Discontinuous Galerkin Method With Upscaling Technique For Multiscale Convection-Diffusion Problem

In this paper, we improve the interior penalty discontinuous Galerkin method at the basis of the upscaling technique for heterogeneous multiscale convection-diffusion problem. The kind of problem describes the flow transport in heterogenous porous me-dia. The traditional finite element method and finite volume method for the multiscale problem require extremely fine mesh to guarantee the accuracy of the numerical solu-tion at the cost of tremendous calculation. Upscaling method could solve multiscale convection-diffusion problem effectively. However, it doesn’t perform well for the convection-dominated problem. Now, we study a combined multiscale method which makes full use of the interior penalty discontinuous Galerkin method with the upscal-ing technique to solve the convection-diffusion problem. We name it IPDG-Upscaling Method. The method incorporates the upscaling technique into IPDG. IPDG method has a lot of advantage, such as local mass conservation and flexibility in mesh par-tition. Upscaling technique could capture the small scale information of the exact solution. Assuming that the oscillating coefficients are periodic, we present particular error analysis and also provide numerical examples to verify the feasibility and effi-ciency. Otherwise, IPDG-Upscaling method contributes to solving the the convection-diffusion problem with random coefficients and the convection-dominated problem. This explains that IPDG-Upscaling method is practical widely.