Abstract: We considered the stability of the weak Galerkin finite element method for solving the second order elliptic equations and Biharmonic equations. Firstly, we introduced  the weak functions and the weak gradient operators to approximate the standard functions and standard differential operators in the weak Galerkin method. Secondly,  we gave the definitions of  norms of |||·||| and |||·|||_-1in the weak  function space, and obtained  the stability of the weak Galerkin finite element method based on the two norms.

Key words: weak differential operator, weak Galerkin finite element method, stability, elliptic equation, Biharmonic equation