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

Qing Xia

TL;DR

This work addresses efficient 3D elliptic PDEs on complex geometries by marrying the Difference Potentials Method with lattice Green's functions, forming an unfitted Boundary Algebraic Equation framework. By replacing finite auxiliary domains with free-space LGFs, the method achieves matrix-free, boundary-reduced computations and natural handling of unbounded domains while preserving accuracy. The authors establish equivalence between direct DP-based and indirect BAE formulations, analyze spectral properties, and show that double-layer formulations yield better conditioning for iterative solvers. Numerical experiments solve Poisson and modified Helmholtz equations on implicit 3D geometries, demonstrating robust second-order convergence and scalable performance, indicating strong potential for unbounded-domain applications and extensions to Helmholtz, Stokes, and high-order schemes.

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.

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

TL;DR

This work addresses efficient 3D elliptic PDEs on complex geometries by marrying the Difference Potentials Method with lattice Green's functions, forming an unfitted Boundary Algebraic Equation framework. By replacing finite auxiliary domains with free-space LGFs, the method achieves matrix-free, boundary-reduced computations and natural handling of unbounded domains while preserving accuracy. The authors establish equivalence between direct DP-based and indirect BAE formulations, analyze spectral properties, and show that double-layer formulations yield better conditioning for iterative solvers. Numerical experiments solve Poisson and modified Helmholtz equations on implicit 3D geometries, demonstrating robust second-order convergence and scalable performance, indicating strong potential for unbounded-domain applications and extensions to Helmholtz, Stokes, and high-order schemes.

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.

Paper Structure

This paper contains 16 sections, 12 theorems, 88 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Difference Potentials Method
  3. Difference Potentials in the finite domain
  4. Difference Potentials in the infinite domain
  5. Boundary Algebraic Equations
  6. Direct formulation
  7. Indirect formulation
  8. Spectral property
  9. Boundary equations
  10. Schur complement
  11. Linear solve
  12. Outline of the developed unfitted BAE
  13. Numerical results
  14. Poisson's equation
  15. Modified Helmholtz equation
  16. ...and 1 more sections

Key Result

Theorem 1

The discrete equation eqn:discrete for points in $M^+$ is equivalent to the following boundary equation with projections: where $P_\gamma u_\gamma:=Tr_{\gamma}G_h[\chi_{M^-}L_hu_\gamma]$ and $G_hf_\gamma:=Tr_\gamma G_h[\chi_{M^+}f_h]$ and $Tr_{\gamma}$ denotes the trace operator on the set $\gamma$, $\chi_{M^-}$ denotes the characteristic function for the set $M^-$.

Figures (5)

  • Figure 1: Example of point sets in 2D (dot: $\gamma_{+}$, circle: $\gamma_-$)
  • Figure 2: Relative residual of GMRES using double layer formulation for Poisson equations
  • Figure 3: Numerical solution and pointwise errors for Poisson equation
  • Figure 4: Relative residual of GMRES using double layer formulation for modified Helmholtz equations
  • Figure 5: Numerical solution and point errors for modified Helmholtz equation

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Theorem 4
  • Theorem 5
  • proof
  • ...and 16 more