Solvability of elliptic systems with square integrable boundary data

We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L ₂ Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when A is either Hermitean, block or constant. Our methods apply to more general systems of partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.     

Static potentials for quarkonia at finite temperatures

We review non-perturbative static potentials commonly used in potential models for quarkonia at finite T. Potentials derived from Polyakov loop correlators are shown to be inappropriate for this purpose. The free energy is physical but has the wrong spatial decay and perturbative limit. The so-called singlet free energy is gauge dependent and unphysical. An appropriate static real time potential can be defined through a generalisation of pNRQCD to finite T. In perturbation theory, its real part reproduces the Debye-screened potential, its imaginary part accounts for Landau damping. Possibilities for its non-perturbative evaluation are discussed.     

Analytische Erweiterungen von Produkträumen

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A survey of multilevel preconditioned iterative methods

We survey multilevel iterative methods applied for solving large sparse systems with matrices, which depend on a level parameter, such as arise by the discretization of boundary value problems for partial differential equations when successive refinements of an initial discretization mesh is used to construct a sequence of nested difference or finite element meshes.We discuss various two-level (two-grid) preconditioning techniques, including some for nonsymmetric problems. The generalization of these techniques to the multilevel case is a nontrivial task. We emphasize several ways this can be done including classical multigrid methods and a recently proposed algebraic multilevel preconditioning method. Conditions for which the methods have an optimal order of computational complexity are presented.On leave from the Institute of Mathematics, and Center for Informatics and Computer Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria. The research of the second author reported here was partly supported by the Stichting Mathematisch Centrum, Amsterdam.     

Model-based optimization of a preparative ion-exchange step for antibody purification

A method using a model-based approach to design and optimize an ion-exchange step in a protein purification process is proposed for the separation of IgG from a mixture containing IgG, BSA and myoglobin. The method consists of three steps. In the first step, the model is calibrated against carefully designed experiments. The chromatographic model describes the convective and dispersive flow in the column, the diffusion in the adsorbent particles, and the protein adsorption using Langmuir kinetics with mobile phase modulators (MPM). In the second step, the model is validated against a validation experiment and analyzed. In the third and final step, the operating conditions are optimized. In the optimization step, the loading volume and the elution gradient are optimized with regard to the most important costs: the fixed costs and the feed cost. The optimization is achieved by maximizing the objective functions productivity (i.e. the production rate for a given amount of stationary phase) and product yield (i.e. the fraction of IgG recovered in the product stream). All optimization is conducted under the constraint of 99% purity of the IgG. The model calibration and the analysis show that this purification step is determined mainly by the kinetics, although as large a protein as IgG is used in the study. The two different optima resulting from this study are a productivity of 2.7 g IgG/(s m3) stationary phase and a yield of 90%. This model-based approach also gives information of the robustness of the chosen operating conditions. It is shown that the bead diameter could only be increased from 15 microm to 35 microm with maximum productivity and a 99% purity constraint due to increased diffusion hindrance in larger beads.     

Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators

Summary. The convergence of the conjugate gradient method is studied for preconditioned linear operator equations with nonsymmetric normal operators, with focus on elliptic convection-diffusion operators in Sobolev space. Superlinear convergence is proved first for equations whose preconditioned form is a compact perturbation of the identity in a Hilbert space. Then the same convergence result is verified for elliptic convection-diffusion equations using different preconditioning operators. The convergence factor involves the eigenvalues of the corresponding operators, for which an estimate is also given. The above results enable us to verify the mesh independence of the superlinear convergence estimates for suitable finite element discretizations of the convection-diffusion problems.Mathematics Subject Classification (2000): 65J10, 65F10, 65N15The second author was supported by the Hungarian Research Grant OTKA No. T. 043765.Dedicated to David M. Young on the occasion of his 80th birthday.     

Incomplete block-matrix factorization iterative methods for convection-diffusion problems

Standard Galerkin finite element methods or finite difference methods for singular perturbation problems lead to strongly unsymmetric matrices, which furthermore are in general notM-matrices. Accordingly, preconditioned iterative methods such as preconditioned (generalized) conjugate gradient methods, which have turned out to be very successful for symmetric and positive definite problems, can fail to converge or require an excessive number of iterations for singular perturbation problems.This is not so much due to the asymmetry, as it is to the fact that the spectrum can have both eigenvalues with positive and negative real parts, or eigenvalues with arbitrary small positive real parts and nonnegligible imaginary parts. This will be the case for a standard Galerkin method, unless the meshparameterh is chosen excessively small. There exist other discretization methods, however, for which the corresponding bilinear form is coercive, whence its finite element matrix has only eigenvalues with positive real parts; in fact, the real parts are positive uniformly in the singular perturbation parameter.In the present paper we examine the streamline diffusion finite element method in this respect. It is found that incomplete block-matrix factorization methods, both on classical form and on an inverse-free (vectorizable) form, coupled with a general least squares conjugate gradient method, can work exceptionally well on this type of problem. The number of iterations is sometimes significantly smaller than for the corresponding almost symmetric problem where the velocity field is close to zero or the singular perturbation parameter =1.The 2 ^nd author's research was sponsored by Control Data Corporation through its PACER fellowship program.The 3 ^rd author's research was supported by the Netherlands organization for scientific research (NWO).On leave from the Institute of Mathematics, Academy of Science, 1090 Sofia, P.O. Box 373, Bulgaria.     

Stability and error estimates for the $\theta$-method for strongly monotone and infinitely stiff evolution equations

Summary.   For evolution equations with a strongly monotone operator we derive unconditional stability and discretization error estimates valid for all . For the -method, with , we prove an error estimate , if , where is the maximal integration step for an arbitrary choice of sequence of steps and with no assumptions about the existence of the Jacobian as well as other derivatives of the operator , and an optimal estimate under some additional relation between neighboring steps. The first result is an improvement over the implicit midpoint method , for which an order reduction to sometimes may occur for infinitely stiff problems. Numerical tests illustrate the results. Received March 10, 1999 / Revised version received April 3, 2000 / Published online February 5, 2001     

High-spin spectroscopy of 63 31Ga32 and 65 31Ga34

High-spin states in the nuclei ⁶³Ga and ⁶⁵Ga were studied following the reaction ³²S + ⁴⁰Ca at a reaction energy of 125 MeV and using the GASP γ-ray spectrometer in combination with the ISIS charged-particle detector system. In addition to low-lying negative-parity states with single-particle character, rotational-like cascades built on the 9/2⁺ and 19/2^- states have been observed in both nuclei. Sidebands with negative parity in ⁶³Ga and positive parity in ⁶⁵Ga could be established. The experimental results are compared with Total Routhian Surface calculations which describe the nuclei as moderately deformed (β₂≈ 0.25) and γ soft at low rotational frequencies. The highest experimentally observed positive-parity state in ⁶⁵Ga (41/2⁺) is in good agreement with the calculated crossing of the collective band with a non-collective one terminating at this spin. Received: 20 December 2000 / Accepted: 2 May 2001