Sergio Gómez

Sergio Gómez

This work extends the discrete compactness results of Walkington (SIAM J. Numer. Anal., 47(6):4680--4710, 2010) for high-order discontinuous Galerkin time discretizations of parabolic problems to more general function space settings. In particular, we show a discrete version of the Aubin--Lions--Simon lemma that holds for general Banach spaces $X$, $B$, and $Y$ satisfying $X \hookrightarrow B$ compactly and $B \hookrightarrow Y$ continuously. Our proofs rely on the properties of a time reconstruction operator and remove the need for quasi-uniform time partitions assumed in previous works. Thus, we provide a useful and flexible tool for the analysis of high-order discontinuous Galerkin time discretizations of complex nonlinear partial differential equations.

Lourenço Beirão da Veiga, Sergio Gómez, Ilaria Perugia et al.

We propose and analyze a class of finite element methods for the time-dependent incompressible magnetohydrodynamics system based on $H(\mathrm{curl})$-conforming discretizations for both the velocity and the magnetic field. This choice is guided by the aim of developing methods that are also suitable for the types of solutions arising in problems posed on nonconvex domains. Within this framework, we introduce three stabilized formulations, and study how the stabilization mechanisms employed influence their structural properties. In particular, we focus on suitability for nonconvex polyhedral domains, the need for Lagrange multipliers for the magnetic field, pressure-robustness, and quasi-robustness with respect to both the fluid and magnetic Reynolds numbers. The proposed formulations are further assessed through numerical experiments, highlighting their practical performance.