Unfitted boundary algebraic equation method based on difference potentials and lattice Green's function in 3D

TL;DR

This paper introduces an unfitted boundary algebraic equation method leveraging difference potentials and lattice Green's functions for efficient 3D elliptic PDE solutions on complex geometries, enabling matrix-free computation and improved conditioning.

Contribution

It develops a novel unfitted boundary algebraic equation approach using free-space lattice Green's functions, with theoretical analysis and validation demonstrating efficiency and optimal convergence.

Findings

Matrix-free implementation reduces computational cost.

Double layer formulations offer better conditioning for iterative solvers.

Method achieves optimal convergence rates in 3D complex geometries.

Abstract

This work presents an unfitted boundary algebraic equation (BAE) method for solving three-dimensional elliptic partial differential equations on complex geometries using finite difference on structured meshes. We demonstrate that replacing finite auxiliary domains with free-space LGFs streamlines the computation of difference potentials, enabling matrix-free implementations and significant cost reductions. We establish theoretical foundations by showing the equivalence between direct formulations in difference potentials framework and indirect single/double layer formulations and analyzing their spectral properties. The spectral analysis demonstrates that discrete double layer formulations provide better-conditioned systems for iterative solvers, similarly as in boundary integral method. The method is validated through matrix-free numerical experiments on both Poisson and modified Helmholtz equations in 3D implicitly defined geometries, showing optimal convergence rates and computational efficiency. This framework naturally extends to unbounded domains and provides a foundation for applications to more complex systems like Helmholtz and Stokes equations.