Gabriel R. Barrenechea

Gabriel R. Barrenechea, Andreas Wachtel

Uniform LBB conditions are desirable to approximate the solution of Navier-Stokes, Oseen, and Stokes equations on anisotropic meshes and to enable anisotropic refinements. We prove such conditions for the second order Taylor-Hood pairs $\mathbb{Q}_2 \times \mathbb{Q}_1$ and $\mathbb{P}_2 \times \mathbb{P}_1$ on a class of anisotropic meshes. These meshes may contain refined edge and corner patches. To this end, we generalise Verfürth's trick and recent results by some of the authors.

Gabriel R. Barrenechea, Michał Bosy, Victorita Dolean

In several studies it has been observed that, when using stabilised $\mathbb{P}_k^{}\times\mathbb{P}_k^{}$ elements for both velocity and pressure, the error for the pressure is smaller, or even of a higher order in some cases, than the one obtained when using inf-sup stable $\mathbb{P}_k^{}\times\mathbb{P}_{k-1}^{}$ (although no formal proof of either of these facts has been given). This increase in polynomial order requires the introduction of stabilising terms, since the finite element pairs used do not stability the inf-sup condition. With this motivation, we apply the stabilisation approach to the hybrid discontinuous Galerkin discretisation for the Stokes problem with non-standard boundary conditions.

Gabriel R. Barrenechea, Michał Bosy, Victorita Dolean et al.

Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this paper we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with non standard boundary conditions. This analysis is supported by numerical evidence. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.

Gabriel R. Barrenechea, Erik Burman, Fotini Karakatsani

For the case of approximation of convection--diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions.