A Functionally Connected Element Method for Solving Boundary Value Problems
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866907826804228096 |
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| author | Yang, Jielin Dong, Suchuan |
| author_facet | Yang, Jielin Dong, Suchuan |
| contents | We present the general forms of piece-wise functions on partitioned domains satisfying an intrinsic $C^0$ or $C^1$ continuity across the sub-domain boundaries. These general forms are constructed based on a strategy stemming from the theory of functional connections, and we refer to partitioned domains endowed with these general forms as functionally connected elements (FCE). We further present a method, incorporating functionally connected elements and a least squares collocation approach, for solving boundary and initial value problems. This method exhibits a spectral-like accuracy, with the free functions involved in the FCE form represented by polynomial bases or by non-polynomial bases of quasi-random sinusoidal functions. The FCE method offers a unique advantage over traditional element-based methods for boundary value problems involving relative boundary conditions. A number of linear and nonlinear numerical examples in one and two dimensions are presented to demonstrate the performance of the FCE method developed herein. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06393 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Functionally Connected Element Method for Solving Boundary Value Problems Yang, Jielin Dong, Suchuan Numerical Analysis Fluid Dynamics We present the general forms of piece-wise functions on partitioned domains satisfying an intrinsic $C^0$ or $C^1$ continuity across the sub-domain boundaries. These general forms are constructed based on a strategy stemming from the theory of functional connections, and we refer to partitioned domains endowed with these general forms as functionally connected elements (FCE). We further present a method, incorporating functionally connected elements and a least squares collocation approach, for solving boundary and initial value problems. This method exhibits a spectral-like accuracy, with the free functions involved in the FCE form represented by polynomial bases or by non-polynomial bases of quasi-random sinusoidal functions. The FCE method offers a unique advantage over traditional element-based methods for boundary value problems involving relative boundary conditions. A number of linear and nonlinear numerical examples in one and two dimensions are presented to demonstrate the performance of the FCE method developed herein. |
| title | A Functionally Connected Element Method for Solving Boundary Value Problems |
| topic | Numerical Analysis Fluid Dynamics |
| url | https://arxiv.org/abs/2403.06393 |