Fourier-based potential theory without an explicit Green's function
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910125148602368 |
|---|---|
| author | Fryklund, Fredrik |
| author_facet | Fryklund, Fredrik |
| contents | Integral equation methods provide an effective framework for solving partial differential equations, but their applicability typically relies on the availability of explicit free-space Green's functions. For coupled systems arising in multiphysics applications, such Green's functions are generally not available, limiting the scope of classical potential theory-based approaches. In this work, we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regularization of the symbol, which yields a decomposition of the solution into a smooth, nonlocal component and a spatially localized residual. For the localized component, we derive explicit asymptotic expansions for volume, single layer, and double layer potentials in powers of a length scale parameter $\varepsilon$. The coefficients are expressed in terms of local geometric quantities and derivatives of the source data. The derivation is carried out entirely in the Fourier domain and applies to the Poisson equation in two and three dimensions, as well as to a class of coupled strongly elliptic systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11436 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fourier-based potential theory without an explicit Green's function Fryklund, Fredrik Analysis of PDEs Numerical Analysis 31A10, 65E05, 35S30, 65R10, 45M05 Integral equation methods provide an effective framework for solving partial differential equations, but their applicability typically relies on the availability of explicit free-space Green's functions. For coupled systems arising in multiphysics applications, such Green's functions are generally not available, limiting the scope of classical potential theory-based approaches. In this work, we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regularization of the symbol, which yields a decomposition of the solution into a smooth, nonlocal component and a spatially localized residual. For the localized component, we derive explicit asymptotic expansions for volume, single layer, and double layer potentials in powers of a length scale parameter $\varepsilon$. The coefficients are expressed in terms of local geometric quantities and derivatives of the source data. The derivation is carried out entirely in the Fourier domain and applies to the Poisson equation in two and three dimensions, as well as to a class of coupled strongly elliptic systems. |
| title | Fourier-based potential theory without an explicit Green's function |
| topic | Analysis of PDEs Numerical Analysis 31A10, 65E05, 35S30, 65R10, 45M05 |
| url | https://arxiv.org/abs/2604.11436 |