A family of quadratic finite volume method for solving the Stokes equation. (1st July 2022)

• Record Type: Journal Article
• Authors: Zhang, Jiehua
• Is Part Of: Computers & mathematics with applications. Volume 117(2022)
• Page Start: 155
• Page End: 186
• Publication Date: 2022-07-01
• Subjects: Finite volume method -- Stokes equation -- Mapping -- Stability; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2022.04.014
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: This paper presents a family of stable quadratic finite volume methods for solving the Stokes equation over triangular meshes. We introduce a novel mapping, which connects trial spaces with test spaces of the finite volume methods, and impose a geometric condition on the triangular meshes of primary partitions to ensure the ellipticity of discrete elliptic bilinear forms in the finite volume methods. By using the abstract theories of generalized saddle-point problems, the stability (or inf-sup condition in general) of the proposed finite volume methods could be established. The resulting finite volume methods have the optimal H 1 -norm error estimate for velocity of the Stokes problem without constraints on dual meshes of dual partitions of the finite volume methods, and have the optimal L 2 -norm error estimate for velocity with a specially selected dual mesh of the dual partitions. In addition, by adopting another appropriate dual mesh of the dual partitions, the finite volume methods may be identified as the corresponding finite element methods with differences only in their right-hand items. As a result, the optimal-order error estimate could be obtained in the H 1 -norm for velocity without any restriction on mesh geometry of the primary partitions. Finally, we investigate numerically the imposed geometric conditions on meshes of the primary partitions for ensuring the ellipticity of the discrete elliptic bilinear forms and we validate our theoretical error estimates through some numerical experiments.