An exponential integration generalized multiscale finite element method for parabolic problems.

We consider linear and semilinear parabolic problems posed in high-contrast multiscale media in two dimensions. The presence of high-contrast multiscale media adversely affects the accuracy, stability, and overall efficiency of numerical approximations such as finite elements in space combined with some time integrator. In many cases, implementing time discretizations such as finite differences or exponential integrators may be impractical because each time iteration needs the computation of matrix operators involving very large and ill-conditioned sparse matrices. Here, we propose an efficient Generalized Multiscale Finite Element Method (GMsFEM) that is robust against the high-contrast diffusion coefficient. We combine GMsFEM with exponential integration in time to obtain a good approximation of the final time solution. Our approach is efficient and practical because it computes matrix functions of small matrices given by the GMsFEM method. We present representative numerical experiments that show the advantages of combining exponential integration and GMsFEM approximations. The constructions and methods developed here can be easily adapted to three-dimensional domains.