Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation

We continue the study of the nonconforming multiscale finite element method (Ms- FEM) introduced in 17, 14 for second order elliptic equations with highly oscillatory coefficients. The main difficulty in MsFEM, as well as other numerical upscaling 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(Є^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

An isogeometric analysis for elliptic homogenization problems

A novel and efficient approach which is based on the framework of isogeometric analysis for elliptic homogenization problems is proposed. These problems possess highly oscillating coefficients leading to extremely high computational expenses while using traditional finite element methods. The isogeometric analysis heterogeneous multiscale method (IGA-HMM) investigated in this paper is regarded as an alternative approach to the standard Finite Element Heterogeneous Multiscale Method (FE-HMM) which is currently an effective framework to solve these problems. The method utilizes non-uniform rational B-splines (NURBS) in both macro and micro levels instead of standard Lagrange basis. Beside the ability to describe exactly the geometry, it tremendously facilitates high-order macroscopic/microscopic discretizations thanks to the flexibility of refinement and degree elevation with an arbitrary continuity level provided by NURBS basis functions. A priori error estimates of the discretization error coming from macro and micro meshes and optimal micro refinement strategies for macro/micro NURBS basis functions of arbitrary orders are derived. Numerical results show the excellent performance of the proposed method