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Dive into the research topics where Assyr Abdulle is active.

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Featured researches published by Assyr Abdulle.


Acta Numerica | 2012

The heterogeneous multiscale method

Assyr Abdulle; Weinan E; Björn Engquist; Eric Vanden-Eijnden

The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.


SIAM Journal on Scientific Computing | 2001

Fourth Order Chebyshev Methods with Recurrence Relation

Assyr Abdulle

In this paper, a new family of fourth order Chebyshev methods (also called stabilized methods) is constructed. These methods possess nearly optimal stability regions along the negative real axis and a three-term recurrence relation. The stability properties and the high order make them suitable for large stiff problems, often space discretization of parabolic PDEs. A new code ROCK4 is proposed, illustrated at several examples, and compared to existing programs.


Multiscale Modeling & Simulation | 2005

On a priori error analysis of fully discrete heterogeneous multiscale FEM

Assyr Abdulle

Heterogeneous multiscale methods have been introduced by E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87--132] as a methodology for the numerical computation of problems with multiple scales. Analyses of the methods for various homogenization problems have been done by several authors. These results were obtained under the assumption that the microscopic models (the cell problems in the homogenization context) are analytically given. For numerical computations, these microscopic models have to be solved numerically. Therefore, it is important to analyze the error transmitted on the macroscale by discretizing the fine scale. We give in this paper H1 and L2 a priori estimates of the fully discrete heterogeneous multiscale finite element method. Numerical experiments confirm that the obtained a priori estimates are sharp.


Numerische Mathematik | 2001

Second order Chebyshev methods based on orthogonal polynomials

Assyr Abdulle; Alexei A. Medovikov

Summary. Stabilized methods (also called Chebyshev methods) are explicit Runge-Kutta methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. The aim of this paper is to show that with the use of orthogonal polynomials, we can construct nearly optimal stability polynomials of second order with a three-term recurrence relation. These polynomials can be used to construct a new numerical method, which is implemented in a code called ROCK2. This new numerical method can be seen as a combination of van der Houwen-Sommeijer-type methods and Lebedev-type methods.


Journal of Computational Physics | 2003

Finite difference heterogeneous multi-scale method for homogenization problems

Assyr Abdulle; Weinan E

In this paper, we propose a numerical method, the finite difference heterogeneous multi-scale method (FD-HMM), for solving multi-scale parabolic problems. Based on the framework introduced in [Commun. Math. Sci. 1 (1) 87], the numerical method relies on the use of two different schemes for the original equation, at different grid level which allows to give numerical results at a much lower cost than solving the original equations. We describe the strategy for constructing such a method, discuss generalization for cases with time dependency, random correlated coefficients, nonconservative form and implementation issues. Finally, the new method is illustrated with several test examples.


Multiscale Modeling & Simulation | 2011

Finite element heterogeneous multiscale method for the wave equation

Assyr Abdulle; Marcus J. Grote

A finite element heterogeneous multiscale method is proposed for the wave equation with highly oscillatory coefficients. It is based on a finite element discretization of an effective wave equation at the macro scale, whose a priori unknown effective coefficients are computed on sampling domains at the micro scale within each macro finite element. Hence the computational work involved is independent of the highly heterogeneous nature of the medium at the smallest scale. Optimal error estimates in the energy norm and the L2 norm and convergence to the homogenized solution are proved, when both the macro and the micro scales are refined simultaneously. Numerical experiments corroborate the theoretical convergence rates and illustrate the behavior of the numerical method for periodic and heterogeneous media.


Multiscale Modeling & Simulation | 2005

Heterogeneous Multiscale FEM for Diffusion Problems on Rough Surfaces

Assyr Abdulle; Christoph Schwab

We present a finite element method for the numerical solution of diffusion problems on rough surfaces. The problem is transformed to an elliptic homogenization problem in a two dimensional parameter domain with a rapidly oscillating diffusion tensor and source term. The finite element method is based on the heterogeneous multiscale methods of E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87--132]. For periodic surface roughness of scale


SIAM Journal on Scientific Computing | 2008

S-ROCK: Chebyshev Methods for Stiff Stochastic Differential Equations

Assyr Abdulle; Stéphane Cirilli

\varepsilon


Multiscale Modeling & Simulation | 2008

Finite element heterogeneous multiscale methods with near optimal computational complexity

Assyr Abdulle; Björn Engquist

and amplitude


Mathematical Models and Methods in Applied Sciences | 2006

Analysis of a heterogeneous multiscale FEM for problems in elasticity

Assyr Abdulle

O(\varepsilon)

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Gilles Vilmart

École Normale Supérieure

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Yun Bai

École Polytechnique Fédérale de Lausanne

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Ondrej Budáč

École Polytechnique Fédérale de Lausanne

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Orane Camille Jecker

École Polytechnique Fédérale de Lausanne

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Martin Ernst Huber

École Polytechnique Fédérale de Lausanne

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Timothée Noé Pouchon

École Polytechnique Fédérale de Lausanne

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Patrick Henning

Royal Institute of Technology

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