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Numerical solution of the boundary value problem of elliptic equation by Levi function scheme

Jinchao Pan, Jijun Liu

TL;DR

This work addresses solving elliptic PDEs with variable coefficients using Levi-function representations by introducing a modified volume-volume and surface integral formulation restricted to an interior subdomain $\Omega_{\epsilon_0}$ to ensure well-posedness. It proves existence and uniqueness of the density pair in both continuous and $L^2$ spaces, and develops two numerical strategies: an adaptive self-discretization (ADS) that enforces uniform accuracy across the domain and a dual reciprocity method (DRM) that recasts volume potentials as boundary integrals, enabling mesh-free computation. Numerical experiments in 2D and 3D demonstrate accurate solutions at interior points with competitive costs compared to FEM, validating the Levi-function based approach for inhomogeneous media. The methods offer robust, scalable tools for efficiently solving variable-coefficient elliptic problems through boundary-domain integral representations.

Abstract

For boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the Levi function can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the Levi function framework. The well-posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the Levi function, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme (ADS) for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method (DRM) which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2-dimensional and 3-dimensional cases are presented to show the validity of the proposed schemes.

Numerical solution of the boundary value problem of elliptic equation by Levi function scheme

TL;DR

This work addresses solving elliptic PDEs with variable coefficients using Levi-function representations by introducing a modified volume-volume and surface integral formulation restricted to an interior subdomain to ensure well-posedness. It proves existence and uniqueness of the density pair in both continuous and spaces, and develops two numerical strategies: an adaptive self-discretization (ADS) that enforces uniform accuracy across the domain and a dual reciprocity method (DRM) that recasts volume potentials as boundary integrals, enabling mesh-free computation. Numerical experiments in 2D and 3D demonstrate accurate solutions at interior points with competitive costs compared to FEM, validating the Levi-function based approach for inhomogeneous media. The methods offer robust, scalable tools for efficiently solving variable-coefficient elliptic problems through boundary-domain integral representations.

Abstract

For boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the Levi function can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the Levi function framework. The well-posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the Levi function, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme (ADS) for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method (DRM) which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2-dimensional and 3-dimensional cases are presented to show the validity of the proposed schemes.
Paper Structure (4 sections, 2 theorems, 92 equations, 3 figures, 1 table)

This paper contains 4 sections, 2 theorems, 92 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Integral representation of the solution
  3. Self-adaptive discretization for the volume potential
  4. The dual reciprocity method

Key Result

Theorem 2.1

Assume that the specified conductivity in liu11-01 meets $0<\sigma_0\le \sigma\in C^1(\overline\Omega)$ and Then for any $(F,f)\in C(\overline\Omega)\times C(\partial\Omega)$, there exists a unique solution $(\tilde{\mu},\tilde{\psi})\in C(\overline\Omega_{\epsilon_0})\times C(\partial\Omega)$ to liu22-07, where $\Omega_{\epsilon_0}:=\{x\in\Omega:\; \hbox{dist}(x,\partial\Omega)\ge\epsilon_0\}$ w

Figures (3)

  • Figure 1: $\Pi_i$ is transformed as $\Omega_i$ by $\Omega_i=\tilde{p}(\Pi_i)$.
  • Figure 2: Refinement of the interval $[0,1]$ as $2N$ subintervals and reordering the partition points.
  • Figure 3: The adaptive partition scheme at each layer.

Theorems & Definitions (6)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 3.1