Iterative Splitting Methods for Multiscale Problems

One motivation to solve multiscale problems arises from dynamical problems in fluids and plasmas. For numerical methods to be suitable for such problems it is important that one consider the coupling involved in the multiscale problems. We consider the coupling of two different scales, e.g., the microand macro-scales, and accelerate the standard splitting schemes via novel schemes based on the idea of embedding the microscales into the macro-scale or by reconstructing the macro-scale with partially micro-scale computations. We concentrate on a recent modification of a standard iterative splitting scheme with respect to the micro -- macro coupling (interpolation) and macro -- micro coupling (restriction) and the equilibration of the scales. The convergence of the novel multiscale iterative splitting scheme (MISS) is discussed, as well as its algorithmical implementation. Applications of such splitting schemes in space and time are presented, at first for simple fluid dynamic problems and stochastic problems. At the end of the paper, we summarize our results and present some ideas for future research.