An improved convergence analysis of smoothed aggregation algebraic multigrid

We present an improved analysis of the smoothed aggregation algebraic multigrid method extending the original proof in [Numer. Math. 2001; 88 :559–579] and its modification in [Multilevel Block Factorization Preconditioners. Matrix‐based Analysis and Algorithms for Solving Finite Element Equations. Springer: New York, 2008]. The new result imposes fewer restrictions on the aggregates that makes it easier to verify in practice. Also, we extend a result in [Appl. Math. 2011] that allows us to use aggressive coarsening at all levels. This is due to the properties of the special polynomial smoother that we use and analyze. In particular, we obtain bounds in the multilevel convergence estimates that are independent of the coarsening ratio. Numerical illustration is also provided. Copyright © 2011 John Wiley & Sons, Ltd.     

Copper(II) complexes of the antihypertensive drug nadolol

The complexation of the non-selective β-blocker nadolol, HL, 1 with copper(II) leads to formation of mono-and dinuclear complexes depending mainly on the metal-to-ligand molar ratio. The mononuclear violet complex CuL₂·2Solv, 2, was obtained in a soluble form at metal-to-ligand molar ratio Cu(II): HL ≤ 1: 10 in methanolic or slightly alkaline aqueous solutions. The dinuclear green complex Cu₂L₂Cl₂·H₂O, 3 was synthesized at Cu(II): HL ≥ 1: 2 molar ratio in methanolic solutions. The complexes were studied using spectral (UV-Vis, FT-IR, EPR), magnetochemical, thermogravimetric methods and elemental analysis. In the complexes nadolol acts as a monoanionic bidentate ligand coordinated to copper(II) through the NH-and the deprotonated OH-groups of its aminoalcohol fragment.      

Mesh‐independent convergence of the modified inexact Newton method for a second order non‐linear problem

In this paper, we consider an inexact Newton method applied to a second order non‐linear problem with higher order non‐linearities. We provide conditions under which the method has a mesh‐independent rate of convergence. To do this, we are required, first, to set up the problem on a scale of Hilbert spaces and second, to devise a special iterative technique which converges in a higher than first order Sobolev norm. We show that the linear (Jacobian) system solved in Newton's method can be replaced with one iterative step provided that the initial non‐linear iterate is accurate enough. The closeness criteria can be taken independent of the mesh size. Finally, the results of numerical experiments are given to support the theory. Published in 2005 by John Wiley & Sons, Ltd.     

Multilevel iterative methods for mixed finite element discretizations of elliptic problems

Summary For solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed. The methods are within the general framework of the product (multiplicative) scheme for operators in a Hilbert space, proposed recently by Bramble, Pasciak, Wang, and Xu [5,6,26,27] and make use of certain multilevel decomposition of the corresponding spaces for the flux variable.     

Computational scales of Sobolev norms with application to preconditioning

This paper provides a framework for developing computationally efficient multilevel preconditioners and representations for Sobolev norms. Specifically, given a Hilbert space and a nested sequence of subspaces , we construct operators which are spectrally equivalent to those of the form . Here , , are positive numbers and is the orthogonal projector onto with . We first present abstract results which show when is spectrally equivalent to a similarly constructed operator defined in terms of an approximation of , for . We show that these results lead to efficient preconditioners for discretizations of differential and pseudo-differential operators of positive and negative order. These results extend to sums of operators. For example, singularly perturbed problems such as can be preconditioned uniformly independently of the parameter . We also show how to precondition an operator which results from Tikhonov regularization of a problem with noisy data. Finally, we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition.

     

Complexes of copper(II) with the β-blocker atenolol

Mono- and binuclear copper(II) complexes with atenolol (HAt) can be obtained, depending on the reaction conditions. The mononuclear violet complex cation has the general formula Cu(HAt)₄ ²⁺ with an elongated octahedral geometry. The two ligands in the equatorial plane are bound in a bidentate fashion through the hydroxyl oxygen and amino nitrogen, while the other two atenolol molecules in axial position are coordinated in a monodentate way. The binuclear green complex Cu₂At₂Cl₂, is neutral, where atenolol acts as a bidentate (O^–, NH) bridging ligand. The bridge between the two Cu atoms is realized by the deprotonated oxygen of the alcohol group.     

A monotone finite volume method for advection–diffusion equations on unstructured polygonal meshes

We present a new second-order accurate monotone finite volume (FV) method for the steady-state advection–diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes and guarantees solution non-negativity. The interpolation-free approximation of the diffusive flux uses the nonlinear two-point stencil proposed in Lipnikov [23]. Approximation of the advective flux is based on the second-order upwind method with a specially designed minimal nonlinear correction. The second-order convergence rate and monotonicity are verified with numerical experiments.     

Mixed finite element methods for incompressible flow: Stationary Stokes equations

In this article, we develop and analyze a mixed finite element method for the Stokes equations. Our mixed method is based on the pseudostress‐velocity formulation. The pseudostress is approximated by the Raviart‐Thomas (RT) element of index k ≥ 0 and the velocity by piecewise discontinuous polynomials of degree k. It is shown that this pair of finite elements is stable and yields quasi‐optimal accuracy. The indefinite system of linear equations resulting from the discretization is decoupled by the penalty method. The penalized pseudostress system is solved by the H(div) type of multigrid method and the velocity is then calculated explicitly. Alternative preconditioning approaches that do not involve penalizing the system are also discussed. Finally, numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010