Proper Generalized Decomposition Method Based On Reduced Multiscale Finite Element And Its Applications For The Elliptic Equations

With the rapidly development of the applied science and engineering technology based on mathematics,many felt a good need for model construction and numerical simulation,and typically the model equips large amounts of complex data and high-dimensional parameters.Proper generalized decomposition(PGD)becomes an effec-tive method to approximate parameterized multiscale model,and further the PGD could be viewed as a particularly efficient a priori model reduction strategy.The solu-tion is sought as a finite sum of terms,each one involving the product of functions of each coordinate.In order to efficiently approximate the multiscale model,generalized multiscale finite element method(GMsFEM)is introduced,the multiple scales can be captured by constructing the multiscale basis in the coarse grid instead of solving the model in a very fine mesh.In the framework of GMsFEM,cross-validation is applied to identify the optimal generalized multiscale finite element basis,thus a reduced or-der multiscale model is obtained by projecting the original full order model onto the reduced multiscale finite element space,that is,Reduced Generalized Multiscale Finite Method based on Cross-Validation(CV-RGMsFEB).For each non-linear iteration of PGD,reduced order multiscale model is used to approximate the spatial space,which is the main idea of proper generalized decomposition(PGD)based on reduced multi-scale finite element method.To illustrate the efficacy of the proposed methods,a few numerical examples for elliptic partial differential equations(PDEs)with multsicale and random inputs are presented.