A Mixed Finite Element Method for Stationary Magneto-Heat Coupling System with Variable Coefficients.

Area of Science:

• Computational fluid dynamics
• Magnetohydrodynamics (MHD)
• Numerical analysis

Background:

• Magnetohydrodynamics (MHD) problems involve complex interactions between fluid flow and magnetic fields.
• Temperature-dependent physical parameters increase the nonlinearity and difficulty of solving MHD systems.
• Existing numerical methods may struggle with the low regularity of solutions in such problems.

Purpose of the Study:

• To develop and analyze a mixed finite element method for thermally coupled, stationary incompressible MHD problems.
• To handle systems with temperature-dependent coefficients and address increased nonlinearity.
• To establish rigorous error estimates for the proposed numerical scheme.

Main Methods:

• A mixed finite element method is employed for the analysis.
• Nédélec edge elements are used for the magnetic equation approximation.
• Mixed finite elements approximate the thermal and Navier-Stokes equations.
• The scheme is designed for coefficients dependent on temperature.

Main Results:

• Optimal error estimates are rigorously established for velocity, pressure, temperature, magnetic induction, and the Lagrange multiplier.
• The analysis accommodates solutions with low regularity.
• A numerical experiment validates the performance and convergence rates of the scheme.

Conclusions:

• The proposed mixed finite element method is effective for thermally coupled, stationary incompressible MHD problems with temperature-dependent parameters.
• The numerical scheme provides accurate results and demonstrates good convergence properties.
• This work contributes to the reliable numerical simulation of complex MHD phenomena.