Forgivable variational crimes

Abstract

It is common use to call variatonal crimes those applications of the variational method or more generally, of the Galerkin method, where not all the assumptions needed to validate the method are exactly satisfied. This covers a.o. the use of trial functions for second order elliptic problems that are only approximately continuous along element boundaries or the use of quadrature formulas to only approximately compute the entries of the stiffness matrix or the replacement of the exact domain boundary by an approximate one, to cite the main examples. Classical techniques based on bounding the so-called consistency error terms have been used to analyse and most often absolve such crimes. In the present contribution, we propose an alternate approach for a restricted class of variational crimes such that there is no approximation on the representation of the RHS of the exact equation and such that a generalized variational principle can be introduced in such way that both the exact and the approximate problems appear as Galerkin approximations of this generalized problem. This reduces their analysis to successive applications of the variational method itself and produces essentially the same error bounds as perfectly legal applications of the variational method, whence our suggestion to consider such crimes as forgivable. Examples of applications including and generalizing the PCD method presented elsewhere in this conference are considered by way of illustration. © Springer-Verlag Berlin Heidelberg 2003.