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1.
Yalchin Efendiev Juan Galvis Eduardo Gildin 《Journal of computational physics》2012,231(24):8100-8113
In this paper, we study model reduction for multiscale problems in heterogeneous high-contrast media. Our objective is to combine local model reduction techniques that are based on recently introduced spectral multiscale finite element methods (see [19]) with global model reduction methods such as balanced truncation approaches implemented on a coarse grid. Local multiscale methods considered in this paper use special eigenvalue problems in a local domain to systematically identify important features of the solution. In particular, our local approaches are capable of homogenizing localized features and representing them with one basis function per coarse node that are used in constructing a weight function for the local eigenvalue problem. Global model reduction based on balanced truncation methods is used to identify important global coarse-scale modes. This provides a substantial CPU savings as Lyapunov equations are solved for the coarse system. Typical local multiscale methods are designed to find an approximation of the solution for any given coarse-level inputs. In many practical applications, a goal is to find a reduced basis when the input space belongs to a smaller dimensional subspace of coarse-level inputs. The proposed approaches provide efficient model reduction tools in this direction. Our numerical results show that, only with a careful choice of the number of degrees of freedom for local multiscale spaces and global modes, one can achieve a balanced and optimal result. 相似文献
2.
Stefanie Sonner Messoud A. Efendiev Hermann J. Eberl 《Mathematical Methods in the Applied Sciences》2011,34(13):1667-1684
We analyze a system of reaction–diffusion equations that models quorum‐sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium‐type degeneracy and super diffusion. We prove the well‐posedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
3.
The fractal dimension of the global attractors of porous mediaequations in bounded domains is studied. The conditions whichguarantee this attractor to be finite dimensional are foundand the examples of infinite-dimensional attractors that donot satisfy these conditions are constructed. The upper andlower bounds for the Kolmogorov -entropy of infinite-dimensionalattractors are also obtained. 相似文献
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M. A. Efendiev J. Fuhrmann S. V. Zelik 《Mathematical Methods in the Applied Sciences》2004,27(8):907-930
For the Boussinesq approximation of the equations of coupled heat and fluid flow in a porous medium we show that the corresponding system of partial differential equations possesses a global attractor. We give lower and upper bounds of the Hausdorff dimension of the attractor depending on a physical parameter of the system, namely the Rayleigh number of the flow. Numerical experiments confirm the theoretical findings and raise new questions on the structure of the solutions of the system. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
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B. I. Efendiev 《Differential Equations》2014,50(4):562-567
We obtain necessary initial conditions for a continuous second-order differential equation. 相似文献
9.
Messoud A. Efendiev Mitsuharu Ôtani 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2011,28(4):565
In this paper we give a detailed study of the global attractors for parabolic equations governed by the p-Laplacian in a heterogeneous medium. Not only the existence but also the infinite dimensionality of the global attractors is presented by showing that their ε-Kolmogorov entropy behaves as a polynomial of the variable 1/ε as ε tends to zero, which is not observed for non-degenerate parabolic equations. The upper and lower bounds for the Kolmogorov ε-entropy of infinite-dimensional attractors are also obtained. 相似文献
10.
A simple method is proposed which allows obtaining distributed-feedback laser action simultaneously at several spectral lines of any wavelength with independent tuning within the gain profile of the dye. 相似文献