A linearized energy preserving finite element method for the dynamical incompressible magnetohydrodynamics equations

TL;DR

This paper introduces a linearized finite element method for incompressible magnetohydrodynamics that preserves energy, is suitable for complex domains, and offers optimal error estimates with efficient linear solves.

Contribution

A novel linearized mixed finite element method that preserves energy and is effective on non-smooth, multi-connected domains for MHD equations.

Findings

Energy-preserving property of the method

Optimal error estimates established

Numerical experiments confirm theoretical results

Abstract

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.