A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media

  • Hou T
  • Wu X
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A direct numerical solution of the multiple scale prob-lems is difficult even with modern supercomputers. The In this paper, we study a multiscale finite element method for solving a class of elliptic problems arising from composite materials major difficulty of direct solutions is the scale of computa-and flows in porous media, which contain many spatial scales. The tion. For groundwater simulations, it is common to have method is designed to efficiently capture the large scale behavior millions of grid blocks involved, with each block having a of the solution without resolving all the small scale features. This dimension of tens of meters, whereas the permeability is accomplished by constructing the multiscale finite element base functions that are adaptive to the local property of the differential measured from cores is at a scale of several centimeters operator. Our method is applicable to general multiple-scale prob-[23]. This gives more than 10 5 degrees of freedom per lems without restrictive assumptions. The construction of the base spatial dimension in the computation. Therefore, a tremen-functions is fully decoupled from element to element; thus, the dous amount of computer memory and CPU time are re-method is perfectly parallel and is naturally adapted to massively quired, and they can easily exceed the limit of today's parallel computers. For the same reason, the method has the ability to handle extremely large degrees of freedom due to highly hetero-computing resources. The situation can be relieved to some geneous media, which are intractable by conventional finite element degree by parallel computing; however, the size of discrete (difference) methods. In contrast to some empirical numerical problem is not reduced. The load is merely shared by more upscaling methods, the multiscale method is systematic and self-processors with more memory. Some recent direct solu-consistent, which makes it easier to analyze. We give a brief analysis of the method, with emphasis on the ''resonant sampling'' effect. tions of flow and transport in porous media are reported Then, we propose an oversampling technique to remove the reso-in [1, 25, 9, 22]. Whenever one can afford to resolve all the nance effect. We demonstrate the accuracy and efficiency of our small scale features of a physical problem, direct solutions method through extensive numerical experiments, which include provide quantitative information of the physical processes problems with random coefficients and problems with continuous at all scales. On the other hand, from an engineering per-scales. Parallel implementation and performance of the method are also addressed. ᮊ 1997 Academic Press spective, it is often sufficient to predict the macroscopic properties of the multiple-scale systems, such as the effec-tive conductivity, elastic moduli, permeability, and eddy

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  • Thomas Y. Hou

  • Xiao-Hui Wu

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