Abstract
In order to solve numerically a second-order elliptic boundary-value problem in a bounded domain, ..cap omega.., by use of the finite element method, it is often necessary in practice to violate certain assumptions of the standard variational formulation. Two of these variational crimes are considered here. It is shown that optimal L infinity estimates still hold. The first crime occurs when a nonconforming finite element method is employed, so that smoothness requirements are violated at interelement boundaries. The second crime occurs when numerical integration is employed, so that the bilinear form is perturbed. Abstract error estimates are established. These estimates are first applied to treat a nonconforming finite element method and then to treat the case of numerical integration. In both cases, the patch test is crucial to the proof of L infinity estimates, just as it was in the case of mean square estimates.
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