Multigrid methods for the ghost finite element approximation of elliptic problems (2505.05105v1)

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.