On MAC schemes on triangular delaunay meshes, their convergence and application to coupled flow problems

Abstract

We study the convergence of two generalized marker-and-cell covolume schemes for the incompressible Stokes and Navier-Stokes equations introduced by Cavendish, Hall, Nicolaides, and Porsching. The schemes are defined on unstructured triangular Delaunay meshes and exploit the Delaunay-Voronoi duality. The study is motivated by the fact that the related discrete incompressibility condition allows to obtain a discrete maximum principle for the finite volume solution of an advection-diffusion problem coupled to the flow.The convergence theory uses discrete functional analysis and compactness arguments based on recent results for finite volume discretizations for the biharmonic equation. For both schemes, we prove the strong convergence in L 2 for the velocities and the discrete rotations of the velocities for the Stokes and the Navier-Stokes problem. Further, for one of the schemes, we also prove the strong convergence of the pressure in L 2 .These predictions are confirmed by numerical examples presented in the article.

Stable and convergent approximation of two-dimensional vector fields on unstructured meshes

“…The same issue is not discussed 85 in our work, because the pressure field disappear in the variational form for the problem. On the other hand, it appears that the results of [31] only apply to triangular-Delaunay meshes, while ours apply to arbitrarily unstructured staggered grids. Due to these differences, we are comfortable in publishing this work.…”

“…Thanks to the weak convergences (3.20) and (3.21) and the strong convergences (3.28) and (3.29), we can pass to the limit in (3.30) and obtain 31) which implies that…”

“…But the current work and the cited work differ in the approaches taken. We utilize an external approximation framework 80 for vector fields for the convergence analysis, while [31] rely on a strong reconstruction operator (to reconstruct the velocity field from either the tangential or normal velocity components) and a compactness result. They also prove the convergence for the pressure field for one version of the MAC schemes, which comes as an extra bonus of their approach.…”

Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations

“…Since then, the TMAC scheme has been investigated using the finite volume methods approach [13,21,37], the finite element methods approach [18–20,23,24], and the discountinuous Galerkin (DG) approach [7,14,15,45,46]. The MAC scheme can be interpreted within these approaches when the underlying grids are rectangular [23,26,30,34,37].…”

Superconvergence analysis of the MAC scheme for the two dimensional stokes problem

“…This analysis allows to obtain the optimal convergence order not only for the velocity approximation, but for the pressure approximation as well. The analysis structure maybe used as a base to form a general framework for second order and high order MAC schemes, in particular for MAC schemes based on triangular partitions (cf., [23]). Second, our results are obtained under a weak requirement of the solution regularity that the exact velocity is in W 3,∞ ( ).…”