A Bernstein–Bézier Lagrange–Galerkin method for three-dimensional advection-dominated problems

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

Abstract

We present a high-order Bernstein–Bézier finite element discretization to accurately solve three-dimensional advection-dominated problems on unstructured tetrahedral meshes. The key idea consists of implementing a modified method of characteristics to discretize the advection terms in a Bernstein–Bésier finite element framework. The proposed Bernstein–Bézier Lagrange–Galerkin method has been designed so that the Courant–Friedrichs–Lewy condition is strongly relaxed using semi-Lagrangian time discretization. A low complexity procedures in building finite element matrices and load vectors is also achieved in the present work by both the analytical rule and the sum factorization method using the tensorial feature of Bernstein polynomials. 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 Lagrange–Galerkin finite element method for three-dimensional advection-dominated problems.

Citation

El-Amrani, M., Kacimi, A. E., Khouya, B., & Seaid, M. (2023). A Bernstein–Bézier Lagrange–Galerkin method for three-dimensional advection-dominated problems. Computer Methods in Applied Mechanics and Engineering, 403, Article 115758. https://doi.org/10.1016/j.cma.2022.115758

• Journal Article Type: Article
• Acceptance Date: Oct 29, 2022
• Online Publication Date: Nov 15, 2022
• Publication Date: Jan 1, 2023
• Deposit Date: Apr 5, 2023
• Journal: Computer Methods in Applied Mechanics and Engineering
• Print ISSN: 0045-7825
• Electronic ISSN: 1879-2138
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 403
• Article Number: 115758
• DOI: https://doi.org/10.1016/j.cma.2022.115758
• Public URL: https://durham-repository.worktribe.com/output/1175376