L. Beirão da Veiga

L. Beirão da Veiga, S. Gómez, K. B. Haile

We propose and analyze a space-time discontinuous Galerkin method for the incompressible Stokes equations on moving domains within the arbitrary Lagrangian-Eulerian setting. We use a contravariant Piola map in the definition of the discrete velocity space to preserve the pointwise divergence-free property on the discrete level. We show that the method is inf-sup stable, with no constraints on the spatial mesh or the time partition. We also establish a priori error estimates in the energy norm for arbitrary degrees of approximation in space and time. For piecewise-constant and piecewise-linear approximations in time, we show that the method is also robust at low viscosity regimes, and provide numerical evidence suggesting that this property extends to high-order cases as well. We present several numerical experiments to validate our theoretical findings.

L. Beirão da Veiga, F. Dassi, A. Russo et al.

The Virtual Element Method (VEM) is a well-established framework for solving partial differential equations on polygonal and polyhedral meshes. In this paper, we introduce a novel hybrid VEM that integrates both conforming and nonconforming virtual spaces. We apply this formulation to a three-dimensional linear elasticity problem, providing rigorous theoretical analysis to demonstrate optimal convergence rates. Furthermore, we explore the extension of this approach to domains with curved boundaries.