Monolithic multigrid for the marker-and-cell discretization of the Stokes–Darcy equations

Abstract

We consider the marker-and-cell scheme for numerically solving the Stokes–Darcy equations. The corresponding discrete system has a double saddle-point structure. Designing a fast solver for such a problem is challenging o the different scales of the physical parameters. We propose a monolithic multigrid solver with a block-lower-triangular smoother based on the block-LU decomposition of the coefficient matrix, which requires solving a Poisson-type equation with a scalar Laplacian and two Schur complement systems. We demonstrate the robustness of a sparse approximate inverse smoother for the Laplacian, and we handle the Schur complement systems by applying simple relaxation schemes as smoothers. The proposed scheme is economical, and yet it is robust with respect to the mesh size and the physical parameters.