A stochastic heterogeneous multiscale method for porous media flow

A new multiscale algorithm is introduced based on the framework of the heterogeneous multiscale method. The mixed finite element method used ensures continuity of the flux within the entire domain. This method is shown to be free of “resonance error” and uses less memory than the mixed multiscale finite element method. To account for the highstochastic dimensionality of the permeability field, we employ the recently developed stochastic high dimensional model representation (HDMR) technique for the solution of stochastic PDEs. This method efficiently decomposes the original high-dimensional problem into several lower dimensional sub-problems which can be efficiently solved using adaptive sparse grid collocation. The accuracy and efficiency of the method is demonstrated through a number of benchmark examples.