An Equation-Free Approach for Second Order Multiscale Hyperbolic Problems in Non-Divergence Form

The present study concerns the numerical homogenization of second order hyperbolic equations in non-divergence form, where the model problem includes a rapidly oscillating coefficient function. These small scales influence the large scale behavior, hence their effects should be accurately modelled in a numerical simulation. A direct numerical simulation is prohibitively expensive since a minimum of two points per wavelength are needed to resolve the small scales. A multiscale method, under the equation free methodology, is proposed to approximate the coarse scale behaviour of the exact solution at a cost independent of the small scales in the problem. We prove convergence rates for the upscaled quantities in one as well as in multi-dimensional periodic settings. Moreover, numerical results in one and two dimensions are provided to support the theory

Numerical Homogenization of the Acoustic Wave Equations with a Continuum of Scales

In this paper, we consider numerical homogenization of acoustic wave equations with heterogeneous coefficients, namely, when the bulk modulus and the density of the medium are only bounded. We show that under a Cordes type condition the second order derivatives of the solution with respect to harmonic coordinates are $L^2$ (instead $H^{-1}$ with respect to Euclidean coordinates) and the solution itself is in $L^{\infty}(0,T,H^2(\Omega))$ (instead of $L^{\infty}(0,T,H^1(\Omega))$ with respect to Euclidean coordinates). Then, we propose an implicit time stepping method to solve the resulted linear system on coarse spatial scales, and present error estimates of the method. It follows that by pre-computing the associated harmonic coordinates, it is possible to numerically homogenize the wave equation without assumptions of scale separation or ergodicity.Comment: 27 pages, 4 figures, Submitte

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

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).Comment: Accepted for publication in SIAM MM

A Generalized Multiscale Finite Element Method for the Brinkman Equation

In this paper we consider the numerical upscaling of the Brinkman equation in the presence of high-contrast permeability fields. We develop and analyze a robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for the Brinkman model. In the fine grid, we use mixed finite element method with the velocity and pressure being continuous piecewise quadratic and piecewise constant finite element spaces, respectively. Using the GMsFEM framework we construct suitable coarse-scale spaces for the velocity and pressure that yield a robust mixed GMsFEM. We develop a novel approach to construct a coarse approximation for the velocity snapshot space and a robust small offline space for the velocity space. The stability of the mixed GMsFEM and a priori error estimates are derived. A variety of two-dimensional numerical examples are presented to illustrate the effectiveness of the algorithm.Comment: 22 page

Interplay of Theory and Numerics for Deterministic and Stochastic Homogenization

The workshop has brought together experts in the broad field of partial differential equations with highly heterogeneous coefficients. Analysts and computational and applied mathematicians have shared results and ideas on a topic of considerable interest both from the theoretical and applied viewpoints. A characteristic feature of the workshop has been to encourage discussions on the theoretical as well as numerical challenges in the field, both from the point of view of deterministic as well as stochastic modeling of the heterogeneities

Generalized multiscale finite element methods for wave propagation in heterogeneous media

Numerical modeling of wave propagation in heterogeneous media is important in many applications. Due to the complex nature, direct numerical simulations on the fine grid are prohibitively expensive. It is therefore important to develop efficient and accurate methods that allow the use of coarse grids. In this paper, we present a multiscale finite element method for wave propagation on a coarse grid. The proposed method is based on the Generalized Multiscale Finite Element Method (GMsFEM). To construct multiscale basis functions, we start with two snapshot spaces in each coarse-grid block where one represents the degrees of freedom on the boundary and the other represents the degrees of freedom in the interior. We use local spectral problems to identify important modes in each snapshot space. These local spectral problems are different from each other and their formulations are based on the analysis. To our best knowledge, this is the first time where multiple snapshot spaces and multiple spectral problems are used and necessary for efficient computations. Using the dominant modes from local spectral problems, multiscale basis functions are constructed to represent the solution space locally within each coarse block. These multiscale basis functions are coupled via the symmetric interior penalty discontinuous Galerkin method which provides a block diagonal mass matrix, and, consequently, results in fast computations in an explicit time discretiza- tion. Our methods' stability and spectral convergence are rigorously analyzed. Numerical examples are presented to show our methods' performance. We also test oversampling strategies. In particular, we discuss how the modes from different snapshot spaces can affect the proposed methods' accuracy