A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D

Abstract

In this paper, we propose a Newton–Krylov solver and a Picard–Krylov solver for finite element discrete problem of stationary incompressible magnetohydrodynamic equations in three dimensions. Using a mixed finite element method, we discretize the velocity and the pressure by H1(Ω)-conforming finite elements and discretize the magnetic field by H(curl,Ω)-conforming edge elements. An efficient preconditioner is proposed to accelerate the convergence of GMRES method for solving linearized discrete problems. By extensive numerical experiments, we demonstrate the robustness of the Newton–Krylov solver for relatively large physical parameters and the optimality with respect to the number of degrees of freedom. Moreover, the numerical experiments show that the Newton–Krylov solver is more robust than the Picard–Krylov solver for large Reynolds number.