Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multigrid methods for the ghost finite element approximation of elliptic problems

Hridya Dilip, Armando Coco

TL;DR

This work develops multigrid solvers for elliptic PDEs on arbitrary domains using the nodal ghost-FEM with level-set defined geometry and Nitsche-based Dirichlet enforcement. A key contribution is explicit, geometry-based stabilization parameter formulas that guarantee coercivity while preserving MG efficiency, including $\lambda(K)=\gamma C(K)$ and, in 1D, $C=1/(\theta_1 h)$ so that $\lambda=\gamma/(\theta_1 h)$. The authors show that residual transfers across grids need not rely on splitting the interior and boundary contributions in ghost-FEM, thanks to the variational formulation, and they demonstrate that coarse-grid operators can be obtained via Galerkin projection or direct coarse-grid assembly with identical results. In 2D, local per-cut-cell stabilization coupled with selective smoothing on cut cells yields optimal MG convergence ($\rho\approx0.1$) across diverse geometries, with W-cycles matching two-grid performance and substantial computational savings over direct solvers.

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.

Multigrid methods for the ghost finite element approximation of elliptic problems

TL;DR

This work develops multigrid solvers for elliptic PDEs on arbitrary domains using the nodal ghost-FEM with level-set defined geometry and Nitsche-based Dirichlet enforcement. A key contribution is explicit, geometry-based stabilization parameter formulas that guarantee coercivity while preserving MG efficiency, including and, in 1D, so that . The authors show that residual transfers across grids need not rely on splitting the interior and boundary contributions in ghost-FEM, thanks to the variational formulation, and they demonstrate that coarse-grid operators can be obtained via Galerkin projection or direct coarse-grid assembly with identical results. In 2D, local per-cut-cell stabilization coupled with selective smoothing on cut cells yields optimal MG convergence () across diverse geometries, with W-cycles matching two-grid performance and substantial computational savings over direct solvers.

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.
Paper Structure (18 sections, 1 theorem, 55 equations, 11 figures, 1 algorithm)

This paper contains 18 sections, 1 theorem, 55 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The ghost finite element method
  3. Multigrid method
  4. Two-grid correction scheme in 1D
  5. V-cycle and W-cycle
  6. One-dimensional case
  7. Smoother
  8. Transfer operators
  9. Brief overview of the Splitting Strategy in FDM
  10. Transfer operators in ghost-FEM
  11. Coarse grid operator
  12. Optimal stabilization parameter for MG efficiency
  13. Numerical tests in 1D
  14. Two-dimensional case
  15. Transfer operators
  16. ...and 3 more sections

Key Result

Proposition 1

In the context of ghost-fem, the splitting and the standard strategies are equivalent, namely $\mathbf{r}^\text{SPLIT}_{2h} = \mathbf{r}_{2h}$, where $\mathbf{r}^\text{SPLIT}_{2h}$ is given by eq:rsplit and $\mathbf{r}_{2h} = \mathcal{I}^h_{2h} \mathbf{r}_h$ corresponds to the standard (non-splittin

Figures (11)

  • Figure 1: Setup of a one-dimensional grid: boundary points and basis functions.
  • Figure 2: Plots of the convergence factor $\rho$ against the position of the Dirichlet boundary $\theta_1$ for different mesh sizes $h$ using the tgcs and W-cycle.
  • Figure 3: Plots of $C$ for a global generalized eigenvalue problem against the Dirichlet boundary configuration $\theta_1$ and mesh size $h$ on a log-log scale.
  • Figure 4: Different configurations of cut regions for 2D geometries.
  • Figure 5: 2D domains used for the numerical experiments.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Proposition