The Theory And Application Of The Least-Squares Mixed Finite Element Methods

In this paper, the following second-order elliptic problemis simulated by a new method, which is a combination of least-squares and expanded mixed finite element, least-squares expanded mixed finite element method. Our analysis shows that the new method inherits all the advantages of both least-squares and expanded mixed finite element method, i.e., it can explicitly approximate three variables: the scalar unknown, its gradient and its flux, the finite element spaces are not subject to the LBB consistency condition, and the resulting system is symmetric and positive definite. The stability, existence and uniqueness of the solution for the weak form are proved. Optimal error estimates for the scalar unknown, its gradient and its flux in H~1, L~2 and L~2-space are derived. Numerical tests confirm the efficiency of the new method.Then we simulate the following second-order parabolic problem by least-squares expanded mixed finite method. The semi discretization and fully discretization least-squares expanded mixed finite element procedures for this problemare proposed. And we prove the solvability and stability of the procedures we proposed. Numerical analysis shows that the method can approximate the scalar unknown, its gradient and its flux well. Numerical tests confirm the efficiency of the method.