A Fourth-Order Cut-cell Multigrid Method for Generic 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 all forms of elliptic equations 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 ${\cal C}^1$ 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.