Uniform error estimates for certain narrow Lagrange finite elements

Abstract:

Error estimates of Dupont and Scott are used to derive uniform error estimates for Lagrange finite elements in ${\Re ^n}\;(n \geq 2)$ under the following conditions: (1) The elements can be arbitrarily narrow in any coordinate direction such that a sufficient number of interpolation points are grouped on lines parallel to that coordinate axis, and (2) the space of approximating functions ${F_T}$ in each element T must include the space of polynomials of degree $\leq m - 1$ for some $m \geq 1 + n/2$. If n is odd, this does not cover elements of lowest degree that are normally considered with the shape regularity requirement that the ratio of their outer and inner diameters be bounded. For example, if $n = 3$, the usual requirement with shape regularity is that each ${F_T}$ contain all first-degree polynomials. The result of this paper requires that each ${F_T}$ contain all quadratic polynomials, and consequently does not apply to linear (Courant) elements in tetrahedrons or trilinear (tensor) elements in rectangular boxes. Counterexamples in these two cases are included.