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

TL;DR

This paper introduces a Green's function-based framework for accurately enclosing solutions of Poisson's equation, overcoming limitations of traditional methods especially in complex geometries and with discontinuous sources.

Contribution

It develops the concept of Green-representable solutions, enabling rigorous pointwise enclosures for Poisson's equation in challenging domains.

Findings

Provides explicit test function constructions for 1D problems.

Employs the Method of Fundamental Solutions for 2D polygonal domains.

Demonstrates strict, accurate enclosures even with singularities and discontinuities.

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.