Convergence analysis of a fully discrete finite element method for thermally coupled incompressible MHD problems with temperature-dependent coefficients

Qianqian Ding,Xiaonian Long,Shipeng Mao

doi:10.1051/m2an/2022028

Abstract

In this paper, we study a fully discrete finite element scheme of thermally coupled incompressible magnetohydrodynamic with temperature-dependent coefficients in Lipschitz domain. The variable coefficients in the MHD system and possible nonconvex domain may cause nonsmooth solutions. We propose a fully discrete Euler semi-implicit scheme with the magnetic equation approximated by Nédélec edge elements to capture the physical solutions. The fully discrete scheme only needs to solve one linear system at each time step and is unconditionally stable. Utilizing the stability of the numerical scheme and the compactness method, the existence of weak solution to the thermally coupled MHD model in three dimensions is established. Furthermore, the uniqueness of weak solution and the convergence of the proposed numerical method are also rigorously derived. Under the hypothesis of a low regularity for the exact solution, we rigorously establish the error estimates for the velocity, temperature and magnetic induction unconditionally in the sense that the time step is independent of the spacial mesh size.

Highlights

Magnetohydrodynamic (MHD) is the theory of macroscopic interaction of conductive fluid and electromagnetic induction
The proposed schemes in the above two papers are based on the magnetic induction approximated by Lagrange H1 finite element method and all the error estimates are conducted under sufficiently smooth assumption on the exact solutions
We show the uniqueness of weak solution for incompressible MHD models with temperature-dependent coefficients provided it satisfies a smoother condition, which seems to be new in the literature

Summary

Introduction

Magnetohydrodynamic (MHD) is the theory of macroscopic interaction of conductive fluid and electromagnetic induction. As far as we know, the first work to study error estimates of finite element methods for the thermally coupled MHD equations with temperature-dependent coefficients is given in [47], where a fully discrete Crank–Nicolson scheme is proposed and investigated. The proposed schemes in the above two papers are based on the magnetic induction approximated by Lagrange H1 finite element method and all the error estimates are conducted under sufficiently smooth assumption on the exact solutions. We will give a rigorous convergence analysis and error estimates of a fully discrete finite element method for the MHD system described by (1.1)–(1.7) based on the magnetic induction approximated by H(curl)-conforming Nedelec edge element.

Functional setting for the magneto-heat coupling model

A fully discrete finite element method based on Euler scheme

Well-posedness and convergence of the fully discrete solution

Numerical experiments

Summary