A local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations

Assyr Abdulle, Giacomo Rosilho De Souza
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2022
Article

Résumé

We introduce a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The proposed method is based on a coarse grid and iteratively improves the solution's accuracy by solving local elliptic problems in refined subdomains. For purely diffusion problems, we already proved that this scheme converges under minimal regularity assumptions (Abdulle and Rosilho de Souza, 2019) [1]. In this paper, we provide an algorithm for the automatic identification of the local elliptic problems' subdomains employing a flux reconstruction strategy. Reliable error estimators are derived for the local adaptive method. Numerical comparisons with a classical nonlocal adaptive algorithm illustrate the efficiency of the method. (C) 2021 The Author(s). Published by Elsevier Inc.

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https://infoscience.epfl.ch/record/292920?ln=fr

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Assyr Abdulle, Giacomo Rosilho De Souza
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2022
Article

Résumé

Official source

https://infoscience.epfl.ch/record/292920?ln=fr

À propos de ce résultat

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Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales

Assyr Abdulle

An analysis of a multiscale symmetric interior penalty discontinuous Galerkin finite element method for the numerical discretization of elliptic problems with multiple scales is proposed. This new method, first described in [A. Abdulle, C.R. Acad. Sci. Paris, Ser. I 346 (2008)] is based on numerical homogenization. It allows to significantly reduce the computational cost of a fine scale discontinuous Galerkin method by probing the fine scale data on sampling domains within a macroscopic partition of the computational domain. Macroscopic numerical fluxes, an essential ingredient of discontinuous Galerkin finite elements, can be recovered from the computation on the sampling domains with negligible computation overhead. Fully discrete a priori error bounds are derived in the L-2 and H-1 norms.

2012

Chargement

The aim of this master thesis is to study and develop a stabilized reduced basis method suitable for the approximation of the solution of parametrized advection-diffusion PDEs with high Péclet number, that is, roughly, the ratio between the advection coefficient and the diffusion one. Advection-diffusion equations are very important in many engineering applications, because they are used to model, for example, heat transfer phenomena or the diffusion of pollutants in the atmosphere. In such applications, we often need very fast evaluations of the approximated solution, depending on some input parameters. This happens, for example, in the case of real-time simulation. Moreover, we need rapid evaluations also if we have to perform repeated approximation of the solution, for different input parameters. An important case of this many-query situation is represented by some optimization problems, in which the objective function to optimize depends on the parameters through the solution of a PDE. The reduced basis (RB) method meets our need for rapidity and it is also able to guarantee the reliability of the solution, thanks to sharp a posteriori error bounds. We can find in literature many works about the application of the RB method to advection-diffusion problems but they mainly deal with equations in which the Péclet number is low. The need for stabilization arise from the fact that the finite element (FE) approximated solution - that the RB method aims to recover - shows strong instability problems that have to be fixed. In this work we want to go further in the study of the stabilization of the RB method for advection dominated problem in both steady and unsteady case. As regards the steady case, we first compare two possible stabilization strategies, by testing them on some test problems, in order to design an efficient stabilized reduced basis method. We will then test this method using the piecewise quadratic FE approximation as reference solution, instead of the usual piecewise linear one. We extend the method designed for the steady case to the time dependent case and we will carry out some numerical tests.

2012

Chargement

Fully discrete heterogeneous Multiscale methods and application to transport problems in microarrays

Assyr Abdulle

Physical systems encompassing a variety of strongly coupled scales pose major computational challenges in terms of analysis modeling and simulation. In this report we first discuss a multiscale modeling approach for the transport of particles such as DNA in heterogeneous devices (microarrays). The model involves a multiscale elliptic equation coupled with a multiscale advection-diffusion equation. We introduce a numerical method for the solution of the coupled system of equations. We first discuss a new multiscale finite element method for the solution of the elliptic problem. We then explain how it is possible to use an explicit stabilized method (ROCK) for the numerical solution of the stiff system of ordinary differential equations of large dimension, originating from the method of lines discretization of the advection-diffusion equation.

EMS2005

Chargement

2012