A fourth-order, multigrid cut-cell method for solving Poisson's equation in three-dimensional irregular domains

TL;DR

This paper introduces a high-order, multigrid cut-cell method for efficiently solving Poisson's equations in complex 3D irregular domains, combining advanced discretization, geometry handling, and optimal solver complexity.

Contribution

It presents a novel fourth-order cut-cell method with a new lattice generation algorithm and an optimized multigrid solver for irregular 3D geometries.

Findings

Achieves high-order accuracy in complex geometries

Maintains optimal computational complexity

Demonstrates effectiveness through numerical experiments

Abstract

We propose a fourth-order cut-cell method for solving Poisson's equations in three-dimensional irregular domains. Major distinguishing features of our method include (a) applicable to arbitrarily complex geometries, (b) high order discretization, (c) optimal complexity. Feature (a) is achieved by Yin space, which is a mathematical model for three-dimensional continua. Feature (b) is accomplished by poised lattice generation (PLG) algorithm, which finds stencils near the irregular boundary for polynomial fitting. Besides, for feature (c), we design a modified multigrid solver whose complexity is theoretically optimal by applying nested dissection (ND) ordering method.