A semi-implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations

Abstract

We present and analyze a semi-implicit finite element method (FEM) for the dynamical incompressible magnetohydrodynamics (MHD) equations. The finite element approximation is based on mixed conforming elements, where Taylor–Hood type elements are used for the Navier–Stokes equations and Nédélec edge elements are used for the magnetic equation. The divergence free conditions are weakly satisfied at the discrete level. Due to the use of Nédélec edge element, the proposed method is particularly suitable for problems defined on non-smooth and multi-connected domains. For the temporal discretization, we use a semi-implicit scheme which only needs to solve a linear system at each time step. Moreover, the linearized mixed FEM is energy conserving. We establish an optimal error estimate under a very low assumption on the exact solutions and domain geometries. Numerical results are provided to show its effectiveness and verify the theoretical analysis. In particular, a benchmark lid-driven cavity problem is provided to show that the proposed numerical method produces results as good as that of the ones which are divergence free.