Juan Galvis

Generalized multiscale finite element methods (GMsFEM)

In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re-used for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse-grid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method.

Multiscale finite element methods for high-contrast problems using local spectral basis functions

In this paper we study multiscale finite element methods (MsFEMs) using spectral multiscale basis functions that are designed for high-contrast problems. Multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem. This local spectral problem strongly depends on the choice of initial partition of unity functions. The resulting space enriches the initial multiscale space using eigenvectors of local spectral problem. The eigenvectors corresponding to small, asymptotically vanishing, eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions. Multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small, asymptotically vanishing, eigenvalues. We present a convergence study that shows that the convergence rate (in energy norm) is proportional to (H/??)1/2, where ?? is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space. Thus, we would like to reach to a larger eigenvalue with a smaller coarse space. This is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems. Numerical results are presented to back-up our theoretical results and to show higher accuracy of MsFEMs with spectral multiscale basis functions. We also present a hierarchical construction of the eigenvectors that provides CPU savings.

Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media

In this paper, we study domain decomposition preconditioners for multiscale flows in high-contrast media. We consider flow equations governed by elliptic equations in heterogeneous media with a large contrast in the coefficients. Our main goal is to develop domain decomposition preconditioners with the condition number that is independent of the contrast when there are variations within coarse regions. This is accomplished by designing coarse-scale spaces and interpolators that represent important features of the solution within each coarse region. The important features are characterized by the connectivities of high-conductivity regions. To detect these connectivities, we introduce an eigenvalue problem that automatically detects high-conductivity regions via a large gap in the spectrum. A main observation is that this eigenvalue problem has a few small, asymptotically vanishing eigenvalues. The number of these small eigenvalues is the same as the number of connected high-conductivity regions. The coars...

Domain Decomposition Preconditioners for Multiscale Flows in High Contrast Media: Reduced Dimension Coarse Spaces

In this paper, robust preconditioners for multiscale flow problems are investigated. We consider elliptic equations with highly varying coefficients. We design and analyze two-level domain decomposition preconditioners that converge independent of the contrast in the media properties. The coarse spaces are constructed using selected eigenvectors of a local spectral problem. Our new construction enriches any given initial coarse space to make it suitable for high-contrast problems. Using the initial coarse space we construct local mass matrices for the local eigenvalue problems. We show that there is a gap in the spectrum of the eigenvalue problem when high-conductivity regions are disconnected. The eigenvectors corresponding to small, asymptotically vanishing eigenvalues are chosen to construct an enrichment of the initial coarse space. Only via a judicious choice of the initial space do we reduce the dimension of the resulting coarse space. Classical coarse basis functions such as multiscale or energy mi...

Approximating Infinity-Dimensional Stochastic Darcy's Equations without Uniform Ellipticity

We consider a stochastic Darcys pressure equation whose coefficient is generated by a white noise process on a Hilbert space employing the ordinary (rather than the Wick) product. A weak form of this equation involves different spaces for the solution and test functions and we establish a continuous inf-sup condition and well-posedness of the problem. We generalize the numerical approximations proposed in Benth and Theting [Stochastic Anal. Appl., 20 (2002), pp. 1191-1223] for Wick stochastic partial differential equations to the ordinary product stochastic pressure equation. We establish discrete inf-sup conditions and provide a priori error estimates for a wide class of norms. The proposed numerical approximation is based on Wiener-Chaos finite element methods and yields a positive definite symmetric linear system. We also improve and generalize the approximation results of Benth and Gjerde [Stochastics Stochastics Rep., 63 (1998), pp. 313-326] and Cao [Stochastics, 78 (2006), pp. 179-187] when a (generalized) process is truncated by a finite Wiener-Chaos expansion. Finally, we present numerical experiments to validate the results.

Balancing Domain Decomposition Methods for Mortar Coupling Stokes-Darcy Systems

We consider Stokes equations in the fluid region Ωf and Darcy equations for the filtration velocity in the porous medium Ωp, and coupled at the interface Γ with adequate transmission conditions. Such problem appears in several applications like well-reservoir coupling in petroleum engineering, transport of substances across groundwater and surface water, and (bio)fluid-organ interactions. There are some works that address numerical analysis issues such as inf-sup and approximation results associated to the continuous and discrete formulations Stokes-Darcy systems [8, 7, 6] and Stokes-Laplacian systems [2, 3], mortar discretizations analysis [12, 6], preconditioning analysis for Stokes-Laplacian systems [4, 1]. Here we are interested on preconditionings for Stokes-Mortar-Darcy with flux boundary conditions, therefore the global system as well as the local systems require flux compatibilities. Here we propose two preconditioners based on balancing domain decomposition methods [9, 11, 5]; in the first one the energy of the preconditioner is controlled by the Stokes system while in the second one it is controlled by the Darcy system. The second is more interesting because it is scalable for the parameters faced in practice. Let Ωf , Ωp ⊂ n be polyhedral subdomains, Ω = int(Ωf ∪ Ωp) and Γ = int(∂Ωf ∪ ∂Ωp), with outward unit normal vectors on ∂Ωj denoted by ηj , j = f, p. The tangent vectors of Γ are denoted by τ 1 (n = 2), or τ l, l = 1, 2 (n = 3). Define Γj := ∂Ωj \ Γ , j = f, p. Fluid velocities are denoted by uj : Ωj → , j = f, p. Pressures are pj : Ωj → , j = f, p. We have:

Local-global multiscale model reduction for flows in high-contrast heterogeneous media

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.

A Domain Decomposition Preconditioner for Multiscale High-Contrast Problems

We present a new class of coarse spaces for two-level additive Schwarz preconditioners that yield condition number bound independent of the contrast in the media properties. These coarse spaces are an extension of the spaces discussed in [3]. Second order elliptic equations are considered. We present theoretical and numerical results. Detailed description of the results and numerical studies will be presented elsewhere.

A Systematic Coarse-Scale Model Reduction Technique for Parameter-Dependent Flows in Highly Heterogeneous Media and Its Applications

In this paper, we propose a multiscale approach for solving the parameter-dependent elliptic equation with highly heterogeneous coefficients. In particular, we assume that the coefficients have both small scales and high contrast (where the high contrast refers to the large variations in the coefficients). The main idea of our approach is to construct local basis functions that encode the local features present in the coefficient to approximate the solution of a parameter-dependent flow equation. Constructing local basis functions involves (1) finding initial multiscale basis functions, and (2) constructing local spectral problems for complementing the initial coarse space. We use the reduced basis (RB) approach to construct a reduced dimensional local approximation that allows quickly computing the local spectral problem. This is done following the RB concept by constructing a low dimensional approximation offline. For any online parameter value, we use a reduced dimensional approximation of the local pr...

Spectral Element Agglomerate Algebraic Multigrid Methods for Elliptic Problems with High-Contrast Coefficients

We apply a recently proposed [5] robust overlapping Schwarz method with a certain spectral construction of the coarse space in the setting of element agglomeration algebraic multigrid methods (or agglomeration AMGe) for elliptic problems with high-contrast coefficients. Our goal is to design multilevel iterative methods that converge independent of the contrast in the coefficients. We present simplified bounds for the condition number of the preconditioned operators. These bounds imply convergence that is independent of the contrast. In the presented preliminary numerical tests, we use geometric agglomerates; however, the algorithm is general and offers some simplifications over the previously proposed spectral agglomerate AMGe methods (cf., [3, 2]).