| 1. |
- Abdulle, Assyr, et al.
(författare)
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A parabolic local problem with exponential decay of the resonance error for numerical homogenization
- 2021
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Ingår i: Mathematical Models and Methods in Applied Sciences. - 0218-2025. ; 31:13
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Tidskriftsartikel (refereegranskat)abstract
- This paper aims at an accurate and efficient computation of effective quantities, e.g., the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods are based on a micro-macro coupling, where the macromodel describes the coarse scale behaviour, and the micro model is solved only locally to upscale the effective quantities, which are missing in the macro model. The fact that the micro problems are solved over small domains within the entire macroscopic domain, implies imposing artificial boundary conditions on the boundary of the microscopic domains. A naive treatment of these artificial boundary conditions leads to a first order error in ε/δ, where ε < δ represents the characteristic length ofthe small scale oscillations and δ^d is the size of micro domain. This error dominates all other errors originating from the discretization of the macro and the micro problems, and its reduction is a main issue in today’s engineering multiscale computations. The objective of the present work is to analyse a parabolic approach, first announced in [A. Abdulle,D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019], for computing the homogenized coefficients with arbitrarily high convergence rates in ε/δ. The analysis covers the setting of periodic microstructure,and numerical simulations are provided to verify the theoretical findings for more general settings, e.g. non-periodic micro structures.
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| 2. |
- Abdulle, Assyr, et al.
(författare)
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A parabolic local problem with exponential decay of the resonance error for numerical homogenization
- 2021
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific Pub Co Pte Ltd. - 0218-2025 .- 1793-6314. ; 31:13, s. 2733-2772
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Tidskriftsartikel (refereegranskat)abstract
- This paper aims at an accurate and efficient computation of effective quantities, e.g. the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods are based on a micro–macro-coupling, where the macromodel describes the coarse scale behavior, and the micromodel is solved only locally to upscale the effective quantities, which are missing in the macromodel. The fact that the microproblems are solved over small domains within the entire macroscopic domain, implies imposing artificial boundary conditions on the boundary of the microscopic domains. A naive treatment of these artificial boundary conditions leads to a first-order error in ?/??/δ, where ?<??<δ represents the characteristic length of the small scale oscillations and ??δd is the size of microdomain. This error dominates all other errors originating from the discretization of the macro and the microproblems, and its reduction is a main issue in today’s engineering multiscale computations. The objective of this work is to analyze a parabolic approach, first announced in A. Abdulle, D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019, for computing the homogenized coefficients with arbitrarily high convergence rates in ?/??/δ. The analysis covers the setting of periodic microstructure, and numerical simulations are provided to verify the theoretical findings for more general settings, e.g. non-periodic microstructures.
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| 3. |
- Abdulle, Assyr, et al.
(författare)
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AN ELLIPTIC LOCAL PROBLEM WITH EXPONENTIAL DECAY OF THE RESONANCE ERROR FOR NUMERICAL HOMOGENIZATION
- 2023
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Ingår i: Multiscale Modeling & simulation. - : Society for Industrial & Applied Mathematics (SIAM). - 1540-3459 .- 1540-3467. ; 21:2, s. 513-541
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Tidskriftsartikel (refereegranskat)abstract
- Numerical multiscale methods usually rely on some coupling between a macroscopic and a microscopic model. The macroscopic model is incomplete as effective quantities, such as the homogenized material coefficients or fluxes, are missing in the model. These effective data need to be computed by running local microscale simulations followed by a local averaging of the microscopic information. Motivated by the classical homogenization theory, it is a common practice to use local elliptic cell problems for computing the missing homogenized coefficients in the macro model. Such a consideration results in a first order error O(E/8), where E represents the wavelength of the microscale variations and 8 is the size of the microscopic simulation boxes. This error, called ``resonance error,"" originates from the boundary conditions used in the microproblem and typically dominates all other errors in a multiscale numerical method. Optimal decay of the resonance error remains an open problem, although several interesting approaches reducing the effect of the boundary have been proposed over the last two decades. In this paper, as an attempt to resolve this problem, we propose a computationally efficient, fully elliptic approach with exponential decay of the resonance error.
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| 4. |
- Abdulle, Assyr, et al.
(författare)
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Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
- 2019
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Ingår i: Comptes rendus. Mathematique. - : Elsevier BV. - 1631-073X .- 1778-3569. ; 357:6, s. 545-551
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Tidskriftsartikel (refereegranskat)abstract
- This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell problems, result in an exponential decay of the resonance error.
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| 5. |
- Abdulle, Assyr, et al.
(författare)
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High weak order methods for stochastic differential equations based on modified equations
- 2012
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 34:3, s. A1800-A1823
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Tidskriftsartikel (refereegranskat)abstract
- Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (mean-square stable) stochastic problems, and implicit integrators that exactly conserve all quadratic firstintegrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.
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| 6. |
- Abdulle, Assyr, et al.
(författare)
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Localized orthogonal decomposition method for the wave equation with a continuum of scales
- 2017
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 86:304, s. 549-587
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Tidskriftsartikel (refereegranskat)abstract
- This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with an L2-projection. We derive explicit convergence rates of the method in the L∞(L2)-, W1,∞(L2)-and L∞(H1)-norms without any assumptions on higher order space regularity or scale-separation. The order of the convergence rates depends on further graded assumptions on the initial data. We also prove the convergence of the method in the framework of G-convergence without any structural assumptions on the initial data, i.e. without assuming that it is well-prepared. This rigorously justifies the method. Finally, the performance of the method is demonstrated in numerical experiments.
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| 7. |
- Holst, Henrik
(författare)
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Multiscale Methods for Wave Propagation Problems
- 2011
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Simulations of wave propagation in heterogeneous media and at high frequencies are important in many applications such as seismic-, {electro-magnetic-,} acoustic-, fluid flow problems and others. These are classical multiscale problems and often too computationally expensive for direct numerical simulation. The smallest scales must be well resolved over a computational domain represented by the largest scale and this results in a very high computational cost. We develop and analyze numerical techniques based on the heterogeneous multiscale method (HMM) framework for such wave equations with highly oscillatory solutions $u^{\varepsilon}$ where $\varepsilon$ represents the size of the smallest scale. In these techniques the oscillatory microscale is approximated on small local microproblems of size $\varepsilon$ in spatial and time directions. The solution of the microproblems are then coupled to a global macroscale model in divergence form $u_{tt} = \nabla \cdot F$ where the flux $F$ is obtained from the microproblems. The oscillations can either originate from fluctuations in the velocity coefficients or from high frequency initial and boundary conditions. We have developed algorithms that couple micro and macroscales for both these cases. The choice of macroscale variables is inspired by the analytic theories of homogenization and geometrical optics respectively. In the first case local averages $u \approx u^{\varepsilon}$ are used on the macroscale. In the second case, phase $\phi$ and energy are natural macroscopic variables. There are two major goals of this research. One goal is to develop and analyze algorithms for simulating multiscale wave propagation with low computational complexity, and even independent of $\varepsilon$ for finite time problems. This is seen in many examples in one, two and three dimensions. The other goal is to use wave propagation as a model to better understand the HMM framework. An example in this direction is simulation with oscillatory wave field over long time. The dispersive effects that then occur is well approximated by a HMM method that was originally formulated for finite time where added accuracy is required but no explicit adjustment to include dispersion, an evidence of the robustness of the method.
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