Commun. Comput. Phys., 5 (2009), pp. 928-941.

Numerical Investigation on the Boundary Conditions for the Multiscale Base Functions

Shan Jiang 1, Yunqing Huang 2*

1 Department of Mathematics, Xiangtan University, Xiangtan 411105, China.
2 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Institute for Computational and Applied Mathematics, Xiangtan University, Xiangtan 411105, China.

Received 26 May 2007; Accepted (in revised version) 27 June 2008
Available online 14 October 2008


We study the multiscale finite element method for solving multiscale elliptic problems with highly oscillating coefficients, which is designed to accurately capture the large scale behaviors of the solution without resolving the small scale characters. The key idea is to construct the multiscale base functions in the local partial differential equation with proper boundary conditions. The boundary conditions are chosen to extract more accurate boundary information in the local problem. We consider periodic and non-periodic coefficients with linear and oscillatory boundary conditions for the base functions. Numerical examples will be provided to demonstrate the effectiveness of the proposed multiscale finite element method.

AMS subject classifications: 35B30, 35J25, 65C20, 65N12, 65N30

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Key words: Multiscale finite element method, multiscale base functions, oscillatory boundary condition, periodic coefficient, non-periodic coefficient.

*Corresponding author.
Email: (S. Jiang), (Y. Huang)

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