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Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation. (English) Zbl 1085.65109

Summary: We continue the study of the nonconforming multiscale finite element method (Ms-FEM) introduced by T. Y. Hou and X.-H. Wu [J. Comput. Phys. 134, No. 1, 169–189 (1997; Zbl 0880.73065)] and by Y. R. Efendiev, T. Y. Hou and X.-H. Wu [SIAM J. Numer. Anal. 37, No. 3, 888–910 (2000; Zbl 0951.65105)] for second-order elliptic equations with highly oscillatory coefficients. The main difficulty in MsFEM, as well as other numerical upsealing methods, is the scale resonance effect. It has been show that the leading order resonance error can be effectively removed by using an over-sampling technique. Nonetheless, there is still a secondary cell resonance error of \(O(\varepsilon^2/h^2)\). Here, we introduce a Petrov-Galerkin MsFEM formulation with nonconforming multiscale trial functions and linear test functions. We show that the cell resonance error is eliminated in this formulation and hence the convergence rate is greatly improved. Moreover, we show that a similar formulation can be used to enhance the convergence of an immersed-interface finite element method for elliptic interface problems.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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