  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.
