Two Types Of Mixed Continuous Space-Time Finite Element Methods For Sobolev Equation

On the basis of mixed finite element method,H~1-Galerkin finite element method,and extended mixed finite element method,combined with the splitting scheme,within the space-time finite element framework,the one-dimensional Sobolev equation and the two-dimensional Sobolev equation are numerically solved respectively.This type of numerical method not only has high-order accuracy in time and space,but also avoids the difficulty of solving coupled equations.At the same time,in the process of solving the one-dimensional Sobolev equation,the H~1-Galerkin finite element method,does not need to meet the LBB compatibility condition.In chapter 1,a brief description of the relevant historical background of the finite element method,some finite element methods for solving Sobolev equation,and the main research work of this article are discussed.In chapter 2,the H~1-Galerkin space-time mixed finite element splitting scheme of the one-dimensional Sobolev equation is given,and the existence,uniqueness and stability of its finite element solution are proved,and then for different variables,different space-time projection are introduced.The error estimates of the three variables are given respectively,and we provide a numerical example to verify the validity and feasibility of the format finally.In chapter 3,the space-time extended mixed finite element scheme of the two-dimensional Sobolev equation is given,and the existence and uniqueness of its finite element solution is proved.By introducing a mixed elliptic projection,the error estimate of the numerical solution is given.Finally,we summarize the research work of this article,and briefly explain the future research directions.