A Combined Finite Element And Multiscale Finite Element Method For The Convection-diffusion Equations With Oscillating Coeffi- Cients

The numerical solution of the advection-diffusion transport equation has been the object of intense research for the past few decades. To our knowledge,there are not so many works on how to solve the advection-diffusion transport equation which has the strongly varying diffusion and convection coefficients simultaneously. This paper pro-poses a combined finite element and multiscale finite element method(FE-MsFEM) to solve the convection-diffusion type equation with highly oscillating coefficients.which arises in the studying of groundwater and solute transport in porus media. The key point of the proposed method is to deal with long narrow channels by using the stan-dard finite element method on a fine mesh and the other portions by the oversampling MsFEM and to deal with the convection terms by upwind FEM. The transmission con-ditions across the FE-MsFE interface is treated by the penalty technique. To illustrate this idea,a rigorous error analysis for this FE-MsFEM is given under the assumption that the oscillating coefficients are periodic and scale-separable. While such a sim-plifying assumption is not required by our method,it allows us to use homogenization theory to obtain the asymptotic structure of the solution. Numerical experiments are carried out for the convection-diffusion type elliptic equations with periodic and ran-dom highly oscillating coefficients to demonstrate the accuracy and efficiency of the proposed method.