Application of the generalized multiscale finite element method in parameter-dependent PDE simulations with a variable-separation technique

In this paper, we combine the generalized multiscale finite element method (GMsFEM) with a variable-separation technique to tackle the parameter-dependent partial differential equations (PDEs). The solution is approximated via an expansion series, each term of which lives in the tensor product of the parametric space and the spatial space. Governing equations for each term are derived based on energy minimization. An iterative algorithm is presented to obtain the expansion series, which requires solving parameter-independent PDEs repeatedly. We then present the procedure of GMsFEM and apply it to these parameter-independent PDEs. Numerical examples are presented to demonstrate the effectiveness of the expansion series and the computational efficiency brought by GMsFEM.