Numerical Analysis for a Non-isothermal Incompressible Navier-Stokes-Allen-Cahn System

In this paper we develop the numerical analysis for a non-isothermal diffuse-interface model, in dimension N=2,3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2, 3,$$\end{document} that describes the movement of a mixture of two incompressible viscous fluids. This model consists of modified Navier-Stokes equations coupled with a phase-field equation given by a convective Allen-Cahn equation, and energy transport equation for the temperature; which admits a dissipative energy inequality. We propose an energy stable numerical scheme based on the Finite Element Method, and we analyze optimal weak and strong error estimates, as well as convergence towards regular solutions. In order to construct the numerical scheme, we introduce two extra variables (given by the gradient of the temperature and the variation of the energy with respect to the phase-field function) which allows us to control the strong regularity required by the model, which is one of the main difficulties appearing from the numerical point of view. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed, energy stable and satisfies a set of uniform estimates which allow to analyze the convergence of the scheme. Finally, we present some numerical simulations to validate numerically our theoretical results.