A family of quadratic finite volume method for solving the Stokes equation

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 H1-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 L2-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 H1-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.