A Combined Finite Element And Multiscale Finite Element Method For The Advection-diffusion Transport Equations

The numerical solution of the advection-diffusion transport equation has been the object of intense research for the past few decades. However, there is a fundamental difficulty, which is the need to treat simultaneously hyperbolic terms associated with advection and elliptic terms associated with diffusion terms. In fact, no numerical method has yet fully overcome all these problems, especially the case of advection-dominant transport problems.In this paper, a mixed Lagrangian-Eulerian method, based on FE-MsFEM, up-wind FEM and local zooming approach using particle tracking technique, is presented to solve the advection-diffusion transport equations with multiscale and singularities. The FE-MsFEM is a combined finite element and multiscale finite element method, which is developed to overcome the difficulties while the oversampling multiscale fi-nite element method(MsFEM) is used to solve the multiscale problems with singulari-ties in some portions. It deals with such portions by using the standard finite element method on a fine mesh and the other portions by the oversampling MsFEM.To illustrate this FE-MsFEM-LEZOOM method, we first give a full description, then carry out several numerical experiments to demeonstrate the accuracy and effi-ciency of the proposed method.