SUPG Stabilized Petrov-Galerkin Space-Time Finite Element Methods For Two Kinds Of Equations

A kind of fully discrete versions of stabilized Petrov-Galerkin space-time finite element approximate schemes for the convection equations and the convection-diffusionreaction equations was established by combining the space-time finite element discretization with SUPG method.The formulations discussed in this paper were different from traditional SUPG method.The discrete variation forms were used in both time and space directions in order to derive high order accuracy scheme,especially high accuracy in time.The theoretical proofs of this kind of scheme were difficult to find in relative references,although there are some simulations in practical applications studied by engineers.We focused on presenting the proofs of stability and error estimates of the approximate solutions.The techniques of combing the Gauss-Legendre and the Gauss-Lobatto integration rules with the finite element method were used.The conditions on the space-time meshes were removed,and the space and time variables were decoupled.And the idea of the proof without introducing dual problem presented here will build up the theoretical foundations of the stabilized SUPG space-time finite element scheme.And a numerical example was given to verify the validity of the SUPG stabilized Petrov-Galerkin space-time finite element method to solve the convection problem.