(January 28) Rodolfo Rodriguez: Numerical analysis of a FEM to compute damped vibrations - We deal with the computation of the damped vibration modes of a coupled system consisting of a linear elastic structure and an ideal acoustic fluid, both separated by a thin layer of viscoelastic material with a given frequency dependent impedance. Much attention has been recently paid to this kind of problems, mostly related to the goal of decreasing the level of noise in aircrafts and cars. We consider a displacement formulation for both, the fluid and the solid. Standard finite elements for the structure combined with Raviart-Thomas elements for the fluid has been shown to provide a reliable method to compute the vibration modes of the coupled undamped system. In this paper we consider a simplified mathematical model for the damped transmission conditions which consists of adequately relaxing the kinematic constrain. We provide the mathematical and numerical analysis of the quadratic eigenvalue problem arising from the application of this method in the simpler case of the damped vibrations of a fluid contained in a rigid cavity with its walls covered by absorbing material.

Finally the method is used to compute the vibrations modes of a fluid-solid damped system and these modes are related with the corresponding response peaks.

(February 2) David Silvester: A posteriori error estimation for elliptic PDEs - In this talk we consider the design of robust adaptive methods for solving the Navier-Stokes equations governing steady laminar flow of anincompressible fluid. The essential ingredient is a ``natural'' low order (P₁-P₀ mixed finite element) spatial discretisation, together with efficient a posteriori error indicators which provide upper and lower bounds for the exact spatial error.

Two important subproblems are discussed in detail; incompressibleStokes flow and scalar convection-diffusion. We show that a posteriori error indicators computed by solving local Poisson problems lead to efficient estimators, in both cases. We also present theoretical and numerical results which illustrate the importance of stabilising the constituent finite element approximations when local refinement is used as the basis for mesh adaptivity.

(February 11) Howard Elman: A Multigrid Method for the Discrete Indefinite Helmholtz Equation - Standard multigrid algorithms have proven ineffective for the solution ofdiscretizations of Helmholtz equations. These problems are difficultbecause they are indefinite and because boundary conditions cause them tobe complex and nonselfadjoint. In this work, we examine how the basictools of multigrid, smoothing algorithms and coarse grid correction, areaffected by these issues, and we show how multigrid can be augmented withKrylov subspace iteration to address the difficulties. We demonstrate thenew algorithm's effectiveness through theoretical analysis of a modelproblem and experimental results.

This is joint work with Oliver Ernst and Dianne O'Leary.

(February 25) Andy Wathen: Numerical simulation of the motion of particles in a viscous fluid - an example of the importance of iterative linear algebra in Scientific Computing - We will describe the numerical approximation of equations describing themotion of a large number of interacting particles in a viscousincompressible fluid using an Adaptive Eulerian Lagrangian mixed finiteelement method. The discretisation leads to large linearised systems ofequations for which preconditioned iterative solvers are required. We willdiscuss various preconditioning strategies and show how techniques forpreconditioning the incompressible Navier-Stokes equations form a keycomponent.

(March 4) Kunibert G. Siebert: Concepts of the Finite Element Toolbox ALBERT and Applications to CFD in Combination with Free Boundary Problems - ALBERT is an Adaptive multi-Level finite element toolbox usingBisectioning refinement and Error control by Residual Techniques. Its design is based on an appropriate concept of data structures holdinggeometric, finite element, and algebraic information. Using this concept,abstract adaptive methods for stationary and time dependent problems,assembly tools for discrete systems, and dimension dependent tasks like meshmodifications can be provided in a library. This allows dimension-independentdevelopment and programming of a general class of applications.

Having access to such a finite element toolbox, the implementation timefor adaptive solvers of specialized (non)linear problems is now reducedtremendously. We will present two applications: the computation of theso called "edge tone" in a flue pipe modeled by the incompressibleNavier-Stokes equations and a simulation of the vertical Bridgmanmethod which is used in the industrial growth of crystals. The modelfor the Bridgman method is a coupled system of a Stefan equation forthe temperature in melt and solid crystal and the incompressibleNavier-Stokes-equations for the velocity in the melt, leading toa discretization of the Navier-Stokes-equations on a time depended domain.

(March 11) William F. Mitchell: Full Domain Partitions to Reduce Communication in Parallel PDE Solvers - Adaptive multilevel methods, which combine adaptive mesh refinement withmultigrid solution techniques, have been shown to be very efficient methodsfor the numerical solution of partial differential equations on sequentialcomputers. The use of these methods on parallel computers is currently aresearch topic. In particular, effective use of these techniques in ahigh-latency/low-bandwidth environment, like a network of workstations, isespecially challenging. In this talk we will present a parallel adaptivemultilevel method for elliptic partial differential equations that obtainshigh parallel efficiency in this environment. Recent developments inparallelizing the method via overlapping subdomains on each refinement levelwill be featured.

