A linearized energy preserving finite element method for the dynamical incompressible magnetohydrodynamics equations
This work provides a numerically stable and efficient method for solving incompressible MHD problems on non-smooth domains, but it is an incremental contribution building on existing mixed FEM frameworks.
The paper presents a linearized finite element method for the dynamical incompressible MHD equations that is energy preserving and achieves optimal error estimates under low regularity assumptions. Numerical results on a benchmark problem confirm its effectiveness.
We present and analyze a linearized 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 Nedelec edge elements are used for the magnetic equation. The divergence free conditions are weakly satisfied at the discrete level. Due to the use of Nedelec 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 linearized scheme which only needs to solve a linear system at each time step. Moreover, the linearized mixed FEM is energy preserving. We establish an optimal error estimate under a very low assumption on the exact solutions and domain geometries. Numerical results which includes a benchmark lid-driven cavity problem are provided to show its effectiveness and verify the theoretical analysis.