Conforming P3 Divergence-Free Finite Elements for the Stokes Equations on Subquadrilateral Triangular Meshes

doi:10.1007/s42967-023-00335-0

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (2): 426-441.doi: 10.1007/s42967-023-00335-0

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Conforming P₃ Divergence-Free Finite Elements for the Stokes Equations on Subquadrilateral Triangular Meshes

Shangyou Zhang

Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

收稿日期:2023-03-03 修回日期:2023-08-02 接受日期:2023-10-03 出版日期:2025-06-20 发布日期:2025-04-21
通讯作者: Shangyou Zhang,szhang@udel.edu E-mail:szhang@udel.edu

Conforming P₃ Divergence-Free Finite Elements for the Stokes Equations on Subquadrilateral Triangular Meshes

Shangyou Zhang

Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

Received:2023-03-03 Revised:2023-08-02 Accepted:2023-10-03 Online:2025-06-20 Published:2025-04-21

摘要/Abstract

摘要： The continuous P₃ and discontinuous P₂ finite element pair is stable on subquadrilateral triangular meshes for solving 2D stationary Stokes equations. By putting two diagonal lines into every quadrilateral of a quadrilateral mesh, we get a subquadrilateral triangular mesh. Such a velocity solution is divergence-free point wise and viscosity robust in the sense the solution and the error are independent of the viscosity. Numerical examples show an advantage of such a method over the Taylor-Hood P₃-P₂ method, where the latter deteriorates when the viscosity becomes small.

关键词: Divergence-free, Stokes equations, Finite element, Triangular mesh, Quadrilateral mesh

Abstract: The continuous P₃ and discontinuous P₂ finite element pair is stable on subquadrilateral triangular meshes for solving 2D stationary Stokes equations. By putting two diagonal lines into every quadrilateral of a quadrilateral mesh, we get a subquadrilateral triangular mesh. Such a velocity solution is divergence-free point wise and viscosity robust in the sense the solution and the error are independent of the viscosity. Numerical examples show an advantage of such a method over the Taylor-Hood P₃-P₂ method, where the latter deteriorates when the viscosity becomes small.

Key words: Divergence-free, Stokes equations, Finite element, Triangular mesh, Quadrilateral mesh

中图分类号:

65N15
65N30

Shangyou Zhang. Conforming P₃ Divergence-Free Finite Elements for the Stokes Equations on Subquadrilateral Triangular Meshes[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 426-441.

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Conforming P₃ Divergence-Free Finite Elements for the Stokes Equations on Subquadrilateral Triangular Meshes

Conforming P₃ Divergence-Free Finite Elements for the Stokes Equations on Subquadrilateral Triangular Meshes

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