A $p$-adaptive treecode solution of the Poisson equation in the general domain
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909443925475328 |
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| author | Cui, Zixuan Yang, Lei |
| author_facet | Cui, Zixuan Yang, Lei |
| contents | Raising the order of the multipole expansion is a feasible approach for improving the accuracy of the treecode algorithm. However, a uniform order for the expansion would result in the inefficiency of the implementation, especially when the kernel function is singular. In this paper, a $p$-adaptive treecode algorithm is designed to resolve the efficiency issue for problems defined on a general domain. Such a $p$-adaptive implementation is realized through i). conducting a systematical error analysis for the treecode algorithm, ii). designing a strategy for a non-uniform distribution of the order of multipole expansion towards a given error tolerance, and iii). employing a hierarchy geometry tree structure for coding the algorithm. The proposed $p$-adaptive treecode algorithm is validated by a number of numerical experiments, from which the desired performance is observed successfully, i.e., the computational complexity is reduced dramatically compared with the uniform order case, making our algorithm a competitive one for bottleneck problems such as the demagnetizing field calculation in computational micromagnetics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20434 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A $p$-adaptive treecode solution of the Poisson equation in the general domain Cui, Zixuan Yang, Lei Numerical Analysis Raising the order of the multipole expansion is a feasible approach for improving the accuracy of the treecode algorithm. However, a uniform order for the expansion would result in the inefficiency of the implementation, especially when the kernel function is singular. In this paper, a $p$-adaptive treecode algorithm is designed to resolve the efficiency issue for problems defined on a general domain. Such a $p$-adaptive implementation is realized through i). conducting a systematical error analysis for the treecode algorithm, ii). designing a strategy for a non-uniform distribution of the order of multipole expansion towards a given error tolerance, and iii). employing a hierarchy geometry tree structure for coding the algorithm. The proposed $p$-adaptive treecode algorithm is validated by a number of numerical experiments, from which the desired performance is observed successfully, i.e., the computational complexity is reduced dramatically compared with the uniform order case, making our algorithm a competitive one for bottleneck problems such as the demagnetizing field calculation in computational micromagnetics. |
| title | A $p$-adaptive treecode solution of the Poisson equation in the general domain |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2412.20434 |