A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Some Nonoverlapping Domain Decomposition Methods
SIAM Review
Convergence of a Nonconforming Multiscale Finite Element Method
SIAM Journal on Numerical Analysis
A mixed multiscale finite element method for elliptic problems with oscillating coefficients
Mathematics of Computation
Multi-scale finite-volume method for elliptic problems in subsurface flow simulation
Journal of Computational Physics
SIAM Journal on Scientific Computing
Accurate multiscale finite element methods for two-phase flow simulations
Journal of Computational Physics
Domain decomposition for multiscale PDEs
Numerische Mathematik
Journal of Computational and Applied Mathematics
Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations (Lecture Notes in Computational Science and Engineering)
A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks
Journal of Computational Physics
Local-global multiscale model reduction for flows in high-contrast heterogeneous media
Journal of Computational Physics
Generalized multiscale finite element methods (GMsFEM)
Journal of Computational Physics
Ensemble level multiscale finite element and preconditioner for channelized systems and applications
Journal of Computational and Applied Mathematics
Application of a conservative, generalized multiscale finite element method to flow models
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
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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/@L"*)^1^/^2, where @L"* 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.