Moving mesh method with variational multiscale finite element method for convection-diffusion-reaction equations

The solutions tend to have large gradients, discontinuities, or sharp layers for convection-dominated convection-diffusion-reaction equations. Many studies have demonstrated the advantages of moving mesh and variational multiscale finite element method (VMFEM) in solving such problems. In this paper, we propose a new approach called the moving mesh variational multiscale finite element method (MM-VMFEM), which combines the benefits of both moving mesh and variational multiscale methods for solving convection-diffusion-reaction equations with dominant convection effects. The moving mesh method decouples the mesh equations from the underlying partial differential equations (PDEs) and constructs a connection between the physical and logical domains using harmonic mapping. This moving mesh method can change only the underlying PDEs algorithm partly when solving different differential equations and can keep the mesh topology unchanged after multiple numerical integration. The VMFEM decomposes the unknown scalar and test functions into coarse- and fine-scales, and the use of bubble functions to automatically obtain the stabilization parameters allows the VMFEM to be naturally combined with mesh adaptivity. The effectiveness of the proposed MM-VMFEM is verified by four numerical examples. The experimental results show that the MM-VMFEM can improve the computational accuracy and reduce the numerical spurious oscillations for convection-dominated problems.