Error estimates for 3-d narrow finite elements

Vile obtain error estimates for finite element approximations of the lowest degree valid uniformly for a class of three-dimensional narrow elements. F irst, for the Lagrange interpolation we prove optimal error estimates, both in order and regularity, in L-p for, p > 2. For p = 2 it is known that thi s result is not true. Applying extrapolation results ive obtain an optimal order error estimate for functions sligthly more regular than H-2. These re sults are valid both for tetrahedral and rectangular elements. Second, for the case of rectangular elements, we obtain optimal, in order and regularit y, error estimates for an average interpolation valid for functions in W-1s,W-p with 1 less than or equal to p less than or equal to infinity and 0 l ess than or equal to s less than or equal to 1.