RBF-based multiscale control volume method for second order elliptic problems with oscillatory coefficients

Abstract

Many important engineering problems have multiple-scale solutions. Thermal conductivity of composite materials, flow in porous media, and turbulent transport in high Reynolds number flows are examples of this type. Direct numerical simulations for these problems typically require extremely large amounts of CPU time and computer memory, which may be too expensive or impossible on the present supercomputers. In this paper, we develop a high order computational method, based on multiscale basis function approach and integrated radial-basis-function (IRBF) approximant, for the solution of multiscale elliptic problems with reduced computational cost. Unlike other methods based on multiscale basis function approach, sets of basis and correction functions here are obtained through $C^2$-continuous IRBF element formulations. High accuracy and efficiency of this method are demonstrated by several one- and two-dimensional examples.