Mofdi El-Amrani
Bernstein-Bézier Galerkin-Characteristics Finite Element Method for Convection-Diffusion Problems
El-Amrani, Mofdi; El-Kacimi, Abdellah; Khouya, Bassou; Seaid, Mohammed
Abstract
A class of Bernstein-Bézier basis based high-order finite element methods is developed for the Galerkin-characteristics solution of convection-diffusion problems. The Galerkin-characteristics formulation is derived using a semi-Lagrangian discretization of the total derivative in the considered problems. The spatial discretization is performed using the finite element method on unstructured meshes. The Lagrangian interpretation in this approach greatly reduces the time truncation errors in the Eulerian methods. To achieve high-order accuracy in the Galerkin-characteristics solver, the semi-Lagrangian method requires high-order interpolating procedures. In the present work, this step is carried out using the Bernstein-Bézier basis functions to evaluate the solution at the departure points. Triangular Bernstein-Bézier patches are constructed in a simple and inherent manner over finite elements along the characteristics. An efficient preconditioned conjugate gradient solver is used for the linear systems of algebraic equations. Several numerical examples including advection-diffusion equations with known analytical solutions and the viscous Burgers problem are considered to illustrate the accuracy, robustness and performance of the proposed approach. The computed results support our expectations for a stable and highly accurate Bernstein-Bézier Galerkin-characteristics finite element method for convection-diffusion problems.
Citation
El-Amrani, M., El-Kacimi, A., Khouya, B., & Seaid, M. (2022). Bernstein-Bézier Galerkin-Characteristics Finite Element Method for Convection-Diffusion Problems. Journal of Scientific Computing, 92(2), Article 58. https://doi.org/10.1007/s10915-022-01888-7
| Journal Article Type | Article |
|---|---|
| Acceptance Date | Apr 25, 2022 |
| Online Publication Date | Jul 6, 2022 |
| Publication Date | 2022-08 |
| Deposit Date | Jul 27, 2022 |
| Publicly Available Date | Jul 27, 2022 |
| Journal | Journal of Scientific Computing |
| Print ISSN | 0885-7474 |
| Electronic ISSN | 1573-7691 |
| Publisher | Springer |
| Peer Reviewed | Peer Reviewed |
| Volume | 92 |
| Issue | 2 |
| Article Number | 58 |
| DOI | https://doi.org/10.1007/s10915-022-01888-7 |
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This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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