A Green's Function-Based Enclosure Framework for Poisson's Equation and Generalized Sub- and Super-Solutions
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914283737055232 |
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| author | Tanaka, Kazuaki Iwanami, Ryoga Matsue, Kaname Ochiai, Hiroyuki |
| author_facet | Tanaka, Kazuaki Iwanami, Ryoga Matsue, Kaname Ochiai, Hiroyuki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19682 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Green's Function-Based Enclosure Framework for Poisson's Equation and Generalized Sub- and Super-Solutions Tanaka, Kazuaki Iwanami, Ryoga Matsue, Kaname Ochiai, Hiroyuki Numerical Analysis Analysis of PDEs 65N15, 65N80, 35J05 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. |
| title | A Green's Function-Based Enclosure Framework for Poisson's Equation and Generalized Sub- and Super-Solutions |
| topic | Numerical Analysis Analysis of PDEs 65N15, 65N80, 35J05 |
| url | https://arxiv.org/abs/2601.19682 |