Stability and Convergence of Stabilized Finite Volume Iterative Methods for Steady Incompressible MHD Flows with Different Viscosities

Abstract

Three finite volume iterative schemes for steady incompressible magnetohydrodynamic problems are studied. The theoretical analysis of finite volume methods is more challenging than that of finite element methods because of the presence of a trilinear form and the difficulties with the treatment of nonlinear terms. Nevertheless, we prove the uniform stability of the methods and establish error estimates. It is worth noting that the Newton iterative scheme converges exponentially under viscosity related requirements, while the Oseen iterative method is unconditionally stable and convergent under the uniqueness condition. Some numerical examples confirm the theoretical findings and demonstrate a good performance of the methods under consideration.