The approach produces a full domain partition, in which the usual subdomainon each processor is extended to cover the full domain. This facilitatesparallel algorithms for adaptive refinement, grid partitioning and multigridsolution that require only a few communication steps each.

(March 16) Cleve Moler: MATLAB Math - A survey of the numerical analytic algorithms currently used in MATLAB, including elementary mathematical functions, direct and iterative matrix computation, and ordinary differential equations.

(March 18) Alan Berger: Applied Math. in a Solid State Nuclear Magnetic Resonance Spectroscopy Procedure for Obtaining the Conformation at a Labeled Site in a Peptide - The function of a biomolecule is governed by its three dimensional structure. Hence methods for determining structure are valuable in biochemistry and molecular biology and provide information useful in the rational design of pharmaceuticals. One such technique for obtaining insights into structure is nuclear magnetic resonance (NMR) spectroscopy. One particular NMR procedure (solid state magic angle sample spinning) produces data in the form of a set of discrete peaks which decay away from the central peak. This data is related to the principal values of the chemical shift tensor for the atom under study which in turn is related to the chemical (electronic) environment of that nucleus. The dependence of the peak heights on the chemical shift tensor was expressed in terms of Bessel function expansions which were cast into a more readily evaluated form where use of spline interpolation made it tractable to extract the principal values of the chemical shift tensor from experimental data. A more direct recasting of the peak heights using Fourier series was noted in [3], and analysis of a corresponding "two-dimensional" experiment was carried out in [4]. This type of experiment allows one, in particular, to examine the phi-psi backbone torsion angles (i.e., local conformation) at a labeled site in a protein. The idea of weighting the signal from each molecule depending on its orientation relative to the applied magnetic field was developed in [5], where symmetry conditions which lead to admissible weighting functions were derived. Whether such orientational weighting will lead to enhanced ability to determine structure is at this point still an open question.

(April 1) L. Rukhovets: Fictitious domain method for elliptic boundary value problems with nonlocal boundary conditions in multiply connected domain -

In many problems of continuum mechanics it is necessary to solvetwo dimensional boundary value problems for elliptic second-orderequations in multiply connected domains. For example, there is theproblem of the integral water transport in a basin (a lake, a sea, anocean) with islands. If the domain where boundary value problems aresolved is multiply connected, additional boundary conditions should beimposed. In addition to ordinary boundary conditions the nonlocalboundary conditions are imposed on the solutions of the ellipticequations. They ensure the construction of some single-valuedfunctions, which describe the behaviour of continuum medium, using thesolution of elliptic equation. For example, in the problem of integralwater transport there are some conditions that ensure thesingle-valuedness of the deviation function (constructed with the aidof the stream function) of the fluid level from the equilibrium state.

The talk deals with the finite element method of solving theseboundary value problems. The original problem is approximated by anordinary boundary value problem in a simply connected domain forequation with discontinuous coefficients using the fictitious domainmethod. For example, the application of the fictitious domain methodto the problem of determining the stream function has a transparentphysical interpretation.

The second problem of the discussion is devoted to the methods ofsolving the grid equations of the finite element method (FEM) forboundary value problems with nonlocal boundary conditions. The FEMuses nonlocal trial functions. This is investigated by the fictitiouscomponent method for solving the system of such grid equations. Therate of convergence is independent from the system order. At each stepof iteration process it is necessary to solve an ordinary grid systemin a rectangle that encloses the domain. The talk also deals with asimple iterative method with ordinary rate of convergence. This methodallows to construct an iterative process without using nonlocal trialfunctions.

(April 8) Martin Stynes: The intrinsic norm finite element method for convection-diffusion problems - We give a new variational analysis for convection-diffusion problems thatleads to a new finite element method with a remarkable property.

(April 15) Isaak Mayergoyz: Mathematical Models of Hysteresis - The main phenomenological manifestation of hysteresis is history dependentbranching. It turns out that rate independent hysteretic nonlinearitieshave discrete memories which are formed by the past extremum values ofinput. To model hysteretic nonlinearities, we need a special mathematicaltool which itself (due to its structure) detects and accumulates the past input extrema and chooses the appropriate branch of hysteretic nonlinearity according to the accumulated past history.

