T. Chacón Rebollo

Bio: T. Chacón Rebollo is an academic researcher from University of Seville. The author has contributed to research in topics: Numerical analysis & Finite element method. The author has an hindex of 11, co-authored 33 publications receiving 311 citations. Previous affiliations of T. Chacón Rebollo include University of Bordeaux & Pierre-and-Marie-Curie University.

Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L 1

Abstract: In this paper we consider, in dimension d≥ 2, the standard $$\mathbb{P}_{1}$$ finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in L ∞(Ω) which generalizes Laplace’s equation. We assume that the family of triangulations is regular and that it satisfies an hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs to L 1(Ω), we prove that the unique solution of the discrete problem converges in $$W^{1,q}_0(\Omega)$$ (for every q with $${1 \leq q 1.

A high order term-by-term stabilization solver for incompressible flow problems

Abstract: In this paper, we introduce a low-cost, high-order stabilized method for the numerical solution of incompressible flow problems. This is a particular type of projection-stabilized method where each targeted operator, such as the pressure gradient or the convection, is stabilized by least-squares terms added to the Galerkin formulation. The main methodological originality is that we replace the projection-stabilized structure by an interpolation-stabilized structure, with reduced computational cost for some choices of the interpolation operator. This stabilization has one level, in the sense that it is defined on a single mesh. We prove the stability of our formulation by means of a specific inf–sup condition, which is the main technical innovation of our paper. We perform a convergence and error estimates analysis, proving the optimal order of accuracy of our method. We include some numerical tests that confirm our theoretical expectations.

A Multilayer Method for the Hydrostatic Navier-Stokes Equations: A Particular Weak Solution

Abstract: In this work we present a multilayer approach to the solution of non-stationary 3D Navier–Stokes equations. We use piecewise smooth weak solutions. We approximate the velocity by a piecewise constant (in z) horizontal velocity and a linear (in z) vertical velocity in each layer, possibly discontinuous across layer interfaces. The multilayer approach is deduced by using the variational formulation and by considering a reduced family of test functions. The procedure naturally provides the mass and momentum interfaces conditions. The mass and momentum conservation across interfaces is formulated via normal flux jump conditions. The jump conditions associated to momentum conservation are formulated by means of an approximation of the vertical derivative of the velocity that appears in the stress tensor. We approximate the multilayer model for hydrostatic pressure, by using a polynomial viscosity matrix finite volume scheme and we present some numerical tests that show the main advantages of the model: it improves the approximation of the vertical velocity, provides good predictions for viscous effects and simulates re-circulations behind solid obstacles.

A Model for Two Coupled Turbulent Fluids Part II: Numerical Analysis of a Spectral Discretization

Abstract: We consider a system of equations that models the stationary flow of two immiscible turbulent fluids on adjacent subdomains. The equations are coupled by nonlinear boundary conditions on the interface which is here a fixed given surface. We propose a spectral discretization of this problem and perform its numerical analysis. The convergence of the method is proven in the two-dimensional case, together with optimal error estimates.

A Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows

Abstract: Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: they are based on the variation al formulation of the incompressible Navier–Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used math ematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

A Nonlinear Dual-Domain Decomposition Method: Application to Structural Problems with Damage

Abstract: A dual domain decomposition method dedicated to nonlinear problems is presented. The decomposition is introduced in the nonlinear formulation and the nonlinear problem is first condensed on the interface then solved by a Newton-type method. Considering the specificities of the introduced operators, the algorithm can be interpreted as a local/global strategy with global Newton-type iterations and nonlinear relocalizations per subdomain. Such a strategy is particularly interesting in cases where the nonlinearity is localized. First results are presented on structural problems with damage.