A comparison of the Coco-Russo scheme and $\protect\mathghost$-FEM for elliptic equations in arbitrary domains

TL;DR

This paper compares two unfitted approaches for the Poisson equation with mixed boundary conditions on arbitrary domains: the Coco-Russo finite-difference scheme and the ghost-nodal finite-element method ($\mathghost$-FEM). By combining analytical results and extensive numerical experiments on multiple geometries, it benchmarks accuracy, conditioning, and solver performance, highlighting that Coco-Russo attains strong conditioning and second-order boundary interpolation via a 9-point stencil, while $\mathghost$-FEM offers robust $L^2$ convergence but requires care with the penalty parameter and small-cut phenomena. The study demonstrates how a snapping strategy and tailored solvers can mitigate conditioning issues in unfitted methods, providing practical guidance for method selection in complex geometries. Overall, the work advances the understanding of unfitted schemes for elliptic problems and clarifies the trade-offs between FD and FEM formulations in arbitrary domains.

Abstract

In this paper, a comparative study between the Coco-Russo scheme (based on finite-difference scheme) and the $\mathghost$-FEM (based on finite-element method) is presented when solving the Poisson equation in arbitrary domains. The comparison between the two numerical methods is carried out by presenting analytical results from the literature \cite{cocoStissi,astuto2024nodal}, together with numerical tests in various geometries and boundary conditions.

Figures (14)

• Figure 1: (a): Representation of the domain $\Omega\subset R$ and of the normal vector $\widehat{n}$ to the boundary $\Gamma$. The classification is the following: internal points (blue points), ghost points (red circles), and inactive points (small black dots). Furthermore, we show the different distribution of ghost points for the Coco-Russo scheme in panel (b) and for the $\mathghost$-FEM in (c).
• Figure 2: We show the nine-point stencil for the interpolation operator for Coco-Russo scheme. G (red circle) is the ghost point, B (black square) is the closest point to G that belongs to $\Gamma$ and the eight blue points complete the nine-point stencil. The empty black circles are the points of interpolation in x direction and the empty black squares the ones in y direction. The two quantities $\vartheta_x$ and $\vartheta_y$ are defined in Eq.\ref{['expr_theta']}.
• Figure 3: (a): Ghost value extrapolation would be ill-conditioned for the external grid point $F$. However, $F$ is not a ghost point of the finite-difference method. The extrapolation of the ghost value $G$ would be ill-conditioned if the boundary condition is enforced to the orange boundary point and a linear interpolation stencil (orange dashed rectangle) is used, as in Gibou2002. The orthogonal projection and the bilinear interpolation stencil (green dashed rectangle) adopted in the Coco-Russo method improves the conditioning. (b): The ill-conditioned extrapolation of the ghost value $G$ is overcome by enlarging the interpolation stencil (green squares).
• Figure 4: Shape of the domains considered in our tests. We have the circular (a), the rotated leaf- (b), the flower- (c) and the hourglass-shaped domain (d).
• Figure 5: Comparison of the error behavior and of the conditioning number between the two different numerical methods, for the circular domain and Dirichlet boundary conditions: relative error of the numerical solutions (a), of the gradient (b) and conditioning number of the linear systems in (c); snapping exponent $\alpha = 2$.