Continuous Space-time Finite Element Method For The Parabolic Equation

Parabolic equation arise from some practical problems of engineering and physics,which is a kind of partial differential equation.It is important to analyse and solve numerically this kind of equation.At present,many scholars have studied the numerical solution of partial differential equations,most of them focus on discontinuous elements and mixed discontinuous finite elements.However,there are not many articles on the study of partial differential equations by using the nonstandard finite element method.The continuous space-time element method main feature is that the spatial and the temporal variables are considered simultaneously.Therefore,the advantages of the finite element are simultaneously applied to the space-time variables.In this paper,two-dimensional semi-linear parabolic differential equation is analyzed and studied by using the continuous space-time element method.The existence and uniqueness of the finite element solution are proved and the theoretical error estimates are given.In the first chapter of this paper,we give the space-time finite element method;Sec-ondly,we give the finite element formulation of the equation in the continuous space-time finite element space,the temporal projection operators are defined and their properties are proved;Then we prove the existence and uniqueness of the continuous space-time finite element solution of the equation by using the Brouwer fixed point theorem.At last,we give the space-time error estimates case,which is proved by using the temporal projection operators and their properties;Thirdly,the numerical example is provided to validate the effectiveness.