María Macarena Gómez Mármol

On a certified smagorinsky reduced basis turbulence model

In this work we present a reduced basis Smagorinsky turbulence model for steady flows. We approximate the nonlinear eddy diffusion term using the empirical interpolation method (cf. [M. A. Grepl et al., ESAIM Math. Model. Numer. Anal., 41 (2007), pp. 575--605; Barrault et al., C. R. Acad. Sci. Paris Ser. I Math., 339 (2004), pp. 667--672]) and the velocity-pressure unknowns by an independent reduced-basis procedure. This model is based upon an a posteriori error estimation for a Smagorinsky turbulence model. The theoretical development of the a posteriori error estimation is based on [S. Deparis, SIAM J. Sci. Comput., 46 (2008), pp. 2039--2067] and [A. Manzoni, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 1199--1226], according to the Brezzi--Rappaz--Raviart stability theory, and adapted for the nonlinear eddy diffusion term. We present some numerical tests, programmed in FreeFem++ (cf. [F. Hecht, J. Numer. Math., 20 (2012), pp. 251--265]), in which we show a speedup on the computation by factor larger...

A Bochev-Dohrmann-Gunzburger stabilization method for the primitive equations of the ocean

Abstract We introduce a low-order stabilized discretization of the primitive equations of the ocean with highly reduced computational complexity. We prove stability through a specific inf–sup condition, and weak convergence to a weak solution. We also perform some numerical tests for relevant flows.

ERROR ANALYSIS OF A SUBGRID EDDY VISCOSITY MULTI-SCALE DISCRETIZATION OF THE NAVIER–STOKES EQUATIONS

We propose a finite element discretization of the Navier-Stokes equations that relies on the variational multi-scale approach together with the addition of a Smagorinsky type viscosity, in order to take into account possible subgrid turbulence. We recall that the discrete problem admits a solution and prove a priori error estimates. Next we perform the a posteriori analysis of the discretization. Some numerical experiments justify the interest of this approach.