Projection geometry and relaxed quasi-orthogonality for inf-sup stable Galerkin methods

RESUMO

We give an elementary, coordinate-free proof that uniformly inf-sup stable nested Petrov-Galerkin methods on Hilbert spaces satisfy relaxed general quasi-orthogonality. More precisely, the accumulated squared Galerkin increments over any window of $N$ consecutive levels are bounded by the squared error at the beginning of the window times $N^\sigma$, where $\sigma<1$, and both $\sigma$ and the constant prefactor depend explicitly only on a uniform bound for the Galerkin projections. The proof uses the Hilbert space angle between consecutive blocks of a uniformly bounded compatible projection chain, together with a dyadic decomposition and duality. It avoids matrix representations, wavelet bases, and LU-factorization. For symmetric indefinite problems, we relate the argument to the positive and negative spectral splittings of the Galerkin detail spaces and obtain a valid finite-window version of the sign-decomposition approach. We also construct a fixed self-adjoint involution and a fixed nested, uniformly inf-sup stable Galerkin sequence for which full general quasi-orthogonality fails. The same construction yields, for every $0<\alpha<1$, finite-support targets whose full-tail ratios grow at least like $N^\alpha$. Thus uniform inf-sup stability guarantees sublinear finite-window quasi-orthogonality, whereas full quasi-orthogonality requires additional hierarchical information in general.