A Fourth-Order Cut-cell Multigrid Method for Solving Elliptic Equations on Arbitrary Domains

Abstract

To numerically solve a generic elliptic equation on two-dimensional domains with rectangular Cartesian grids, we propose a cut-cell geometric multigrid method that features (1) general algorithmic steps that apply to two-dimensional constant-coefficient elliptic equations with both divergence and non-divergence forms and all types of boundary conditions, (2) the versatility of handling both regular and irregular domains with arbitrarily complex topology and geometry, (3) the fourth-order accuracy even at the presence of ${\mathcal{C}}^1$ discontinuities on the domain boundary, and (4) the optimal complexity of $O(h^{-2})$.Test results demonstrate the generality, accuracy, efficiency, robustness, and excellent conditioning of the proposed method.