Galerkin method and optimal error algorithms in Hilbert spaces (Q1120253)

Operator equations of the form \(Lu=f\) are considered, where L belongs to the class \({\mathcal L}\) of linear, bounded (by a constant M) and coercive (with a constant m) operators from a Hilbert space V onto its dual \(V^*\) and f belongs to a Hilbert space \(W\subset V^*\). Optimality of the Galerkin method \(P^*_ nLu_ n=P^*_ nf\) is studied, where \(u_ n\in V_ n\), \(V_ n\) is subspace of V, \(P_ n\) is the orthogonal projector onto \(V_ n\) and \(P^*_ n\) is dual to \(P_ n\). It is shown that the Galerkin method is quasi-optimal independently of the choice of the subspace \(V_ n\) and the space W if \(M>m\). In the case \(M=m\), optimality of this method depends strongly on the choice of \(V_ n\) and W. Therefore a new algorithm is defined which is always optimal (independently of the choice of \(V_ n\) and W and relations between M and m).