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A Green's Function-Based Enclosure Framework for Poisson's Equation and Generalized Sub- and Super-Solutions

Kazuaki Tanaka, Ryoga Iwanami, Kaname Matsue, Hiroyuki Ochiai

Abstract

This paper presents a novel framework for enclosing solutions of Poisson's equation based on generalized sub- and super-solutions constructed using fundamental solutions. The conventional definition of sub- and super-solutions based on variational inequalities often fails for natural function classes such as piecewise linear functions and encounters theoretical difficulties in non-convex polygonal domains, where H^2 regularity is lost because of corner singularities. To overcome these limitations, we introduce the concept of ``Green-representable solutions'' utilizing test functions constructed from fundamental solutions. This framework enables a new formulation of sub- and super-solutions that permits rigorous pointwise evaluation. For one-dimensional problems, we derive explicit constructions of the test functions. For two-dimensional polygonal domains, we employ the Method of Fundamental Solutions to generate test functions. The approach is validated through numerical experiments in both settings, including non-convex polygons. The results demonstrate that the proposed method yields strict and accurate pointwise enclosures of the true solution, even for problems with discontinuous source terms or geometric singularities.

A Green's Function-Based Enclosure Framework for Poisson's Equation and Generalized Sub- and Super-Solutions

Abstract

This paper presents a novel framework for enclosing solutions of Poisson's equation based on generalized sub- and super-solutions constructed using fundamental solutions. The conventional definition of sub- and super-solutions based on variational inequalities often fails for natural function classes such as piecewise linear functions and encounters theoretical difficulties in non-convex polygonal domains, where H^2 regularity is lost because of corner singularities. To overcome these limitations, we introduce the concept of ``Green-representable solutions'' utilizing test functions constructed from fundamental solutions. This framework enables a new formulation of sub- and super-solutions that permits rigorous pointwise evaluation. For one-dimensional problems, we derive explicit constructions of the test functions. For two-dimensional polygonal domains, we employ the Method of Fundamental Solutions to generate test functions. The approach is validated through numerical experiments in both settings, including non-convex polygons. The results demonstrate that the proposed method yields strict and accurate pointwise enclosures of the true solution, even for problems with discontinuous source terms or geometric singularities.
Paper Structure (25 sections, 13 theorems, 95 equations, 19 figures, 3 tables, 2 algorithms)

This paper contains 25 sections, 13 theorems, 95 equations, 19 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Solution Representation via fundamental Solutions
  3. Fundamental solutions and integrability conditions
  4. Construction of test functions
  5. Local Green-representability
  6. Global Green-representability
  7. Globally Green-representable solutions in special cases
  8. Green-representable sub- and super-solutions
  9. Definitions of Green-representable sub- and super-solutions
  10. Comparison theorem
  11. Practical implications and boundary conditions
  12. Application to One-dimensional Problems
  13. Solution representation and test function construction
  14. Green-representable sub- and super-solutions
  15. Numerical experiments
  16. ...and 10 more sections

Key Result

Lemma 2.2

Under Assumption assum:f-integrability, the pairing $\langle f,\Gamma_{s_{\rm int}} \rangle$ defined in eq:coupling is well-defined and finite for any $s_{\rm int} \in \Omega$.

Figures (19)

  • Figure 1: Linear functions do not satisfy the super-solution condition. Blue: true solution of \ref{['eq:poisson-strong']} when $f=1$; Green: $\phi(x)=-x(x-0.5)^2(x-1)$; Red: piecewise linear function \ref{['eq:counterexample_supersol']}
  • Figure 2: Enclosure error versus mesh size $h$ for various $c$. Left: $f=1$, right: $f=5$.
  • Figure 3: Error behavior as a function of mesh size $h$ for different values of $C = c/h^2$. Left: $f=1$, right: $f=5$.
  • Figure 4: $h=2^{-3}$
  • Figure 5: $h=2^{-4}$
  • ...and 14 more figures

Theorems & Definitions (28)

  • Lemma 2.2: Integrability of the pairing
  • proof
  • Lemma 2.4: Interpretation of the coupling as a Lebesgue integral
  • proof
  • Definition 2.5: Weak Neumann trace
  • Definition 2.6: Local Green-representability
  • Theorem 2.7: Local Green-representability for regular data
  • proof
  • Remark 2.8
  • Theorem 2.9
  • ...and 18 more