Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast

Abstract

We construct finite-dimensional approximations of solution spaces of divergence form operators with $L^{\infty }$-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in $H^1$ if source terms are in the unit ball of $L^2$ instead of the unit ball of $H^{-1}$. Approximation spaces are generated by solving elliptic PDEs on localized sub-domains with source terms corresponding to approximation bases for $H^2$. The $H^1$-error estimates show that $\mathcal{O}(h^{-d})$-dimensional spaces with basis elements localized to sub-domains of diameter $\mathcal{O}(h^{\alpha }\ln \frac{1}{h})$ (with $\alpha \in [1/2,1)$) result in an $\mathcal{O}(h^{2-2\alpha })$ accuracy for elliptic, parabolic and hyperbolic problems. For high-contrast media, the accuracy of the method is preserved provided that localized sub-domains contain buffer zones of width $\mathcal{O}(h^{\alpha }\ln \frac{1}{h})$ where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).

Accepted for publication in SIAM MMS

References in corpus (5)

• A non-adapted sparse approximation of PDEs with stochastic inputs
• An optimal error estimate in stochastic homogenization of discrete elliptic equations
• Corrector theory for MsFEM and HMM in random media
• Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers
• Sweeping Preconditioner for the Helmholtz Equation: Hierarchical Matrix Representation

Cited by in corpus (7)

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• Asymptotic expansions for high-contrast elliptic equations
• A reduced basis localized orthogonal decomposition
• Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis
• Bayesian Numerical Homogenization
• Space-time FLAVORS: finite difference, multisymlectic, and pseudospectral integrators for multiscale PDEs