The stability of velocity and pressure mixed finite-element approximations in general meshes of the hydrostatic Stokes problem is studied, where two "inf-sup" conditions appear associated to the two constraints of the problem; namely incompressibility and hydrostatic pressure. Since these two constraints have different properties, it is not easy to choose finite element spaces satisfying both. From the analytical point of view, two main results are established; the stability of an anisotropic approximation of the velocity (using different spaces for horizontal and vertical velocities) with piecewise constant pressures, and the unstability of standard (isotropic) approximations which are stable for the Stokes problem, like the mini-element or the Taylor-Hood element. Moreover, we give some numerical simulations, which agree with the previous analytical results and allow us to conjecture the stability of some anisotropic approximations of the velocity with continuous piecewise linear pressure in unstructured meshes.

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This paper studies the stability of velocity-pressure mixed approximations of the Stokes problem when different finite element (FE) spaces for each component of the velocity field are considered. We consider some new combinations of continuous FE reducing the number of degrees of freedom in some velocity components. Although the resulting FE combinations are not stable in general, by using the Stenberg's macro-element technique, we show their stability in a wide family of meshes (namely, in uniformly unstructured meshes). Moreover, a post-processing is given in order to convert any mesh family in an uniformly unstructured mesh family. Finally, some 2D and 3D numerical simulations are provided agree with the previous analysis.

Some new stable finite element (FE) schemes are presented for the hydrostatic Stokes system or primitive equations of the ocean. It is known that the stability of the mixed formulation approximation for primitive equations requires the well-known Ladyzhenskaya-Babuška-Brezzi condition related to the Stokes problem and an extra inf-sup condition relating the pressure and the vertical velocity [F. Guillén-González and J. R. Rodríguez-Galván, Numer. Math., 130 (2015), pp. 225-256]. The main goal of this paper is to avoid this extra condition by adding a residual stabilizing term to the vertical momentum equation. Then, the stability for Stokes-stable FE combinations is extended to the primitive equations and some error estimates are provided using Taylor-Hood P 2 -P 1 or minielement (P 1 +bubble)-P 1 FE approximations, showing the optimal convergence rate in the P 2 -P 1 case. These results are also extended to the anisotropic (nonhydrostatic) problem. On the other hand, by adding another residual term to the continuity equation, a better approximation of the vertical derivative of pressure is obtained. In this case, stability and error estimates including this better approximation are deduced, where optimal convergence rate is deduced in the (P 1 +bubble)-P 1 case. Finally, some numerical experiments are presented supporting previous results.

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The design of numerical approximations of the Cahn-Hilliard model preserving the maximum principle is a challenging problem, even more if considering additional transport terms. In this work, we present a new upwind discontinuous Galerkin scheme for the convective Cahn-Hilliard model with degenerate mobility which preserves the maximum principle and prevents non-physical spurious oscillations. Furthermore, we show some numerical experiments in agreement with the previous theoretical results. Finally, numerical comparisons with other schemes found in the literature are also carried out.

A numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced. The numerical method consists of a stabilized finite element method together with a mass lumping technique and an extra stabilizing term plus a semi-implicit Euler time integration. Then we carry out a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that the sequence of finite element approximations converges toward the unique weak solution of the model at hands. In doing so, nonnegativity is attained due to the stabilizing term and the acuteness on partitions of the computational domain, and hence a priori energy estimates of finite element approximations are established. As we deal with a nonlinear problem, some form of strong convergence is required. The key compactness result is obtained via an adaptation of a Riesz-Fréchet-Kolmogorov criterion by perturbation. A numerical example is also presented.