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This paper studies the mixed element method for the boundary value problem of the biharmonic equationΔ  2   ⁢  u   =  f{\Delta^{2}u=f}    in two dimensions. We start from au  ∼   ∇  ⁡  u   ∼    ∇  2   ⁡  u   ∼   div  ⁢    ∇  2   ⁡  u{u\sim\nabla u\sim\nabla^{2}u\sim\operatorname{div}\nabla^{2}u}    formulation that is discussed in [ 4 ] and construct its stability onH  0  1   ⁢   (  Ω  )    ×    H  ~   0  1    ⁢   (  Ω  )    ×    L  ¯   sym  2    ⁢   (  Ω  )    ×   H   -  1⁢   (  div  ,  Ω  ){H^{1}_{0}(\Omega)\times\tilde{H}^{1}_{0}(\Omega)\times\bar{L}_{\mathrm{sym}}^% {2}(\Omega)\times H^{-1}(\operatorname{div},\Omega)}   . Then we utilise the Helmholtz decomposition ofH   -  1    ⁢   (  div  ,  Ω  ){H^{-1}(\operatorname{div},\Omega)}    and construct a new formulation stable on first-order and zero-order Sobolev spaces. Finite element discretisations are then given with respect to the new formulation, and both theoretical analysis and numerical verification are given.

A Stable Mixed Element Method for the Biharmonic Equation with First-Order Function Spaces