Toni Sayah

A second order accuracy for a full discretized time-dependent Navier–Stokes equations by a two-grid scheme

We study a second-order two-grid scheme fully discrete in time and space for solving the Navier–Stokes equations. The two-grid strategy consists in discretizing, in the first step, the fully non-linear problem, in space on a coarse grid with mesh-size H and time step Δt and, in the second step, in discretizing the linearized problem around the velocity uH computed in the first step, in space on a fine grid with mesh-size h and the same time step. The two-grid method has been applied for an analysis of a first order fully-discrete in time and space algorithm and we extend the method to the second order algorithm. This strategy is motivated by the fact that under suitable assumptions, the contribution of uH to the error in the non-linear term, is measured in the L2 norm in space and time, and thus has a higher-order than if it were measured in the H1 norm in space. We present the following results: if h2 = H3 = (Δt)2, then the global error of the two-grid algorithm is of the order of h2, the same as would have been obtained if the non-linear problem had been solved directly on the fine grid.

A posteriori analysis of iterative algorithms for a nonlinear problem

A posteriori error indicators have been studied in recent years owing to their remarkable capacity to enhance both speed and accuracy in computing. This work deals with a posteriori error estimation for the finite element discretization of a nonlinear problem. For a given nonlinear equation considering finite elements we solve the discrete problem using iterative methods involving some kind of linearization. For each of them, there are actually two sources of error, namely discretization and linearization. Balancing these two errors can be very important, since it avoids performing an excessive number of iterations. Our results lead to the construction of computable upper indicators for the full error. Several numerical tests are provided to evaluate the efficiency of our indicators.

Finite element methods for Darcy’s problem coupled with the heat equation

In this article, we study theoretically and numerically the heat equation coupled with Darcy’s law by a nonlinear viscosity depending on the temperature. We establish existence of a solution by using a Galerkin method and we prove uniqueness. We propose and analyze two numerical schemes based on finite element methods. An optimal a priori error estimate is then derived for each numerical scheme. Numerical experiments are presented that confirm the theoretical accuracy of the discretization.

Parallel computing investigations for the projection method applied to the interface transport scheme of a two-phase flow by the method of characteristics

This paper deals with the discretization of the problem of mould filling in iron foundry and its numerical solution using a Schwarz domain decomposition method. An adapted technique for domain decomposition methods that suits the discretization in time by the method of characteristics is introduced. Furthermore, the projection method is used to reduce the computation time. Finally, numerical experiments show and validate the effectiveness of the proposed scheme.

Convergence Analysis of Two Numerical Schemes Applied to a Nonlinear Elliptic Problem

For a given nonlinear problem discretized by standard finite elements, we propose two iterative schemes to solve the discrete problem. We prove the well-posedness of the corresponding problems and their convergence. Next, we construct error indicators and prove optimal a posteriori estimates where we treat separately the discretization and linearization errors. Some numerical experiments confirm the validity of the schemes and allow us to compare them.