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Abstract: In [J.H. Bramble, I.E. Pasciak, A.T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34 (1997) 1072–1092], a nonlinear Uzawa algorithm for solving generalized saddle point problems iteratively was considered. In [Z. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math. 46 (2003) 157–171], Cao gave another nonlinear Uzawa algorithm in order to accelerate convergence. In this paper, a new nonlinear Uzawa method, which is defined by two nonlinear approximate inverses, is proposed and its convergence result is deduced. At the same time, we analyze convergence results of three nonlinear Uzawa methods. The results of numerical experiments are presented when we apply them to Stokes equations discretized by mixed finite element. [Copyright &y& Elsevier]

Following a general framework of Coxeter spectral analysis of signed graphs Δ and finite posets I introduced by Simson (SIAM J. Discrete Math. 27:827–854, 2013) we present efficient numerical algorithms for the Coxeter spectral study of finite posets I = ({ 1 , ... , n } , ⪯ I) that are positive in the sense that the symmetric Gram matrix G I : = 1 2 (C I + C I t r) ∈ M n (Q) is positive definite, where C I ∈ M n (Z) is the incidence matrix of I encoding the relation ⪯ I. In the framework of scientific computing we present a complete Coxeter spectral classification of finite positive posets I of size n = | I | < 20. It extends one of the main results obtained in Ga̧siorek et al. (Eur. J. Comb. 48:127–142, 2015) for posets of size n ≤ 10. We also show that the connectivity of such posets I is determined by the complex Coxeter spectrum specc I ⊆ C ; equivalently, by the Coxeter polynomial cox I (t) ∈ Z t od I. One of the main results of the paper is a new technique to compute in a polynomial time a Z -invertible matrix B I ∈ M n (Z) such that B I t r · C I · B I = G ˇ D I , where G ˇ D I ∈ M n (Z) is a non-symmetric Gram matrix of a simply-laced Dynkin diagram D I ∈ { A n , D n , E 6 , E 7 , E 8 } associated with a finite positive poset I. [ABSTRACT FROM AUTHOR]