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Publication# Multilevel preconditioners for elliptic problems with multiple scales

Résumé

The purpose of this thesis is to investigate methods for the solution of multiscale problems both from the mathematical and numerical point of view, with a particular concern on applications to flows through heterogeneous porous media. After an overview of the recent developments in this vast field, a study of two representative multiscale techniques is carried out. The multiscale finite element method and a conjugate gradient iterative method preconditioned with an overlapping Schwarz domain decomposition preconditioner are compared in the one-dimensional case. Both methods are well suited for parallel environments and have comparable performance and accuracy. Then, we focus our attention on an aggregation-based two-level overlapping Schwarz domain decomposition preconditioner. We study its theoretical properties and show its robustness to mesh refinement as well as to strong variations in the multiscale coefficients of our model problem. We carry out a convergence analysis to give upper bounds for the condition number of the preconditioned linear system arising from the finite element discretization of the problem at hand. We make explicit the relation between the multiscale coefficient and the coarse space basis functions and show that the condition number can be bounded independently of the ratio of the values of the multiscale coefficient even when the discontinuities in the coefficient are not resolved by the coarse mesh. A new aggregation algorithm is proposed according to the suggestions issuing from the theory which builds a coarse space able to cope with the multiscale coefficient. Numerical experiments on various configurations show that the bounds are sharp and that the method is robust with respect to strong variations. Finally, an application of this preconditioning technique to a two-phase flow problem is presented in order to investigate its performance.

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Vincent Maronnier, Marco Picasso, Jacques Rappaz

A numerical model is presented for the simulation of complex fluid flows with free surfaces. The unknowns are the velocity and pressure fields in the liquid region, together with a function defining the volume fraction of liquid. Although the mathematical formulation of the model is similar to the volume of fluid (VOF) method, the numerical schemes used to solve the problem are different. A splitting method is used for the time discretization. At each time step, two advection problems and a generalized Stokes problem are to be solved. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small rectangular cells, using a forward characteristic method. The generalized Stokes problem is solved using a finite element method on a fixed, unstructured mesh. Numerical results are presented for several test cases: the filling of an S-shaped channel, the filling of a disk with core, the broken dam in a confined domain. (C) 1999 Academic Press.

1999Engineers rely on efficient simulations that provide them with reliable data in order to make proper engineering design decisions. The purpose of this thesis is to design adaptive numerical methods for multiscale problems in this spirit. We consider elliptic homogenization problems discretized by the finite element heterogeneous multiscale method (FE-HMM). Unlike standard (single-scale) finite element methods, our multiscale discretization scheme relies on coupled macro and micro finite elements. The framework of the HMM allows to design an algorithm that follows the classical finite element structure on the macro level. The fine scales of the multiscale problems are taken into account by replacing the element-wise numerical integration over unknown macroscopic data by a numerical integration over suitably averaged micro solutions. These micro solutions are obtained from micro FE problems on sampling domains within the macro elements. This thesis is divided into two parts. In the first part, we discuss a short and versatile FE implementation of the multiscale algorithm. The implementation is flexible, easy to use and to modify and can handle simplicial or quadrilateral FE and various macro-micro coupling conditions for the constrained micro problems. The implementation of time-dependent problems is also discussed. Numerical examples including three dimensional problems are presented and demonstrate the efficiency and the versatility of the computational strategy. In the second part (the main part of this thesis), we present an a posteriori error analysis for the FE-HMM. The a posteriori analysis enables us to estimate the accuracy of a numerical solution (and therefore its reliability) and further it allows for the design of adaptive numerical methods, which are the most efficient. The crucial component for the design of an adaptive multiscale method is the introduction of appropriate error indicators. As the error indicators depend on macroscopic data (such as the macroscopic diffusion tensor) that are not readily available, we construct error indicators that only depend on the available macro and micro FE solutions, available from previous computations. We provide a posteriori estimates for the upper and lower bound in the energy norm. The corresponding macroscopic mesh refinement strategy is therefore both reliable and efficient. The microscopic mesh is refined simultaneously and – under appropriate assumptions – optimally with the macroscopic mesh. This means that the strategy reduces the macro and micro error at the same rate. In the case of a uniformly oscillating tensor and exact micro computations, the standard a posteriori error estimates for the FEM applied to the homogenized problem are recovered. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method and demonstrate the optimality of the chosen macro-micro coupling. We extend the adaptive FE-HMM to higher order FE. We further derive a posteriori estimates for the error in quantities of interest that are needed to make certain design decisions; the quantity of interest is represented by a linear functional. We derive and analyze a multiscale counterpart to the classical dual-weighted residual method and design a corresponding goal-oriented adaptive multiscale method. The efficiency of the method is shown in numerical experiments.

The aim of this work is the development of a geometrical multiscale framework for the simulation of the human cardiovascular system under either physiological or pathological conditions. More precisely, we devise numerical algorithms for the partitioned solution of geometrical multiscale problems made of different heterogeneous compartments that are implicitly coupled with each others. The driving motivation is the awareness that cardiovascular dynamics are governed by the global interplay between the compartments in the network. Thus, numerical simulations of stand-alone local components of the circulatory system cannot always predict effectively the physiological or pathological states of the patients, since they do not account for the interaction with the missing elements in the network. As a matter of fact, the geometrical multiscale method provides an automatic way to determine the boundary (more precisely, the interface) data for the specific problem of interest in absence of clinical measures and it also offers a platform where to study the interaction between local changes (due, for instance, to pathologies or surgical interventions) and the global systemic dynamics. To set up the framework an abstract setting is devised; the local specific mathematical equations (partial differential equations, differential algebraic equations, etc.) and the numerical approximation (finite elements, finite differences, etc.) of the heterogeneous compartments are hidden behind generic operators. Consequently, the resulting global interface problem is formulated and solved in a completely transparent way. The coupling between models of different dimensional scale (three-dimensional, one-dimensional, etc.) and type (Navier-Stokes, fluid-structure interaction, etc.) is addressed writing the interface equations in terms of scalar quantities, i.e., area, flow rate, and mean (total) normal stress. In the resulting flexible framework the heterogeneous models are treated as black boxes, each one equipped with a specific number of compatible interfaces such that (i) the arrangement of the compartments in the network can be easily manipulated, thus allowing a high level of customization in the design and optimization of the global geometrical multiscale model, (ii) the parallelization of the solution of the different compartments is straightforward, leading to the opportunity to make use of the latest high-performance computing facilities, and (iii) new models can be easily added and connected to the existing ones. The methodology and the algorithms devised throughout the work are tested over several applications, ranging from simple benchmark examples to more complex cardiovascular networks. In addition, two real clinical problems are addressed: the simulation of a patient-specific left ventricle affected by myocardial infarction and the study of the optimal position for the anastomosis of a left ventricle assist device cannula.