Schur complements on Hilbert spaces and applications
Constantin Bacuta

Abstract: We review the classical Babusca- Brezzi theory and introduce a different approach in analyzing Arrow-Hurvitz-Uzawa like algorithms. The new approach is based on using natural Schur operators. For any continuous bilinear form defined on a pair of Hilbert Spaces satisfying the compatibility B-B condition, symmetric Schur type operators are defined on each of the two Hilbert spaces. The spectrum of the new operators can be bound only in terms of the compatibility B-B constant and the continuity constant of the given form. We find the convergence factors for two Arrow-Hurvitz-Uzawa algorithms using the spectral properties of the Schur complements. Our analysis, combined with standard techniques of discretization and a posteriori error estimates, could lead to new and efficient (adaptive) algorithms for solving saddle point systems.