Convergence of the Finite Volume Method on Unstructured Meshes for a 3D Phase Field Model of Solidification

TL;DR

This paper proves the convergence of the finite volume method on unstructured meshes for a 3D phase field model of solidification, encompassing heat and phase equations with general reaction terms.

Contribution

It develops new interpolation and a priori estimate techniques to establish convergence for complex phase field models on unstructured meshes.

Findings

Convergence of the finite volume method is proven for the model.

The approach handles general reaction terms in the phase field equations.

Boundedness of key terms is established through new estimates.

Abstract

We present a convergence result for the finite volume method applied to a particular phase field problem suitable for simulation of pure substance solidification. The model consists of the heat equation and the phase field equation with a general form of the reaction term which encompasses a variety of existing models governing dendrite growth and elementary interface tracking problems. We apply the well known compact embedding techniques in the context of the finite volume method on admissible unstructured polyhedral meshes. We develop the necessary interpolation theory and derive an a priori estimate to obtain boundedness of the key terms. Based on this estimate, we conclude the convergence of all of the terms in the equation system.