Tetrahedral quadratic finite volume method schemes for the Stokes equation

A family of quadratic finite volume methods is proposed in this paper for solving the Stokes equation over three-dimensional tetrahedral meshes, where the velocity is approximated by continuous piecewise Lagrange quadratic polynomials while the pressure is approximated by continuous piecewise linear polynomials on the same meshes. By introducing a map with a non-zero coefficient, who connects the trial space with the test space of the finite volume methods, an equivalence relationship is founded between the traditional finite volume method schemes, the classical finite volume method schemes, and the particular finite volume method schemes. By analyzing the affine matrix induced by tetrahedral meshes and establishing the equivalent discrete norms over tetrahedral meshes, it is discovered that the stability of the finite volume method schemes relies on the geometric shape conditions of tetrahedra. Under certain constraints on the geometric shape requirements, the stability of the finite volume method schemes is certificated by the Lax Milgram theorem of Babuska's generalization. Based on the stability, when selecting the dual partitions of tetrahedrons that satisfies the so-called orthogonal conditions, the Aubin-Nitsche technique is applied to derive the error estimates of the optimal L2-norm with regard to the velocity. Finally, some numerical tests are presented to demonstrate the accuracy and efficiency for the proposed methods.