Analysis of a space-time continuous Galerkin method for convection-dominated Sobolev equations

The convergence of space-time continuous Galerkin (STCG) method for the Sobolev equations with convection-dominated terms is studied in this article. It allows variable time steps and the change of the spatial mesh from one time interval to the next, which can make this method suitable for numerical simulations on unstructured grids. We prove the existence and uniqueness of the approximate solution and get the optimal convergence rates in L∞(H1) norm which do not require any restriction assumptions on the space and time mesh size. Finally, some numerical examples are designed to validate the high efficiency of the method showed herein and to confirm the correctness of the theoretical analysis.