Max—norm Estimates

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.