Anisotropic Error Estimates And Space Adaptivity For A Semidiscrete Finite Element Approximation Of The Transient Transport Equation

Marco Picasso
Siam Publications, 2013
Article

Résumé

A stabilized semidiscrete finite element discretization of the transient transport equation is studied in the framework of anisotropic meshes. A priori and a posteriori error estimates are derived, the involved constants being independent of the mesh aspect ratio, only space discretization being considered. Numerical results on nonadapted, anisotropic meshes and small time steps confirm the sharpness of the theoretical predictions. An anisotropic, adaptive finite element algorithm is then proposed with the goal to control the L-2 error in space at final time, the time step being kept constant. Numerical results are then presented on anisotropic, adapted meshes and small time steps. Three different methods are proposed to interpolate the solution between two adapted meshes. Numerical results indicate that both the conservative interpolation method of [F. Alauzet and M. Mehrenberger, Internat. J. Numer. Methods Engrg., 84 (2010), pp. 1552-1588] or the L-2 projection advocated in [P. E. Farrell et al., Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 2632-2642] yield accurate results.

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Marco Picasso
Siam Publications, 2013
Article

Résumé

Official source

https://infoscience.epfl.ch/record/189653?ln=fr

À propos de ce résultat

Concepts associés (11)

L’analyse numérique est une discipline à l'interface des mathématiques et de l'informatique. Elle s’intéresse tant aux fondements qu’à la mise en pratique des méthodes permettant de résoudre, par des

En analyse numérique, la méthode des éléments finis (MEF, ou FEM pour finite element method en anglais) est utilisée pour résoudre numériquement des équations aux dérivées partielles. Celles-ci pe

L'espace se présente dans l'expérience quotidienne comme une notion de géométrie et de physique qui désigne une étendue, abstraite ou non, ou encore la perception de cette étendue. Conceptuellement,

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Numerical Study Of An Anisotropic Error Estimator In The L-2(H-1) Norm For The Finite Element Discretization Of The Wave Equation

Marco Picasso

An anisotropic a posteriori error estimate is derived for a finite element discretization of the wave equation in two space dimensions. Only the error due to space discretization is considered, and the error estimates are derived in the nonnatural L-2(0, T; H-1(Omega)) norm using elliptic reconstruction. A numerical study of the effectivity index on unstructured, nonadapted, anisotropic meshes confirms the sharpness of the error estimator, provided the error due to time discretization is negligible compared to the finite element error. An anisotropic, adaptive finite element algorithm is then presented to control the finite element error in the L-2(0, T; H-1(Omega)) norm. Numerical results on adapted meshes indicate that the error estimator slightly underestimates the true error. We conjecture that the missing information corresponds to the interpolation error between successive meshes. It is observed that the error estimator becomes sharp again when considering the damped wave equation

2010

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Analysis and numerical simulation of viscoelastic flows

Andrea Bonito

Mathematical and numerical aspects of viscoelastic flows are investigated here. Two simplified mathematical models are considered. They are motivated by a splitting algorithm for solving viscoelastic flows with free surfaces. The first model is a simplified Oldroyd-B model. Existence on a fixed time interval is proved in several Banach spaces provided the data are small enough. Short time existence is also proved for arbitrarily large data in Hölder spaces for the time variable. These results are based on the maximal regularity property of the Stokes operator and on the analycity behavior of the corresponding semi-group. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates, using an implicit function theorem framework. Then, the extension of these results to a stochastic simplified Hookean dumbbells model is discussed. Because of the presence of the Brownian motion, existence in a fixed time interval, provided the data are small enough, is proved only in some of the Banach spaces considered previously. The dumbbells' elongation is split in two parts, one satisfying a standart stochastic differential equation, the other satisfying a partial differential equation with a stochastic source term. A finite element discretization in space is also proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates. A numerical algorithm for solving viscoelastic flows with free surfaces is also described. This algorithm is based on a splitting method in time and two different meshes are used for the space discretization. Convergence of the numerical model is checked for the pure extensional flow and the filling of a pipe. Then, numerical results are reported for the stretching of a filament and for jet buckling.

EPFL2006

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, ,

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.

1999

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