Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them?
(Crossposted from MathOverflow, where it encountered little interest, but probably here I can find more people with a FEM background.)
Chapter 8 of Brenner and Scott's Mathematical Theory of Finite Element Methods is devoted to this subject. In particular, theorem 8.1.11 and the corollary give you that
$ \|u - u_h\|_{W^1_\infty} \le C h^{k - 1}\|u\|_{W^k_\infty}$
for linear elliptic problems with sufficiently smooth coefficients, provided that the finite element space satisfies some other inequalities. They leave it as an exercise (how typical) to verify that this applies to the usual Lagrange, Hermite and Argyris elements.
If you want to go beyond the standard Laplace equation, Alan Demlow (at Texas A&M University) has derived estimates.
