The Mixed Space-time Finite Element Methods For Sobolev Equation

The space-time finite element method is a kind of numerical methods,in which the finite element discretization was used in space direction,and the variational discrete form was applied in time directions.It has formular high order accuracy in both and time variables.And it can deal with problems with complex boundary conditions,easy to accomplish the adaptive algorithm because of the technique of unifying the space and time variables.The mixed finite element method lowered the order of the original problem by introducing the auxiliary variables.A kind of new mixed space time finite element method is constructed by combing the mixed method and space time finite element method,and the constructed scheme has the advantages of both the mixed method and space time method,for example,lowering equation order and space-time high order accuracy.And the new method is used to solve Sobolev equation.The main contents of this paper include following two aspects.Firstly,the high order accuracy scheme of continuous mixed space-time finite elemen-t method for Sobolev equation is constructed by the techniques of introducing auxiliary variable (?).The order of the equation is lowered by mixed method.The finite element discretization is used in both space and time directions,i.e.the space time finite element method is adopted.The stability and existence as well as unique-ness of the mixed space time finite element solutions are proved.The space and time projections are introduced and the according properties of the projections are discussed.The error estimates of the mixed space time finite element solution is derived based on the properties of the space time projection.And the numerical simulations are given to test the efficiency of the approximate scheme.And the numerical results conformed the theoretical analysis.The contents of this paper,including the mixed space-time finite element numerical scheme for Sobolev equation,and the analysis method of introducing space time projections in error estimates is given for the first time.In the meantime,the time basis functions of Lagrange type are constructed with Gauss quadrature points.The numerical solution is represented by this kind of basis functions of time variable.The existence and uniqueness of the fully discrete solution is proved.The error estimates between the fully discrete solution and semi-discrete solution is obtained.And it is a kind of new analysis method of mixed space-time finite element approximate scheme for Sobolev equation.