Méthodes de quadrature de Gauss

Résumé

Dans le domaine mathématique de l'analyse numérique, les méthodes de quadrature sont des approximations de la valeur numérique d'une intégrale. En général, on remplace le calcul de l'intégrale par une somme pondérée prise en un certain nombre de points du domaine d'intégration (voir calcul numérique d'une intégrale pour plus d'informations). La méthode de quadrature de Gauss, du nom de Carl Friedrich Gauss, est une méthode de quadrature exacte pour un polynôme de degré 2n – 1 avec n points pris sur le domaine d'intégration. Les formules de Gauss jouent un rôle fondamental dans la méthode des éléments finis. Principe général On souhaite évaluer numériquement l'intégrale :I = \int_a^b f(x) \varpi(x) ,\mathrm{d}x. Le domaine d'intégration (a, b) couvre plusieurs cas :

intervalles bornés : comme [a, b], [a, b[, etc.
demi-droite réelle : [a, +∞[, ] -∞, b],
la droite réelle tout entière : ℝ.

Les méthodes sont de la forme :I = \int_a^b f(x) \varpi(

Source officielle

https://fr.wikipedia.org/wiki/M%C3%A9thodes_de_quadrature_de_Gauss

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Multiscale methods for Stokes flow in heterogeneous media

Ondrej Budác

Fluid flow in porous media is a multiscale process where the effective dynamics, which is often the goal of a computation, depends strongly on the porous micro structure. Resolving the micro structure in the whole porous medium can, however, be prohibitive. Novel numerical methods that efficiently approximate the effective flow but resolve only a carefully selected reduced portion of the porous structure are of great interest. In this thesis we propose new numerical multiscale methods for Stokes flow in two- and three-scale porous media. First, we propose the Darcy--Stokes finite element heterogeneous multiscale method (DS-FE-HMM). The method is based on solving the Darcy equation on a macroscopic mesh using the finite element method with numerical quadrature, where the unknown permeability is recovered from micro finite element solutions of Stokes problems that are defined in sampling domains centered at macroscopic quadrature points. An adaptive scheme based on a posteriori error analysis is proposed, where micro-macro mesh refinement is driven by residual-based indicators that quantify both the micro and macro errors. Second, to address the increasing cost of solving the micro problems as the macroscopic mesh is refined, we combine the DS-FE-HMM with reduced basis (RB) method and propose a new multiscale method called the RB-DS-FE-HMM. Efficiency and accuracy of the method relies on a parametrization of the micro geometries and on the Petrov-Galerkin RB formulation that provides a stable and fast evaluation of the effective permeability. A residual-based adaptive mesh refinement scheme is proposed for the macroscopic problem. To achieve a conservative approximation we also combine and analyze a coupling of the RB method with a different macroscopic scheme based on the discontinuous Galerkin finite element method (DG-FEM). Finally, we consider a three-scale porous media model with macro, meso, and micro scale. At the intermediate meso scale the medium is composed of fluid and porous parts and the fluid flow is modeled with the Stokes-Brinkman equation. A three-scale numerical method is derived and an efficient algorithm based on the RB method and empirical interpolation method on the micro and meso scale is proposed.

EPFL2016

Chargement

We present a novel high-order discontinuous Galerkin discretization for the spherical shallow water equations, able to handle wetting/drying and non-conforming, curved meshes in a well-balanced manner. This requires a well-balanced discretization, that cannot rely on exact quadrature, due to the curved mesh. Using the strong form of the discontinuous Galerkin discretization, we achieve a splitting of the well-balanced condition into individual problems for the flux and volume terms, which has significant advantages: It allows for the construction of non-conforming, well-balanced flux discretizations, i.e. we can perform non- conforming mesh refinement while preserving the well-balanced property of the scheme. More importantly, this approach enables the development of a new method for handling wet/dry transitions. In contrast to other wetting/drying methods, it is well-balanced and able to handle wetting/drying robustly at any polynomial order, without the introduction of physical model assumptions such as viscosity, artificial porosity or cancellation of gravity. We perform a series of one-dimensional tests and analyze the properties of our scheme. In order to validate our method for the simulation of large-scale tsunami events on the rotating sphere, we perform numerical simulations of the 2011 Tohoku tsunami and compare our results to real-world buoy data. The method is able to predict arrival times and wave amplitudes accurately even over long distances. This indicates that our method accurately captures all physical phenomena relevant to the long-term evolution of tsunami waves.

2018

Chargement

A new multiscale method combined with model order reduction is proposed for flow problems in three-scale porous media. We derive an effective three-scale model that couples a macroscopic Darcy equation, a mesoscopic Stokes-Brinkman equation, and a microscopic Stokes equation. A corresponding three-scale numerical method is then derived using the finite element discretization with numerical quadrature, where the macroscopic and mesoscopic permeability is upscaled at quadrature points from mesoscopic and microscopic problems, respectively. The computational cost of solving numerous mesoscopic and microscopic flow problems is further reduced by applying a Petrov–Galerkin reduced basis method at the mesocopic and microscopic scales. As there is no natural way to obtain an affine decomposition of the mesoscopic problems, which is instrumental for the efficiency of the model order reduction, we derive a mesoscopic solver that makes use of empirical interpolation techniques. A priori and a posteriori error estimates are derived for the new method that is also tested numerically to corroborate the theoretical convergence rates and illustrate its efficiency.

2017

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Multiscale methods for Stokes flow in heterogeneous media

Ondrej Budác

EPFL2016