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

TL;DR

This paper introduces a versatile, high-accuracy multigrid method for solving elliptic equations on complex two-dimensional domains using cut-cell techniques, achieving optimal complexity and robustness.

Contribution

The paper presents a novel fourth-order cut-cell multigrid method capable of handling arbitrary domain geometries and boundary conditions with high accuracy and efficiency.

Findings

Demonstrates fourth-order accuracy even with boundary discontinuities

Achieves optimal computational complexity of O(h^{-2})

Proves robustness and excellent conditioning across tests

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 ${\cal C}^1$ 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.