Multigrid methods for the ghost finite element approximation of elliptic problems

TL;DR

This paper develops multigrid algorithms for solving elliptic PDEs on complex domains using a ghost finite element method, achieving high accuracy and computational efficiency with robust convergence.

Contribution

It introduces a multigrid framework tailored for ghost finite element discretizations, including explicit stabilization parameter derivation and cut cell smoothing techniques.

Findings

Achieves second-order accuracy on arbitrary domains.

Demonstrates robustness and scalability across various geometries.

Reduces computational cost with targeted smoothing on cut cells.

Abstract

We present multigrid methods for solving elliptic partial differential equations on arbitrary domains using the nodal ghost finite element method, an unfitted boundary approach where the domain is implicitly defined by a level-set function. This method achieves second-order accuracy and offers substantial computational advantages over both direct solvers and finite-difference-based multigrid methods. A key strength of the ghost finite element framework is its variational formulation, which naturally enables consistent transfer operators and avoids residual splitting across grid levels. We provide a detailed construction of the multigrid components in both one and two spatial dimensions, including smoothers, transfer operators, and coarse grid operators. The choice of the stabilization parameter plays a crucial role in ensuring well-posedness and optimal convergence of the multigrid method. We derive explicit algebraic expressions for this parameter based on the geometry of cut cells. In the two-dimensional setting, we further improve efficiency by performing additional smoothing exclusively on cut cells, reducing computational cost without compromising convergence. Numerical results validate the proposed method across a range of geometries and confirm its robustness and scalability.