FVMs on Unstructured Meshes

Main narrative

Finite volume methods (FVMs) are widely employed in physics, engineering. However, theoretical results concerning the stability and convergence of these methods are relatively few. One major reason is that, unlike the finite element methods (FEMs), the FVMs do not naturally fit in a theoretical framework that would facilitate the analyses of them. The situation is even worse on unstructured meshes, where one of the major analysis tool for FVMs and Finite Difference Methods (FDMs), namely the Taylor series expansion, is rendered ineffective. The current project aims to change this situation. It starts by constructing a general functional analytic framework for anlayzing FVMs on unstructured meshes, as well as a complete discrete vector calculus that facilitates the derivation and analysis of FVMs on unstructured meshes. Due to our own research interests and bias, the project will primarily focus on fluid problems and the vorticity-divergence based FV schemes.

Chronicle of developments

1. In Chen (2016, JCAM), we established a functional analytic framework for the analysis of FVMs on unstructured meshes. The two essential ingredients of this framework are Cea’s external approximation of functional spaces and the use of the vorticity and divergence as the primal variables. A complete discrete vector calculus on unstructured meshes, in analogue to the analytical vector calculus, was also presented.

2. The framework was employed (Chen 2016, JCAM) to establish the convergence of the Mark-and-Cell scheme for the incompressible Stokes problem on unstructured meshes, without a priori assumptions on the true solutions.

3. The framework was employed to establish error estimates for the MAC schemes for the incompressible Stokes problem on unstructured meshes (Chen 2017, NMPDE). It was shown that on general unstructured meshes, both the velocity and the vorticity converge at the first order. The convergence is one order higher on special meshes such as the perfect quadrilateral or triangular meshes.