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Bernstein-Bézier Galerkin-Characteristics Finite Element Method for Convection-Diffusion Problems

El-Amrani, Mofdi; El-Kacimi, Abdellah; Khouya, Bassou; Seaid, Mohammed

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Authors

Mofdi El-Amrani

Abdellah El-Kacimi

Bassou Khouya



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
Public URL https://durham-repository.worktribe.com/output/1196845

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Publisher Licence URL
http://creativecommons.org/licenses/by/4.0/

Copyright Statement
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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