The Finite Volume method (FVM) is widely adopted in many different applications because of its built-in conservation properties, its ability to deal with arbitrary mesh and its computational efficiency. In this work, we consider the Rhie-Chow stabilized Box Method (RCBM) for the approximation of the Stokes problem. The Box Method (BM) is a piecewise linear Petrov-Galerkin formulation on the Voronoi dual mesh of a Delaunay triangulation, whereas the Rhie-Chow (RC) stabilization is a well known stabilization technique for FVM. The first part of the paper provides a variational formulation of the RC stabilization and discusses the validity of crucial properties relevant for the well-posedeness and convergence of RCBM. Moreover, a numerical exploration of the convergence properties of the method on 2D and 3D test cases is presented. The last part of the paper considers the theoretically justification of the well-posedeness of RCBM and the experimentally observed convergence rates. This latter justification hinges upon suitable assumptions, whose validity is numerically explored.

翻译：有限体积法因其固有的守恒特性、处理任意网格的能力以及计算高效性，被广泛应用于诸多领域。本文研究采用Rhie-Chow稳定化Box方法近似求解Stokes问题。Box方法是基于Delaunay三角剖分Voronoi对偶网格上的分片线性Petrov-Galerkin格式，而Rhie-Chow稳定化则是有限体积法中一种著名的稳定化技术。论文第一部分给出了Rhie-Chow稳定化的变分公式，并讨论了与RCBM适定性和收敛性相关的关键性质的正确性。此外，本文通过二维和三维测试算例对方法的收敛特性进行了数值探索。最后一部分论证了RCBM适定性的理论依据及实验观测到的收敛速率，该论证依赖于若干合适假设，且这些假设的有效性通过数值手段进行了验证。