A convergence analysis of Generalized Multiscale Finite Element Methods

Abstract

In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be extended, without great difficulty, to more sophisticated GMsFEMs. For concreteness, the obtained error estimates generalize and simplify the convergence analysis of [J. Comput. Phys. 230 (2011), 937-955]. The GMsFEM method construct basis functions that are obtained by multiplication of (approximation of) local eigenvectors by partition of unity functions. Only important eigenvectors are used in the construction. The error estimates are general and are written in terms of the eigenvalues of the eigenvectors not used in the construction. The error analysis involve local and global norms that measure the decay of the expansion of the solution in terms of local eigenvectors. Numerical experiments are carried out to verify the feasibility of the approach with respect to the convergence and stability properties of the analysis in view of the good scientific computing practice.

Multiscale Method for Solving High-order BVPs

“…Pezza [17] and Aminikhah [18] proposed a multiscale numerical algorithm for fractional BVPs. In recent years, multiscale finite element method has also been applied to solve the numerical solutions of partial differential equations [19,20]. Reproducing kernel space is an important banach space, which has been used in the field of numerical analysis.…”

On the Conservation Properties in Multiple Scale Coupling and Simulation for Darcy Flow with Hyperbolic-Transport in Complex Flows

“…The new elliptic solver [14,15] is a general tool for imposing local conservation for highorder methods to deal with multiscale permeabilities. In particular is also applicable to the Generalized Multiscale Finite Element Method (GMsFEM); see [16,74,119,75,118] and references therein. On the other hand, the new Lagrangian-Eulerian method seems to be a promissing general approach to capture nonlinear wave interactions linked to multiscale behavior of interactions of waves in a wide range of models and, in particular, for systems (see [17,18,19,20,21,22]).…”

“…Understanding the multiscale properties of subsurface flows is a major problem of modern approaches predicting groundwater level changes and predictive technologies in petroleum reservoir. In this regard, many innovative techniques have been reported as such local-global upscaling approach [63,78,96,53,117,62], multiscale methods [1,32,82,84,91,98,99,79,30], model order reduction techniques [71,83,45,110,55], Twoscale homogenization theory [66,25,26,42,86,112,31]; see papers for a survey on recent development of multiscale computing and modeling approach [110,2,114,76,77,116,80].…”

“…Despite the efforts of many researchers, no universal or unified multiscale modeling and methods for fluid flow through naturally complex geologic reservoirs has been achieved so far (see, e.g., [2,116,46,105,8,110,83,71,98,32,62,117,63]). Under appropriate simplification assumptions compositional and black-oil models can be further simplified for the fundamental multiscale two-phase immiscible displacement with no mass transfer between phases (this is often appropriate in models describing displacements at the length scales associated with reservoir simulation grid blocks, see, e.g., [63]).…”

“…The paper is organized as follows. In Section 2, we construct an embedded high-order model for second-orde elliptic problems with local and global mass conservation, clarifying and simplifying the presentation of its conservative properties in lines as introduced in [14,14,75,3,16]. Next, we construct new a locally conservative Lagrangian-Eulerian method for hyperbolic-transport with focus on a novel approach for conservation properties of the no flow surface region for hyperbolic conservation laws in Section 3.…”