Two‐level stabilized finite volume method for the stationary incompressible magnetohydrodynamic equations

Abstract

In this paper, a two‐level stabilized finite volume method is developed and analyzed for the steady incompressible magnetohydrodynamic (MHD) equations. The linear polynomial space is used to approximate the velocity, pressure and magnetic fields, and two local Gauss integrations are introduced to overcome the restriction of discrete inf‐sup condition. Firstly, the existence and uniqueness of the solution of the discrete problem in the stabilized finite volume method are proved by using the Brouwer's fixed point theorem. H2$$ {H}^2 $$ ‐stability results of numerical solutions are also presented. Secondly, optimal error estimates of numerical solutions in H1$$ {H}^1 $$ and L2$$ {L}^2 $$ ‐norms are established by using the energy method and constructing the corresponding dual problem. Thirdly, the stability and convergence of two‐level stabilized finite volume method for the stationary incompressible MHD equations are provided. Theoretical findings show that the two‐level method has the same accuracy as the one‐level method with the mesh sizes h=𝒪(H2) . Finally, some numerical results are provided to identify with the established theoretical findings and show the performances of the considered numerical schemes.