On MAC Schemes on Triangular Delaunay Meshes, Their Convergence and Application to Coupled Flow Problems

摘要

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.