Jacques Laminie

Differential equations and solution of linear systems

Abstract Many iterative processes can be interpreted as discrete dynamical systems and, in certain cases, they correspond to a time discretization of differential systems. In this paper, we propose to derive iterative schemes for solving linear systems of equations by modeling the problem to solve as a stable state of a proper differential system; the solution of the original linear problem is then computed numerically by applying a time marching scheme. We discuss some aspects of this approach, which allows to recover some known methods but also to introduce new ones. We give convergence results and numerical illustrations.

On the domain decomposition method for the generalized Stokes problem with continuous pressure

In this paper, the generalized Stokes problem αu−νΔu+∇p=f, ∇u=0, is considered, where ν denotes the kinematic viscosity, α is a positive parameter proportional to the inverse of the time-step, and the source term f gathers the external forcing, the nonlinear term, and the explicit part of the linear term. The system is derived from a time discretization for the Navier-Stokes equation governing the motion of a two-dimensional incompressible fluid. A domain decomposition numerical method is developed for solving the system. The original domain is divided into several nonoverlapping subdomains, and in each subdomain the concerned equations are discretized by certain finite element methods. A new feature of the method is that the pressure p is required to be piecewise linear rather than piecewise constant, so that the nonphysical oscillations in the solution using piecewise linear pressure are eliminated. However, a difficulty in using piecewise linear pressure is that the Stokes problem in each subdomain involves Dirichlet boundary conditions on pressure at the interface, which is not a real Stokes problem. To avoid the difficulty, a modification is proposed which removes the continuity conditions on the pressure across the interface. In this way, the Stokes problem in each subdomain has only Dirichlet boundary conditions on the velocity. Formulations and discussions for both the methods are presented, which shows that the modified method is simpler than the original one. Numerical experiments are displayed to show the efficiency of the methods, especially that the modified method has the same accuracy as the original one.

Finite Volume Discretization and Multilevel Methods in Flow Problems

This article is intended as a preliminary report on the implementation of a finite volume multilevel scheme for the discretization of the incompressible Navier–Stokes equations. As is well known the use of staggered grids (e.g. MAC grids, Perić et al. Comput. Fluids, 16(4), 389–403, (1988)) is a serious impediment for the implementation of multilevel schemes in the context of finite differences. This difficulty is circumvented here by the use of a colocated finite volume discretization (Faure et al. (2004a) Submitted, Perić et al. Comput. Fluids, 16(4), 389–403, (1988)), for which the algebra of multilevel methods is much simpler than in the context of MAC type finite differences. The general ideas and the numerical simulations are presented in this article in the simplified context of a two-dimensional Burgers equations; the two-, and three-dimensional Navier–Stokes equations introducing new difficulties related to the incompressibility condition and the time discretization, will be considered elsewhere (see Faure et al. (2004a) Submitted and Faure et al. (2004b), in preparation).

Implementation of finite element nonlinear Galerkin methods using hierarchical bases

The nonlinear Galerkin methods are investigated in the framework of finite element discretization We first describe the theoretical background in relation with multilevel and finite element approximations of attractors. Then on the computational side, we recall the definition of the hierarchical bases and analyze the structure associated to these bases. Finally we present the schemes and report on numerical experiments performed on two-dimensional equations of the Burgers and Navier-Stokes type. Their consistency with the approximation that we make and with the structure of the algorithm is discussed.

A Dynamical Multi-level Scheme for the Burgers Equation: Wavelet and Hierarchical Finite Element

Algorithms issued from the NonLinear Galerkin method have been used in many situations and with different discretizations for the resolution of evolutionary nonlinear equations. The main idea of these methods is to use a splitting of the solution in order to model the equation. According to the splitting of the solution, a splitting of the equation is obtained. The modeling principle is to freeze terms which have a small time variation. In this work we use wavelet discretizations of the 2-D Burgers equations and compare the results with the hierarchical finite elements method. The numerical tests indicate that wavelets give better results than finite elements

On a multilevel approach for the two dimensional Navier–Stokes equations with finite elements

We study if the multilevel algorithm introduced by Debussche, T. Dubois and R. Temam [Theoret. Comput. Fluid Dynam. 7 (1995), no. 4, 279-315; Zbl 838.76060] and Dubois, F. Jauberteau and Temam [J. Sci. Comput. 8 (1993), no. 2, 167-194; MR1242960 (94f:65098)] for the 2D Navier-Stokes equations with periodic boundary conditions and spectral discretization can be generalized to more general boundary conditions and to finite elements. We first show that a direct generalization, as in [C. Calgaro, J. Laminie and R. Temam, Appl. Numer. Math. 23 (1997), no. 4, 403-442; MR1453424 (98d:76124)], for the Burgers equation, would not be very efficient. We then propose a new approach where the domain of integration is decomposed into subdomains. This enables us to define localized small-scale components and we show that, in this context, there is a good separation of scales. We conclude that all the ingredients necessary for the implementation of the multilevel algorithm are present.

Finite element calculation of steady transonic flow in nozzles using primary variables

Steady irrotational-isentropic flow of perfect fluid in a plane converging-diverging nozzle is modelled using a system of two first order partial differential equations in the primary variables. A least square/formulation transforms the first-order system into an equivalent second-order system well adapted to discretization methods and allowing the use of powerful iteration algorithms such as conjugate gradient or Newtons method which yield fast convergence. The other advantage of this variational approach is the direct applicability of the finite element method which is more accurate in this case than the corresponding finite difference method.

Finite element least square method for solving full steady Euler equations in a plane nozzle

A finite element least square method is applied to the steady Euler equations in a nozzle or a channel. For the capture of shock waves an artificial density formula is used. Fast convergence is achieved with I.C.C.G. algorithm.

Three‐dimensional magnetostatic problem

The paper is devoted to an approximation of the solution of Maxwells equations in three dimensional sapce. We present two methods which couple a finite element method inside the magnetic materials with a boundary integral method which uses Poincare-Steklov operator to describe the exterior domain.

An HPF Case Study of a Domain-Decomposition Based Irregular Application

Data-parallel languages, in particular HPF, provide a high-level view of operators overs parallel data structures and hide the details of data partitioning and communication. One of the most difficult issues in compiling such languages is managing irregular data-dependent parallelism. This paper presents the study of a realistic, but non adaptive irregular application. We show that HPF can easily express the natural parallelism of the application. Experimental results and a detailed examination of the compiler process are presented.