LAUSR

A new $$C^0$$ C 0 nonconforming quasi three degree element with 13 freedoms is introduced to solve biharmonic problem. The given finite element space consists of piecewise polynomial space $$P_3$$ P 3 and some bubble functions. Different from non- $$C^0$$ C 0 nonconforming scheme, a smoother discrete solution can be obtained by this method. Compared with the existed 16 freedoms finite element method, this scheme uses less freedoms. As the finite elements are not affine equivalent each other, the associated interpolating error estimation is technically proved by introducing another affine finite elements. With this space to solve biharmonic problem, the convergence analysis is demonstrated between true solution and discrete solution. Under a stronger hypothesis that true solution $$u\in H_0^2(\Omega )\cap H^4(\Omega )$$ u ∈ H 0 2 ( Ω ) ∩ H 4 ( Ω ) , the scheme is of linear order convergence by the measurement of discrete norm $$\Vert \cdot \Vert _h$$ ‖ · ‖ h . Some numerical results are included to further illustrate the convergence analysis.