Convergence analysis of the multiscale method for a class of convection–diffusion equations with highly oscillating coefficients

This paper proposes a kind of multiscale method to solve the convection–diffusion type equation with highly oscillating coefficients, which arises in the studying of groundwater and solute transport in porous media. The introduced method is based on the framework of nonconforming finite element method, which can be considered as a realization of the heterogeneous multiscale method or variational multiscale method. The key point of the proposed method is to define a modified variational bilinear form with appropriate cell problems. Optimal estimate is proved for the error between the solution of the multiscale method and the homogenized solution under the assumption that the oscillating coefficients are periodic. While such a simplifying 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 coefficients to demonstrate the accuracy of the proposed method. Moreover, we successfully use the method to solve the time dependent convection–diffusion equation which models the solute transport in a porous medium with a random log-normal relative permeability.