Communications in Mathematical Sciences

Volume 8 (2010)

Number 4

Heterogeneous multiscale finite element method with novel numerical integration schemes

Pages: 863 – 885



Rui Du

Pingbing Ming


In this paper we introduce two novel numerical integration schemes within the framework of the heterogeneous multiscale method (HMM), when the finite element method is used as the macroscopic solver, to resolve the elliptic problem with a multiscale coefficient. For nonself-adjoint elliptic problems, optimal convergence rate is proved for the proposed methods, which naturally yields a new strategy for refining the macro-micro meshes and a criterion for determining the size of the microcell. Numerical results following this strategy show that the new methods significantly reduce the computational cost without loss of accuracy.


Heterogeneous multiscale method; finite element method; numerical integration schemes; elliptic homogenization problems

2010 Mathematics Subject Classification

39A12, 65N30, 74Q05, 74Q15, 74Q20

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