Superconvergence Of The Split Least-Squares Mixed Finite Element Method For Parabolic Equations

This paper discusses split least-squares mixed finite element method of the parabolic problem.We use fully discrete scheme,list the split least-squares mixed finite element scheme of first order and Crank-Nicolson split least-squares mixed finite element scheme,then obtain the superconvergence result.The split least-squares mixed finite element method can overcome the disadvantages of the mixed finite element method,it doesn't need to satisfy the compatibility conditions LBB,the intermediate variables can be solved separately,thus solving the original function,the efficiency can be improved highly.At last,we give a numerical example to varify the theoretical result is correct and to show the performance of the introduced schemes.The main work of this paper is outlined as follows:First,we use backward euler differentical to discrete the time variable,obtain split least-squares mixed finite element scheme of first order,prove the existence and uniqueness of the shemes' solution,then introduce the interpolation,finally obtain the superconvergence result.Second,we use central differential to discrete the time variable,obtain Crank-Nicolson split least-squares mixed finite element scheme,prove the existence and uniqueness of the shemes' solution,then introduce the interpolation,finally obtain the superconvergence result.Finally,we appropriatly choose some examples to proceed the numberal simulation,varifying the effectness and superconvergence of the two schemes.