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

TL;DR

This paper introduces a new framework for analyzing finite difference finite volume methods on unstructured meshes, focusing on vorticity and divergence to ensure stability and convergence, and demonstrates its application to the Stokes problem.

Contribution

It presents a novel external approximation framework for vector fields on unstructured meshes, enabling stability and convergence analysis of the MAC scheme for incompressible flows.

Findings

Framework successfully applied to the MAC scheme

Proves convergence on unstructured meshes

Ensures stability via external approximation

Abstract

A new framework is proposed for analyzing staggered-grid finite difference finite volume methods on unstructured meshes. The new framework employs the concept of external approximation of function spaces, and gauge convergence of numerical schemes through the quantities of vorticity and divergence, instead of individual derivatives of the velocity components. The construction of a stable and convergent external approximation of a simple but relevant vector-valued function space is demonstrated, and the new framework is applied to establish the convergence of the MAC scheme for the incompressible Stokes problem on unstructured meshes.