A space-time finite element method based on local projection stabilization in space and discontinuous Galerkin method in time for convection-diffusion-reaction equations

Abstract

In this article, we combine the local projection stabilization (LPS) technique in space and the discontinuous Galerkin (DG) method in time to investigate the time-dependent convection-diffusion-reaction problems. This kind of stabilized space-time finite element (STFE) scheme, based on approximation space enriched by bubble functions that can increase stability, is constructed. The existence, uniqueness and stability are proved with the properties of Lagrange interpolation polynomials established on Radau points in time direction. An error estimate in L∞(L2)-norm is given by introducing the elliptic projection operators in space direction. This estimate approach is different from the previous ones that construct a special interpolant into approximation space showing an extra orthogonality property on the projection space. Since the techniques of Lagrange interpolation in time direction decouple time and space variables, the method proposed in this paper has the advantages of reducing calculation and simplifying theoretical analysis. The space and time convergence orders are illustrated in the first numerical example with smooth solutions. A comparison between the traditional STFE scheme and the constructed scheme for the problem having exponential boundary layers is presented in the second numerical example. The simulation results show that the novel method can greatly reduce nonphysical oscillations. And the influences of the stabilization parameters on the behavior of the approximate solution are discussed by some numerical results.