The Rhie-Chow stabilized Box Method for the Stokes problem

TL;DR

This paper introduces a stabilized finite volume method called Rhie-Chow stabilized Box Method for solving the Stokes problem, combining variational formulation, theoretical analysis, and numerical experiments to demonstrate its effectiveness.

Contribution

It presents a novel stabilization technique for the Box Method applied to the Stokes problem, with theoretical and numerical validation of its properties.

Findings

The method achieves stable and convergent solutions for 2D and 3D cases.

Theoretical analysis supports the well-posedness of the stabilized method.

Numerical experiments confirm the expected convergence rates.

Abstract

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.