Optimal approximation of elliptic problems by linear and nonlinear mappings I

We study the optimal approximation of the solution of an operator equation A(u)=f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs that are given by an isomorphism , where s>0 and Ω is an arbitrary bounded Lipschitz domain in Rd. We prove that approximation by linear mappings is as good as the best n-term approximation with respect to an optimal Riesz basis. We discuss why nonlinear approximation still is important for the approximation of elliptic problems.