Error estimates for 3-$\mathrm{d}$ narrow finite elements

We obtain error estimates for finite element approximations of the lowest degree valid uniformly for a class of three-dimensional narrow elements. First, 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 this result is not true. Applying extrapolation results we obtain an optimal order error estimate for functions sligthly more regular than $H^{2}$. These results are valid both for tetrahedral and rectangular elements. Second, for the case of rectangular elements, we obtain optimal, in order and regularity, error estimates for an average interpolation valid for functions in $W^{1+s,p}$ with $1\le p\le \infty$ and $0\le s\le 1$.