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.