Frédéric Hecht

New development in freefem

Abstract -This is a short presentation of the freefem++ software. In Section 1, we recall most of the characteristics of the software, In Section 2, we recall how to to build the weak form of a partial differential equation (PDE) from the strong form. In the 3 last sections, we present different examples and tools to illustrated the power of the software. First we deal with mesh adaptation for problems in two and three dimension, second, we solve numerically a problem with phase change and natural convection, and the finally to show the possibilities for HPC we solve a Laplace equation by a Schwarz domain decomposition problem on parallel computer.

Automatic mesh generator with specified boundary

Abstract For the purpose of finite element computation for 2D or 3D geometry, one needs an appropriate mesh of the considered domain. A class of full automatic methods, derived from Voronois theory, is suitable to generate the mesh of any shape via a set of points which describes the geometry. Such methods providing triangles in 2D and tetrahedra in 3D can be seen, after an adequate initialization, as the merger of each given point in an existing mesh using an updating process. Unfortunately, this mesh which contains all the given points does not contain, in general, the edges (or the faces) of the boundary which are the natural data to be satisfied. The aim of this paper is, after a brief survey of the different steps of the above method, to point out the problem of the exact fitting of the given boundary and to present a method which guarantees this crucial property.

Delaunay mesh generation governed by metric specifications. Part I algorithms

Abstract This paper proposes a Delaunay-type mesh generation algorithm governed by a metric map. The classical method is briefly established and then the different steps it involves are extended. It will be shown that the proposed method applies in three dimensions. The work is divided in two parts. Part I, i.e. the present paper, is devoted to the algorithmical aspects while Part II will present numerous application examples in the context of finite element computations.

MESH GRADATION CONTROL

This paper gives a procedure to control the size variation in a mesh adaption scheme where the size specication (the so-called control space) is used to govern the mesh generation stage. The method consists in replacing the initial control space by a reduced one by means of size or metric. It allows to improve, a priori, the quality of the adapted mesh and to speed up the adaption procedure. Several numerical examples show the eciency of the reduction scheme. ? 1998 John Wiley & Sons, Ltd.

Fully automatic mesh generator for 3D domains of any shape

Abstract Devoted to mesh generation of 3D domains, this paper examines the different approaches actually in progress. A new method is introduced which can be seen as a variant of the Delaunay-Voronoi tesselation coupled with a control of the given boundary used to define the domain under consideration.

A Parareal in Time Semi-implicit Approximation of the Navier-Stokes Equations

The “parareal in time” algorithm introduced in Lions et al. [2001] enables parallel computation using a decomposition of the interval of time integration. In this paper, we adapt this algorithm to solve the challenging Navier-Stokes problem. The coarse solver, based on a larger timestep, may also involve a coarser discretization in space. This helps to preserve stability and provides for more significant savings.

Error indicators for the mortar finite element discretization of the Laplace equation

The mortar technique turns out to be well adapted to handle mesh adaptivity in finite elements, since it allows for working with nonnecessarily compatible discretizations on the elements of a nonconforming partition of the initial domain. The aim of this paper is to extend the numerical analysis of residual error indicators to this type of methods for a model problem and to check their efficiency thanks to some numerical experiments.

Scalable domain decomposition preconditioners for heterogeneous elliptic problems

Domain decomposition methods are, alongside multigrid methods, one of the dominant paradigms in contemporary large-scale partial differential equation simulation. In this paper, a lightweight implementation of a theoretically and numerically scalable preconditioner is presented in the context of overlapping methods. The performance of this work is assessed by numerical simulations executed on thousands of cores, for solving various highly heterogeneous elliptic problems in both 2D and 3D with billions of degrees of freedom. Such problems arise in computational science and engineering, in solid and fluid mechanics. While focusing on overlapping domain decomposition methods might seem too restrictive, it will be shown how this work can be applied to a variety of other methods, such as non-overlapping methods and abstract deflation based preconditioners. It is also presented how multilevel preconditioners can be used to avoid communication during an iterative process such as a Krylov method.

Adaptive multiresolution analysis based on anisotropic triangulations

A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between the approximated function and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the Lp norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of of the approximated function in case of C2 functions).

A POSTERIORI ANALYSIS OF A PENALTY METHOD AND APPLICATION TO THE STOKES PROBLEM

We derive an a posteriori error estimate for an abstract saddle-point problem when a penalty term is added to stabilize the variational formulation, the aim being to optimize the choice of the penalty parameter. As an application, we consider a discretization of the Stokes problem obtained by combining the penalty technique and the finite element method, we perform its a posteriori analysis in a detailed way and present some numerical experiments on adaptive meshes which are in good agreement with the results of the analysis and confirm its interest.