A new \(C^0\) nonconforming quasi three degree element with 13 freedoms is introduced to solve biharmonic problem. The given finite element space consists of piecewise polynomial space \(P_3\) and some bubble functions. Different from non-\(C^0\) 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 \(u\in H_0^2(\Omega )\cap H^4(\Omega )\), the scheme is of linear order convergence by the measurement of discrete norm \(\Vert \cdot \Vert _h\). Some numerical results are included to further illustrate the convergence analysis.

为了解决双调和问题，引入了一种新的具有（13）自由度的（C ^ 0 \）不合格准三度元。给定的有限元空间由分段多项式空间\（P_3 \）和一些气泡函数组成。与非\（C ^ 0 \）不同如果采用非一致性方案，则可以通过这种方法获得更平滑的离散解。与现有的16自由度有限元方法相比，该方案使用的自由度更少。由于有限元不是仿射等效项，因此通过引入其他仿射有限元可以从技术上证明相关的内插误差估计。利用这个空间来解决双调和问题，证明了真解与离散解之间的收敛性分析。在更强的假设为真解\（u_0在H_0 ^ 2（\ Omega）\ cap H ^ 4（\ Omega）\）下，该方案通过测量离散范数\（\ Vert \ cdot \ Vert _h \）。包括一些数值结果以进一步说明收敛分析。