Uniformly Convergent Nonconforming Element for 3-d Fourth Order Elliptic Singular Perturbation Problem

where ∆ is the standard Laplace operator, ∂u/∂n denotes the outer normal derivative on ∂Ω and ε is a small real parameter with 0 < ε ≤ 1. This problem can be considered a gross simplification of the stationary Cahn-Hilliard equation with ε being the length of the transition region of phase separation. In particular, we are interested in the regime when ε tends to zero. Obviously, if ε approaches zero, the differential Eq. (1.1) formally degenerates to Poisson’s equation. For ε=1, that is, the usual fourth order elliptic equation, many works have been done. When a conforming finite element is used, it should consist of piecewise polynomials that are globally continuously differentiable (C). Such elements require polynomials of high degree and even in two dimensions are not easy to construct. To lower the polynomial degree, some macroelements were created on triangle grids, see e.g., [1, 2]. Recently, a macro type of biquadratic C finite element was constructed on rectangle grids [3, 4], which is a rectangular version of the (C) Powell-Sabin element [1]. On the other hand, many lower degree nonconforming elements in the two and three dimensional cases have been constructed and used in practice.