Contractivity results for alternating direction schemes in Hilbert spaces (Q1334841)

Stability and convergence of initial value problems \(u'(t) + Au(t) = 0\), \(t \in [t_ 0, T]\), \(u(t_ 0) = u_ 0 \in D(A)\) in a complex Hilbert space \(H\) are considered. Here \(A: D(A) \subset H \to H\) is an unbounded linear maximal monotone operator on \(H\). With an extension of the von Neumann theorem on the estimate \(f(A)\) with holomorphic function \(f\), the authors derive contractivities of operators \(R(-A_ 1, \dots, -A_ n)\) for maximal monotone operators \(A_ i\), \(i = 1, \dots, n\) on \(H\) and for appropriate rational approximations \(R(z_ 1, \dots, z_ n)\) of \(\exp(z_ 1 + \cdots + z_ n)\), \(z_ i \in \mathbb{C}\). Thereafter, stabilities and convergence of alternating direction schemes are established under proper conditions. Two and three dimensional parabolic differential equations are discussed as numerical examples.