed interface multigrid methods for interface problems

In [L. Adams and Z. Li, SIAM J. Sci. Comput., 24 (2002), pp. 463--479], a multigrid method was designed specifically for interface problems that have been discretized using the methods described in [L. Adams and Z. Li, SIAM J. Sci. Comput., 24 (2002), pp. 463--479] and in [Z. Li and K. Ito, SIAM J. Sci. Comput., 23 (2001), pp. 339--361] for elliptic interface problems using the maximum principle preserving schemes. In [L. Adams and T. P. Chartier, SIAM J. Sci. Comput., 25 (2002), pp. 1516--1533], a new method was introduced that utilizes a new interpolator for grid points near the immersed interface and a new restrictor that guarantees the coarse-grid matrices are M-matrices. This paper compares the immersed interface multigrid methods introduced in [L. Adams and Z. Li, SIAM J. Sci. Comput., 24 (2002), pp. 463--479] and [L. Adams and T. P. Chartier, SIAM J. Sci. Comput., 25 (2002), pp. 1516--1533] with algebraic multigrid, which uses no geometric information to set up the multigrid components for coarse-grid correction. We show that algebraic multigrid is a robust solver for our test problems. It outperforms the method in [L. Adams and Z. Li, SIAM J. Sci. Comput., 24 (2002), pp. 463--479] and performs nearly as well as the method in [L. Adams and T. P. Chartier, SIAM J. Sci. Comput., 25 (2002), pp. 1516--1533] which is shown to be the most efficient for all problem parameters and sizes.