Coupled iterative analysis for the stationary thermally coupled inductionless MHD system based on charge-conservative finite element method

TL;DR

This paper introduces three coupled iterative methods for solving stationary thermally coupled inductionless MHD equations using charge-conservative finite element approximation, with proven convergence and validated numerical results.

Contribution

It presents novel coupled iterative algorithms (Stokes, Newton, Oseen) for thermally coupled MHD equations with rigorous convergence analysis and error estimates independent of potential.

Findings

Convergence and stability of the iterative methods are established.

Error estimates for key variables are independent of electric potential.

Numerical results confirm the effectiveness of the proposed methods.

Abstract

This paper mainly considers three iterations based on charge-conservative finite element approximation in Lipschitz domain for the stationary thermally coupled inductionless MHD equations. Based on the hybrid finite element method, the unknowns of hydrodynamic are discretized by the stable velocity-pressure finite element pair, and the current density along with electric potential are similarly discretized by the comforming finite element pair in $\boldsymbol{H}(\boldsymbol{div}, \Omega)\times L^2(\Omega)$. And on account of the strong nonlinearity of the equations, we present three coupled iterative methods, namely, the Stokes, Newton and Oseen iteration and the convergence and stability under different uniqueness conditions are analyzed strictly. It is proved especially that the error estimates of velocity, current density, temperature and pressure do not depend on potential. The theoretical analysis is validated by the given numerical results, and for the proposed methods, the applicability and effectiveness are demonstrated.