Optimal approximation of elliptic problems by linear and nonlinear mappings IV: Errors in L2L2 and other norms

We study the optimal approximation of the solution of an operator equation A(u)=fA(u)=f by linear and different types of nonlinear mappings. In our earlier papers we only considered the error with respect to a certain HsHs-norm where ss was given by the operator since we assumed that A:H0s(Ω)→H−s(Ω) is an isomorphism. The most typical case here is s=1s=1. It is well known that for certain regular problems the order of convergence is improved if one takes the L2L2-norm. In this paper we study error bounds with respect to such a weaker norm, i.e., we assume that H0s(Ω) is continuously embedded into a space XX and we measure the error in the norm of XX. A major example is X=L2(Ω)X=L2(Ω) or X=Hr(Ω)X=Hr(Ω) with r