A Lightweight, Geometrically Flexible Fast Algorithm for the Evaluation of Layer and Volume Potentials
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929504117587968 |
|---|---|
| author | Fryklund, Fredrik Greengard, Leslie Jiang, Shidong Potter, Samuel |
| author_facet | Fryklund, Fredrik Greengard, Leslie Jiang, Shidong Potter, Samuel |
| contents | Over the last two decades, several fast, robust, and high-order accurate methods have been developed for solving the Poisson equation in complicated geometry using potential theory. In this approach, rather than discretizing the partial differential equation itself, one first evaluates a volume integral to account for the source distribution within the domain, followed by solving a boundary integral equation to impose the specified boundary conditions. Here, we present a new fast algorithm which is easy to implement and compatible with virtually any discretization technique, including unstructured domain triangulations, such as those used in standard finite element or finite volume methods. Our approach combines earlier work on potential theory for the heat equation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT), and the dual-space multilevel kernel-splitting (DMK) framework. It is insensitive to flaws in the triangulation, permitting not just nonconforming elements, but arbitrary aspect ratio triangles, gaps and various other degeneracies. On a single CPU core, the scheme computes the solution at a rate comparable to that of the fast Fourier transform (FFT) in work per gridpoint. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Lightweight, Geometrically Flexible Fast Algorithm for the Evaluation of Layer and Volume Potentials Fryklund, Fredrik Greengard, Leslie Jiang, Shidong Potter, Samuel Numerical Analysis 31A10, 65E05, 35S30, 65R10, 45M05 Over the last two decades, several fast, robust, and high-order accurate methods have been developed for solving the Poisson equation in complicated geometry using potential theory. In this approach, rather than discretizing the partial differential equation itself, one first evaluates a volume integral to account for the source distribution within the domain, followed by solving a boundary integral equation to impose the specified boundary conditions. Here, we present a new fast algorithm which is easy to implement and compatible with virtually any discretization technique, including unstructured domain triangulations, such as those used in standard finite element or finite volume methods. Our approach combines earlier work on potential theory for the heat equation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT), and the dual-space multilevel kernel-splitting (DMK) framework. It is insensitive to flaws in the triangulation, permitting not just nonconforming elements, but arbitrary aspect ratio triangles, gaps and various other degeneracies. On a single CPU core, the scheme computes the solution at a rate comparable to that of the fast Fourier transform (FFT) in work per gridpoint. |
| title | A Lightweight, Geometrically Flexible Fast Algorithm for the Evaluation of Layer and Volume Potentials |
| topic | Numerical Analysis 31A10, 65E05, 35S30, 65R10, 45M05 |
| url | https://arxiv.org/abs/2409.11998 |