Cis‐trans Isomerism in Copper(II) Complexes with N‐acyl Thiourea Ligands

The N‐acyl thiourea complexes bis[N,N‐diethyl‐N′‐(p‐nitrobenzoyl)‐thioureato]copper(II) ( 1a,1b ) and bis(N,N‐diphenyl‐N′‐benzoylthioureato)copper(II) ( 2a,2b ) crystallize in each case in two modifications. X‐ray structural analysis shows that 1a and 1b are cis‐trans isomers. This is very unusual for N‐acyl thioureato complexes because with exception of one platinum(II) complex up to now only cis complexes have been found. In contrast X‐ray structural analysis of both forms 2a and 2b of the other complex shows no cis‐trans pair. Both modifications are cis complexes. In solution both isomers of the copper(II) complexes are observable by EPR spectroscopy.     

On the abstract theory of additive and multiplicative Schwarz algorithms

Summary. In recent years, it has been shown that many modern iterative algorithms (multigrid schemes, multilevel preconditioners, domain decomposition methods etc.) for solving problems resulting from the discretization of PDEs can be interpreted as additive (Jacobi-like) or multiplicative (Gauss-Seidel-like) subspace correction methods. The key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space into a sum of subspaces and the splitting of the variational problem on into auxiliary problems on these subspaces. In this paper, we propose a modification of the abstract convergence theory of the additive and multiplicative Schwarz methods, that makes the relation to traditional iteration methods more explicit. The analysis of the additive and multiplicative Schwarz iterations can be carried out in almost the same spirit as in the traditional block-matrix situation, making convergence proofs of multilevel and domain decomposition methods clearer, or, at least, more classical. In addition, we present a new bound for the convergence rate of the appropriately scaled multiplicative Schwarz method directly in terms of the condition number of the corresponding additive Schwarz operator. These results may be viewed as an appendix to the recent surveys [X], [Ys]. Received February 1, 1994 / Revised version received August 1, 1994