In this paper, we consider a cubic \(H^2\) nonconforming finite element scheme \(B_{h0}^3\) which does not correspond to a locally defined finite element with Ciarlet\('\)s triple but admit a set of local basis functions. For the first time, we deduce and write out the expression of basis functions explicitly. Distinguished from the most nonconforming finite element methods, \((\delta \Delta _h\cdot ,\Delta _h\cdot )\) with non-constant coefficient \(\delta >0\) is coercive on the nonconforming \(B_{h0}^3\) space which makes it robust for numerical discretization. For fourth order eigenvalue problem, the \(B_{h0}^3\) scheme can provide \(\mathcal {O}(h^2)\) approximation for the eigenspace in energy norm and \(\mathcal {O}(h^4)\) approximation for the eigenvalues. We test the \(B_{h0}^3\) scheme on the vary-coefficient bi-Laplace source and eigenvalue problem, further, transmission eigenvalue problem. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed scheme.

在本文中，我们考虑三次\（H ^ 2 \）非协调有限元方案\（B_ {h0} ^ 3 \），该方案不对应于具有Ciarlet \（'\） s三元组的局部定义有限元，但可以接受一组本地基础功能。第一次，我们显式推导并写出基函数的表达式。与最不合格的有限元方法不同，\（（\ delta \ Delta _h \ cdot，\ Delta _h \ cdot）\）具有非恒定系数\（\ delta> 0 \）在非合格\（B_ { h0} ^ 3 \）空间，使其对于数值离散具有鲁棒性。对于四阶特征值问题，\（B_ {h0} ^ 3 \）方案可以提供能量范数中本征空间的\（\ mathcal {O}（h ^ 2）\）近似和本征值的\（\ mathcal {O}（h ^ 4）\）近似。我们在变系数双拉普拉斯源和特征值问题，以及传输特征值问题上测试了\（B_ {h0} ^ 3 \）方案。最后，通过数值算例说明了所提方案的有效性。