A Finite Difference Ghost-Cell Multigrid Approach for Poisson Equation with Mixed Boundary Conditions in Arbitrary Domain

TL;DR

This paper introduces a multigrid method using finite difference and ghost-cell techniques to efficiently solve the Poisson equation with mixed boundary conditions in arbitrary domains, ensuring good convergence.

Contribution

The paper develops a novel restriction operator and boundary smoothing strategy for multigrid methods in arbitrary domains with mixed boundary conditions.

Findings

Good convergence confirmed by numerical tests

Effective handling of boundary conditions in arbitrary domains

Applicable to general problems with non-eliminated boundary conditions

Abstract

In this paper we present a multigrid approach to solve the Poisson equation in arbitrary domain (identified by a level set function) and mixed boundary conditions. The discretization is based on finite difference scheme and ghost-cell method. This multigrid strategy can be applied also to more general problems where a non-eliminated boundary condition approach is used. Arbitrary domain make the definition of the restriction operator for boundary conditions hard to find. A suitable restriction operator is provided in this work, together with a proper treatment of the boundary smoothing, in order to avoid degradation of the convergence factor of the multigrid due to boundary effects. Several numerical tests confirm the good convergence property of the new method.