Offline computation is an essential component in most multiscale model reduction techniques. However, there are multiscale problems in which offline procedure is insufficient to give accurate representations of solutions, due to the fact that offline computations are typically performed locally and global information is missing in these offline information. To tackle this difficulty, we develop an online local adaptivity technique for local multiscale model reduction problems. We design new online basis functions within Discontinuous Galerkin method based on local residuals and some optimally estimates. The resulting basis functions are able to capture the solution efficiently and accurately, and are added to the approximation iteratively. Moreover, we show that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully. Our analysis also gives

Abstract BDDC and FETI-DP algorithms are developed for three-dimensional elliptic problems with adaptively enriched coarse components. It is known that these enriched components are necessary in the development of robust preconditioners. To form the adaptive coarse components, carefully designed generalized eigenvalue problems are introduced for each faces and edges, and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance T O L. Upper bounds for condition numbers of the preconditioned systems are shown to be C T O L, with the constant C depending only on the maximum number of edges and faces per subdomain, and the maximum number of subdomains sharing an edge. Numerical results are presented to test the robustness of the proposed approach.

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale medium property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave

The construction of local reduced-order models via multiscale basis functions has been an area of active research. In this paper, we propose online multiscale basis functions which are constructed using the offline space and the current residual. Online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators. We derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error. This error decrease is independent of physical parameters, such as the contrast and multiple scales in the problem. The offline spaces are constructed using Generalized Multiscale Finite Element Methods (GMsFEM). We show that if one chooses a sufficient number of offline basis functions, one can guarantee that additional online multiscale basis functions will reduce

In this paper, we present a mixed generalized multiscale finite element method (GMsFEM) for solving flow in heterogeneous media. Our approach constructs multiscale basis functions following a GMsFEM framework and couples these basis functions using a mixed finite element method, which allows us to obtain a mass conservative velocity field. To construct multiscale basis functions for each coarse edge, we design a snapshot space that consists of fine-scale velocity fields supported in a union of two coarse regions that share the common interface. The snapshot vectors have zero Neumann boundary conditions on the outer boundaries, and we prescribe their values on the common interface. We describe several spectral decompositions in the snapshot space motivated by the analysis. In the paper, we also study oversampling approaches that enhance the accuracy of mixed GMsFEM. A main idea of