A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales

We present a technique for solving parametrized elliptic partial differential equations with multiple scales. The technique is based on the combination of the reduced basis method [C. Prud’homme, D. Rovas, K. Veroy, Y. Maday, A.T. Patera, G. Turinici, Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods, Journal of Fluids Engineering 124 (1) (2002) 70–80] and the multiscale finite element method [T.Y. Hou, X.H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of Computational Physics 134 (1) (1997) 169–189] to treat problems in which the differential coefficient is characterized by a large number of independent parameters. For the multiscale finite element method, a large number of cell problems has to be solved at the fine local mesh for each new configuration of the differential coefficient. In order to improve the computational efficiency of this method, we construct reduced basis spaces that are adapted to the local parameter dependence of the differential operator. The approximate solutions of the cell problems are computed accurately and efficiently via performing Galekin projection onto the reduced basis spaces and implementing the offline–online computational procedure. Therefore, a large number of similar computations at the fine local mesh can be carried out with lower computational cost for each new configuration of the differential coefficient. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed approach.