The Mathematical Theory of Finite Element Methods pp 215-240 | Cite as

# 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 *L*^{2} 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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