Study On Reduced-basis Continuous Space-time Finite Element Method For Two Kinds Of Equations

In this paper,the continuous space-time finite element method and reduced-basis method are combined to solve parabolic equation and Sobolev equation.The method discussed here not only has high order accuracy in both space and time directions,but also has the virtue of reducing the degree of freedom.In Chapter 1,the historical background of the continuous space-time finite element method and reduced-basis method is given brief introduction,and the definition and symbol of the finite element space about this paper are given.In Chapter 2,the parabolic equation is studied by reduced-basis continuous space-time finite element method.First,the discrete form of reduced-basis continuous space-time finite element method is provided,then,the variational problem of the dual problem of the original problem is introduced by giving the output function.the existence and uniquness of the numerical solution of the parabolic equation and the dual problem is proved.At last,the proof of posterior error estimate is given.In Chapter 3,the Sobolev equation is studied by reduced-basis continuous space-time finite element method.The discrete form of reduced-basis continuous space-time finite element method is provided,the existence and uniquness of the numerical solution is proved.The proof of posterior error estimate is obtained by constructing the output function and dual problem.Finally,the work of this paper is summarized,and the research directions of future are described in detail.