New analysis of Mixed finite element methods for incompressible Magnetohydrodynamics

TL;DR

This paper presents a new, optimal error analysis for mixed finite element methods in stationary incompressible magnetohydrodynamics, demonstrating second-order accuracy for velocity and confirming results through numerical examples.

Contribution

It introduces a modified Maxwell projection to establish optimal error estimates, improving upon previous analyses of mixed FEMs for MHD systems.

Findings

The method achieves second-order accuracy for velocity.

Numerical results confirm theoretical error estimates.

Analysis applies to both convex and nonconvex domains.

Abstract

The paper focuses on a new error analysis of a class of mixed FEMs for stationary incompressible magnetohydrodynamics with the standard inf-sup stable velocity-pressure space pairs to Navier-Stokes equations and the N\'ed\'elec's edge element for the magnetic field. The methods have been widely used in various numerical simulations in the last several decades, while the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order N\'ed\'elec's edge approximation in analysis. In terms of a newly modified Maxwell projection we establish new and optimal error estimates. In particular, we prove that the method based on the commonly-used Taylor-Hood/lowest-order N\'ed\'elec's edge element is efficient and the method provides the second-order accuracy for numerical velocity. Two numerical examples for the problem in both convex and nonconvex polygonal domains are presented. Numerical results confirm our theoretical analysis.