\documentclass{WCCM-ECCOMAS26-abstracts} \usepackage{times} %\usepackage{amsmath} %\usepackage{amsfonts} %\usepackage{amssymb} \title{Pressure and convection robust finite elements for magnetohydrodynamics} \author{Louren\c{c}o Beir\~{a}o da Veiga$^{\dag}$, Franco Dassi$^{\dag}$ Giuseppe Vacca$^{*}$} \address{ $^{\dag}$ Dipartimento di Matematica e Applicazioni, \\ Universit\`a degli Studi di Milano-Bicocca, \\ Via Roberto Cozzi 55 - 20125 Milano, Italy \\ e-mail: lourenco.beirao@unimib.it, franco.dassi@unimib.it \and $^{*}$ Dipartimento di Matematica, \\ Universit\`a degli Studi di Bari, \\ Via Edoardo Orabona 4 - 70125 Bari, Italy \\ e-mail: giuseppe.vacca@uniba.it } \begin{document} \maketitle \begin{center} \fontsize{11}{12}\selectfont \textbf{ABSTRACT}\\ \end{center} In this talk we present two pressure-robust and convection quasi-robust finite element methods for the fully nonlinear, time-dependent magnetohydrodynamics equations. By quasi-robustness we mean that, under suitable regularity assumptions on the solution, the method admits error estimates that remain uniform with respect to large fluid and magnetic Reynolds numbers, in a norm that also captures convection effects. Both methods rely on an $H({\rm div})$-conforming discretization for the fluid velocity, coupled with a suitable pressure space ensuring an exact sequence property, and on $H^1$-conforming finite elements for the magnetic field. Robustness with respect to convection is achieved through the use of DG-type upwind terms and continuous interior penalty (CIP) stabilizations. The two formulations differ in the strategy adopted to enforce the divergence-free constraint on the magnetic field. In the first approach, a three-field formulation, this constraint is imposed via grad–div stabilization. In the second, a four-field formulation, it is enforced by introducing an additional Lagrange multiplier together with suitable stabilization terms. Numerical experiments are presented to illustrate and support the theoretical results. \end{document}