An unfitted hybridizable discontinuous Galerkin method for the Poisson interface problem and its error analysis
In this article, we present and analyse an unfitted mesh method for the Poisson interface problem. By constructing a novel ansatz function in the vicinity of the interface, we are able to derive an extended Poisson problem whose interface fits a given quasi-uniform triangular mesh. Then we adopt a h...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2017
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/85561 http://hdl.handle.net/10220/43727 |
| _version_ | 1826115013975212032 |
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| author | Dong, Haixia Wang, Bo Xie, Ziqing Wang, Li-Lian |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Dong, Haixia Wang, Bo Xie, Ziqing Wang, Li-Lian |
| author_sort | Dong, Haixia |
| collection | NTU |
| description | In this article, we present and analyse an unfitted mesh method for the Poisson interface problem. By constructing a novel ansatz function in the vicinity of the interface, we are able to derive an extended Poisson problem whose interface fits a given quasi-uniform triangular mesh. Then we adopt a hybridizable discontinuous Galerkin method to solve the extended problem with an appropriate choice of flux for treating the jump conditions. In contrast with existing approaches, the ansatz function is designed through a delicate piecewise quadratic Hermite polynomial interpolation with a post-processing via a standard Lagrange polynomial interpolation. Such an explicit function offers a third-order approximation to the singular part of the underlying solution for interfaces of any shape. It is also essential for both stability and convergence of the proposed method. Moreover, we provide rigorous error analysis to show that the scheme can achieve a second-order convergence rate for the approximation of the solution and its gradient. Ample numerical examples with complex interfaces demonstrate the expected convergence order and robustness of the method. |
| first_indexed | 2024-10-01T03:48:36Z |
| format | Journal Article |
| id | ntu-10356/85561 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T03:48:36Z |
| publishDate | 2017 |
| record_format | dspace |
| spelling | ntu-10356/855612020-03-07T12:31:32Z An unfitted hybridizable discontinuous Galerkin method for the Poisson interface problem and its error analysis Dong, Haixia Wang, Bo Xie, Ziqing Wang, Li-Lian School of Physical and Mathematical Sciences Hybridizable Discontinuous Galerkin Method Poisson Interface Equation In this article, we present and analyse an unfitted mesh method for the Poisson interface problem. By constructing a novel ansatz function in the vicinity of the interface, we are able to derive an extended Poisson problem whose interface fits a given quasi-uniform triangular mesh. Then we adopt a hybridizable discontinuous Galerkin method to solve the extended problem with an appropriate choice of flux for treating the jump conditions. In contrast with existing approaches, the ansatz function is designed through a delicate piecewise quadratic Hermite polynomial interpolation with a post-processing via a standard Lagrange polynomial interpolation. Such an explicit function offers a third-order approximation to the singular part of the underlying solution for interfaces of any shape. It is also essential for both stability and convergence of the proposed method. Moreover, we provide rigorous error analysis to show that the scheme can achieve a second-order convergence rate for the approximation of the solution and its gradient. Ample numerical examples with complex interfaces demonstrate the expected convergence order and robustness of the method. MOE (Min. of Education, S’pore) 2017-09-12T05:40:12Z 2019-12-06T16:06:04Z 2017-09-12T05:40:12Z 2019-12-06T16:06:04Z 2017 Journal Article Dong, H., Wang, B., Xie, Z., & Wang, L.-L. (2017). An unfitted hybridizable discontinuous Galerkin method for the Poisson interface problem and its error analysis. IMA Journal of Numerical Analysis, 37(1), 444-476. 0272-4979 https://hdl.handle.net/10356/85561 http://hdl.handle.net/10220/43727 10.1093/imanum/drv071 en IMA Journal of Numerical Analysis © 2016 The Author(s) (published by Oxford University Press on behalf of the Institute of Mathematics and its Applications). |
| spellingShingle | Hybridizable Discontinuous Galerkin Method Poisson Interface Equation Dong, Haixia Wang, Bo Xie, Ziqing Wang, Li-Lian An unfitted hybridizable discontinuous Galerkin method for the Poisson interface problem and its error analysis |
| title | An unfitted hybridizable discontinuous Galerkin method for the Poisson interface problem and its error analysis |
| title_full | An unfitted hybridizable discontinuous Galerkin method for the Poisson interface problem and its error analysis |
| title_fullStr | An unfitted hybridizable discontinuous Galerkin method for the Poisson interface problem and its error analysis |
| title_full_unstemmed | An unfitted hybridizable discontinuous Galerkin method for the Poisson interface problem and its error analysis |
| title_short | An unfitted hybridizable discontinuous Galerkin method for the Poisson interface problem and its error analysis |
| title_sort | unfitted hybridizable discontinuous galerkin method for the poisson interface problem and its error analysis |
| topic | Hybridizable Discontinuous Galerkin Method Poisson Interface Equation |
| url | https://hdl.handle.net/10356/85561 http://hdl.handle.net/10220/43727 |
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