Defect corrected averaging for highly oscillatory problems

Abstract

The accurate solution of partial differential equations with highly oscillatory source terms over long time scales constitutes a challenging problem. There exists a variety of methods dealing with problems where there are processes, equations or variables on fine and coarse scales. Multiscale methods have in common, that they neither fully resolve the fine scale, nor completely ignore it. On the one hand, these methods strive, without significantly sacrificing accuracy or essential properties of the system, to be much more efficient than methods that fully resolve the fine scale. On the other hand, these methods should be considerably more accurate than methods that completely ignore the fine scale. Our defect corrected averaging procedure is based on a modified coarse scale problem, that approximates the solution of the fine scale problem in stroboscopic points. Nevertheless, our approximation process is clearly different from the stroboscopic averaging method. We give an error estimate for the solution of the modified problem. The computational efficiency of the approximation is furthermore improved by the application of preconditioning techniques. Tests on numerical examples show the efficiency and reliability of our approach.