Multigrid Poisson Solver for Complex Geometries Using Finite Difference Method

TL;DR

The paper addresses solving Poisson/Laplace equations on complex geometries by mapping the physical domain to a uniform computational grid via a coordinate transformation, which induces a spatially varying anisotropic permittivity tensor on the grid. This enables the use of a standard, highly efficient Geometric Multigrid solver on the uniform grid, preserving second-order accuracy while avoiding expensive unstructured-mesh assembly. The authors validate the approach with analytical mappings and numerically generated grids, demonstrating second-order convergence and substantial speedups over traditional SOR, including applications to stretched grids and immersed boundaries. The technique offers a practical, scalable path to handle intricate geometries in electrostatic and flow problems, with clear avenues for extension to 3D, higher-order discretizations, and time-evolving domains.

Abstract

We present an efficient numerical method, inspired by transformation optics, for solving the Poisson equation in complex and arbitrarily shaped geometries. The approach operates by mapping the physical domain to a uniform computational domain through coordinate transformations, which can be applied either to the entire domain or selectively to specific boundaries inside the domain. This flexibility allows both homogeneous (Laplace equation) and inhomogeneous (Poisson equation) problems to be solved efficiently using iterative or fast direct solvers, with only the material parameters and source terms modified according to the transformation. The method is formulated within a finite difference framework, where the modified material properties are computed from the coordinate transformation equations, either analytically or numerically. This enables accurate treatment of arbitrary geometric shapes while retaining the simplicity of a uniform grid solver. Numerical experiments confirm that the method achieves second-order accuracy , and offers a straightforward pathway to integrate fast solvers such as multigrid methods on the uniform computational grid.

Figures (16)

• Figure 1: (a) Computational grid, (b) Physical grid , (c) $\varepsilon_{xx}$ , (d) $\varepsilon_{yy}$ .
• Figure 2: (a) Analytical solution, (b) Numerical solution, (c) Speed-up comparison of Multigrid and SOR method, (d) error analysis. The second order accuracy of the scheme can be seen from the subplot (d).
• Figure 3: Circular domain mapping: (a) Uniform computational grid and (b) transformed physical grid.
• Figure 4: Components of the permittivity tensor: (a) $\varepsilon_{xx}$, (b) $\varepsilon_{xy}$, (c) $\varepsilon_{yy}$, and (d) $\varepsilon_{yx}$.
• Figure 5: Results for the circular domain: (a) Analytical solution, (b) Numerical solution using GMG, (c) Runtime comparison of SOR and GMG methods, and (d) Grid convergence in $\mathrm{L_1}$, $\mathrm{L_2}$, and $\mathrm{L_{\infty}}$ norms.