Mejdi Azaïez

Improvements on open and traction boundary conditions for Navier-Stokes time-splitting methods

We present in this paper a numerical scheme for incompressible Navier-Stokes equations with open and traction boundary conditions, in the framework of pressure-correction methods. A new way to enforce this type of boundary condition is proposed and provides higher pressure and velocity convergence rates in space and time than found in the present state of the art. We illustrate this result by computing some numerical and physical tests. In particular, we establish reference solutions of a laminar flow in a geometry where a bifurcation takes place and of the unsteady flow around a square cylinder.

Spectral discretization of Darcy’s equations with pressure dependent porosity

We consider the flow of a viscous incompressible fluid in a rigid homogeneous porous medium provided with boundary conditions on the pressure around a circular well. When the boundary pressure presents high variations, the permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a spectral discretization of the resulting system of equations which takes into account the axisymmetry of the domain and of the flow. We prove optimal error estimates and present some numerical experiments which confirm the interest of the discretization.

Staggered grids hybrid-dual spectral element method for second-order elliptic problems. Application to high-order time splitting methods for Navier-Stokes equations

Abstract This paper is a presentation of a mixed spectral element method, based on staggered grids strategy, used for the discretization of second-order elliptic problems set under hybrid dual formulation. A numerical analysis is performed proving the expected optimality and some numerical tests are described to confirm the predicted convergence rates. The second part deals with a high-order time splitting/spectral scheme for the simulation of unsteady incompressible Navier-Stokes flows. We show how to use such a solver as a filtering tool to obtain divergence free discrete velocity. Some computational examples are reported which quantify the improvements, induced by such a filter, we have observed: a strengthening of the stability and of the accuracy.

On a Stable Spectral Method for the grad(div) Eigenvalue Problem

We present in this paper a stable spectral element for the approximations of the grad(div) eigenvalue problem in two and three-dimensional quadrangular geometry. Spectral approximations based on Gaussian quadrature rules are built in a dual variational approach with Darcy type equations. We prove that spectral convergence can be reached for the irrotational spectrum without the presence of any spurious eigenmodes, provided an adequate choice is made for the quadrature rules.

Recursive POD expansion for reaction-diffusion equation

This paper focuses on the low-dimensional representation of multivariate functions. We study a recursive POD representation, based upon the use of the power iterate algorithm to recursively expand the modes retained in the previous step. We obtain general error estimates for the truncated expansion, and prove that the recursive POD representation provides a quasi-optimal approximation in

Generalized Lagrangian Coordinates for Transport and Two-Phase Flows in Heterogeneous Anisotropic Porous Media

L^2

Using PGD to Solve Nonseparable Fractional Derivative Elliptic Problems

L2 norm. We also prove an exponential rate of convergence, when applied to the solution of the reaction-diffusion partial differential equation. Some relevant numerical experiments show that the recursive POD is computationally more accurate than the Proper Generalized Decomposition for multivariate functions. We also recover the theoretical exponential convergence rate for the solution of the reaction-diffusion equation.

Mercer's spectral decomposition for the characterization of thermal parameters

We consider an inverse problem that arises in the management of water resources and pertains to the analysis of surface water pollution by organic matter. Most physically relevant models used by engineers derive from various additions and corrections to enhance the earlier deoxygenation?reaeration model proposed by Streeter and Phelps in 1925, the unknowns being the biochemical oxygen demand (BOD) and the dissolved oxygen (DO) concentrations. The one we deal with includes Taylor?s dispersion to account for the heterogeneity of the contamination in all space directions. The system we obtain is then composed of two reaction-dispersion equations. The particularity is that both Neumann and Dirichlet boundary conditions are available on the DO tracer while the BOD density is free of any conditions. In fact, for real-life concerns, measurements on the DO are easy to obtain and to save. On the contrary, collecting data on the BOD is a sensitive task and turns out to be a lengthy process. The global model pursues the reconstruction of the BOD density, and especially of its flux along the boundary. Not only is this problem plainly worth studying for its own interest but it could also be a mandatory step in other applications such as the identification of the location of pollution sources. The non-standard boundary conditions generate two difficulties in mathematical and computational grounds. They set up a severe coupling between both equations and they are the cause of the ill-posed data reconstruction problem. Existence and stability fail. Identifiability is therefore the only positive result one can search for; it is the central purpose of the paper. Finally, we have performed some computational experiments to assess the capability of the mixed finite element in missing data recovery.