In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis.

翻译：本文研究了具有非齐次系数的双拉普拉斯问题的数值方法；具体而言，我们分别在矩形网格上提出了针对非齐次四阶椭圆奇异摄动问题和亥姆霍兹传输特征值问题的有限元格式。新方法使用分片二次多项式的简化矩形Morley（简称RRM）元素空间，该空间具有可能的最低阶数。对于该有限元空间，证明了Grisvard等式的离散模拟以解决稳定性问题，并构造了局部平均插值算子以处理逼近问题。证明了这些格式的最优收敛速率，并给出数值实验以验证理论分析。