Decoupled and linearized scalar auxiliary variable finite element method for the time‐dependent incompressible magnetohydrodynamic equations: Unconditional stability and convergence analysis

Abstract

AbstractThis paper considers the stability and convergence of Euler implicit/explicit scheme for the incompressible magnetohydrodynamic (MHD) equations by the scalar auxiliary variable approach. The linear and nonlinear terms are treated by the implicit and explicit schemes, respectively, then the considered problem is split into two constant coefficient linear algebraic equations plus a nonlinear algebraic equation. Compared with the original discrete coefficient matrix, the computational size reduce and we can solve these subproblems conveniently and efficiently. First, an equivalent form of the MHD problem with four variables is developed, the corresponding stability and convergence results of spatial discrete scheme are presented. Second, the decoupled and linearized auxiliary variable finite element method is constructed, the discrete unconditional energy dissipation and stability of and in various norms are established. The optimal error estimates of numerical solutions in and H‐norms are also developed by the energy method and Gronwall lemma. Finally, some numerical examples are presented to verify the established theoretical findings and show the performances of the considered numerical algorithm.