Optimal approximation of elliptic problems by linear and nonlinear mappings IV: Errors in L2 and other norms

We study the optimal approximation of the solution of an operator equation A(u) = f by linear and different types of nonlinear mappings. In our earlier papers we only considered the error with respect to a certain Hs-norm where s was given by the operator since we assumed that A : Hs 0(Ω) → H−s(Ω) is an isomorphism. The most typical case here is s = 1. It is well known that for certain regular problems the order of convergence is improved if one takes the L2-norm. In this paper we study error bounds with respect to such a weaker norm, i.e., we assume that Hs 0(Ω) is continuously embedded into a space X and we measure the error in the norm of X. A major example is X = L2(Ω) or X = Hr(Ω) with r < s. We prove this better rate of convergence also for non-regular problems. AMS subject classification: 41A25, 41A46, 41A65, 42C40, 46E35, 65C99