The finite element approximation is essentially defined by a mean-square projection of the gradient. Thus, it is natural that error estimates for the gradient of the error directly follow in the L2 norm. It is interesting to ask whether such a gradient-projection would also be of optimal order in some other norm, for example L∞. We prove here that this is the case. Although of interest in their own right, such estimates are also crucial in establishing the viability of approximations of nonlinear problems (Douglas & Dupont 1975) as we indicate in Sect. 8.7. Throughout this chapter, we assume that the domain Ω ⊂ IR d is bounded and polyhedral.
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