The Rhie-Chow stabilized Box Method for the Stokes problem

TL;DR

The paper develops the Rhie-Chow stabilized Box Method (RCBM) for the steady Stokes problem by embedding Rhie-Chow stabilization into the Box Method on a Voronoi dual mesh, and provides both empirical convergence data in 2D/3D and a theoretical analysis of well-posedness and convergence under conjectured stabilization properties. It shows that the discrete problem is stable and convergent under a set of properties for the stabilization (consistency, continuity, coercivity, inf-sup), supported numerically and framed as conjectures. Numerical experiments with OpenFOAM confirm a linear-in-$h$ convergence rate for velocity in $H^1$ and pressure in $L^2$, and the theory links these results to a variational formulation that parallels FEM discretizations. The work enables a robust, locally conservative finite-volume-like scheme for Stokes (and potentially Navier–Stokes) on general polyhedral meshes and connects practical CFD implementations (OpenFOAM) with a solid variational framework.

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.

Figures (4)

• Figure 1: Example of a Delaunay triangulation and its Voronoi dual mesh.
• Figure 2: Scheme of dual mesh geometrical quantities.
• Figure 3: Representation of the 2D (left) and the, clipped, 3D (right) Voronoi grids employed in the numerical assessment of convergence properties of the BM.
• Figure 4: Convergence rates of the numerical error of BM solutions. On the left the 2D case errors and on the right the 3D case ones. Numbers are the rates computed using a Least Squares approximation on the log-log plot values.

Theorems & Definitions (30)

• Theorem 2.1: Well-posedness
• Remark 3.1
• Proposition 3.1: $*$-norm properties
• Remark 3.2
• Remark 4.1
• Remark 4.2
• Conjecture 4.1
• Conjecture 4.2: Generalized inf-sup for Rhie-Chow
• Lemma 6.1: $\widetilde{\mathcal{C}}_B$ consistency
• Lemma 6.2: Continuity of $\mathcal{C}_B$