Numerical Solution of Laplace and Poisson Equations Using Finite Difference Grid Methods

Abstract

The Laplace and Poisson equations form the basis for an important area of computational physics and engineering, with massive applications in electrostatics, heat conduction, and fluid flow. Classical Finite Difference Methods (FDM) are very efficient on structured grids, but suffer from low accuracy and flexibly geometric on the irregular domain. In this paper, we propose a unified high-order finite-volume method (FDM) framework that unifies Lagrange interpolation for arbitrary-order discretization without loss of generality and neural-network-assisted coefficient prediction that overcomes the mesh distortion limits. Dirichlet and Neumann boundary problems are used to thoroughly verify the method in both 2D and 3D benchmarks. Maximum error is reduced by two orders of magnitude at equal grid resolutions, with fourth-order convergence results shown. A neural-network component facilitates the ability to operate robustly, quasi-free of a mesh, and with accuracy, on very severely perturbed grids. In addition, due to the much lower degrees of freedom for equivalent level of accuracy, the high-order schemes achieve a 16-fold reduction in computational cost. Our work reconciles traditional simplicity of classical FDM with modern demands for geometric flexibility and automation, and provides a robust and efficient solver End2End with potential applications to arbitrary real-world elliptic PDEs found in the science and engineering applications.