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In this paper, a piecewise quadratic nonconforming finite element method onrectangular grids for a fourth-order elliptic singular perturbation problem ispresented. This proposed method is robustly convergent with respect to theperturbation parameter. Numerical results are presented to verify thetheoretical findings.  The new method uses piecewise quadratic polynomials, and is of the lowestdegree possible. Optimal order approximation property of the finite elementspace is proved by means of a locally-averaged interpolation operator newlyconstructed. This interpolator, however, is not a projection. Indeed, weestablish a general theory and show that no locally defined interpolationassociated with the locally supported basis functions can be projective for thefinite element space in use. Particularly, the general theory gives an answerto a long-standing open problem presented in [Demko, J. Approx. Theory,$\bf{43}$(2):151--156, 1985].

Lowest-degree robust finite element scheme for a fourth-order elliptic singular perturbation problem on rectangular grids