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Analysis of two-scale finite volume element method for elliptic problem. (English) Zbl 1067.65124

A class of finite volume element methods is proposed and analyzed for solving a second order elliptic boundary value problem whose solution is defined in more than one length scales. The method has the ability to incorporate the small scale behaviors of the solution on the large scale one. The same problem is also dealt with in the homogenization theory, but the proposed method is somewhat different from it. This is achieved through the construction of the basis functions on each element that satisfy the homogeneous elliptic differential equation. Furthermore, the method realizes numerical conservation feature which is highly desirable in many applications.
Existing analyses on its finite element counterpart reveal that there exists a resonance error between the mesh size and the small length scale. Such a result motivates an oversampling technique to overcome this drawback. An analysis of the proposed method is developed under the assumption that the coefficients are of two scales and periodic in the small scale.
The theoretical results are confirmed experimentally by several convergence tests. Moreover, an application of the method to flows in porous media is presented. It is also tested bv some other problems in a separate paper.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
76M12 Finite volume methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage

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References:

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