Moving Mesh with Streamline Upwind Petrov-Galerkin (MM-SUPG) Method for Convection-Diffusion Problems

We investigate the effect of the streamline upwind Petrov-Galerkin method (SUPG) as it relates to the moving mesh partial differential equation (MMPDE) method for convection-diffusion problems in the presence of vanishing diffusivity. We first discretize in space using linear finite elements and then use a $\theta$-scheme to discretize in time. On a fixed mesh, SUPG (FM-SUPG) is shown to enhance the stability and resolves spurious oscillations when compared to the classic Galerkin method (FM-FEM) when diffusivity is small. However, it falls short when the layer-gradient is large. In this paper, we develop a moving mesh upwind Petrov-Galerkin (MM-SUPG) method by integrating the SUPG method with the MMPDE method. Numerical results show that our MM-SUPG works well for these types of problems and performs better than FM-SUPG as well as MMPDE without SUPG.