The mathematical modeling of hysteresis can be traced back to thelandmark paper of Preisach. The talk will deal with Preisach-type modelsof hysteresis, which are constructed as continuous "superpositions" ofrectangular loop operators. The necessary and sufficient conditions forthe representation of actual hysteresis nonlinearities by the Preisachmodel will be discussed along with the issues of the identification andexperimental testing of these models. Some applications to nonlineardiffusion will be presented.

(April 22) Ulrich Rüde: Cache-Aware Iterative Methods - The numerical solution of partial differential equations leads to large,sparse systems of equations with up to several millions of unknowns. Fastiterative algorithms for the solution of these systems are often basedon the multilevel principle. Unfortunately, conventional implementationsof iterative solvers exhibit a high overhead on modern computers. Theperformance degradation is partly caused by the effects of non-uniformdata access combined with indirect addressing that prohibits the useof the instruction-level parallelism. This, however, is essential toexploit advanced CPUs. The second, and possibly more fundamental problemarises from the hierarchical memory architecture with several layers ofcaches. Their effective use requires programs with data access locality.Unfortunately, iterative solvers are typically implemented by using globalsweeps over the whole data set, and thus their performance is essentiallylimited by the speed of the memory system. These problems are addressedin the development of our current cache-aware multigrid methods. Ourcurrent top-performing method solves the Poisson equation discretizedwith a million mesh points in below a second on a single processor AlphaPC and thus beats published results by almost two orders of magnitude.

(joint work with Markus Kowarschik)

(April 29) G.I.Shishkin: Parallel Domain Decomposition Methods with High-Order Time-Accuracy for Parabolic Singularly Perturbed Convection-Diffusion Problems - In this lecture we study the discrete approximation of a Dirichlet problemon an interval for a singularly perturbed parabolic PDE with convectiveterms. The highest derivative in the equation is multiplied by anarbitrarily small parameter epsilon. For epsilon=0 the parabolic equationdegenerates into a first-order hyperbolic equation. As epsilon goes to 0, aboundary layer appears in a neighborhood of the outflow boundary that maygive rise to difficulties when standard discretization methods are used. Inparticular, classical difference approximation on uniform meshes yielderror bounds, in the maximum norm, which depend on an inverse power ofepsilon. For such problems it is of interest to develop finite differenceschemes whose accuracy is independent of epsilon, i.e., schemes thatconverge epsilon-uniformly in the l-infinity-norm.

The order of convergence for known epsilon-uniform convergent schemes isone up to a small logarithmic factor and exactly one with respect to thespace and time variable, respectively, in the case of convection-diffusionproblems. Because the amount of computational work is proportional to thenumber of nodes in the time discretization, the higher order of accuracywith respect to the time variable can allow to use larger time-steps.Therefore, it is natural to look for methods that have higher-orderaccuracy in time, without essentially increasing the amount ofcomputational work. In this presentation we consider schemes, based on adefect-correction technique, for which the order of convergence in time canbe arbitrarily large if the solution is sufficiently smooth.

To achieve a high efficiency of the numerical algorithms, we use a domaindecomposition method to construct difference schemes that admit sequentialand parallel computations. The parallel method does not require iterationsat each time level, so the practical efficiency of acceleration isproportional to the number of parallel solvers used. We give the conditionsfor the iterative difference schemes that preserve epsilon-uniformconvergence. To validate the theoretical results, some numerical resultsare presented and discussed.

This is joint work with W. Hemker and L.P. Shishkina.

(May 6) Pedro Morin: Parameter Identification in a Mathematical Model of Shape Memory Alloys using Quasilinearization - We will consider a system of PDE's that arises from the conservationlaws of linear momentum and energy in one-dimensional Shape MemoryAlloys (SMA).This system can be formally written as a semilinear abstract Cauchyproblem in a Hilbert space.A quasilinearization-based algorithm for parameter identification inthis kind of Cauchy problems will be presented, as well as sufficientconditions for the convergence of the algorithm.Finally, numerical examples in which the algorithm is applied torecover the non-physical parameters describing the free energypotential in SMA, will be presented.

(May 13) Ludmil Zikatanov: Edge Average Finite Element scheme for convection diffusion equations - A simple technique will be given in this talk on the construction and analysisof a class of finite element discretizations for convection-diffusion problemsin any spatial dimensions by properly averaging the PDE coefficients on elementedges. The resulting finite element stiffness matrix is an M-matrix under somemild assumption for the underlying (generally unstructured) finite elementgrids. As a consequence the proposed edge-averaged finite element (EAFE)scheme is particularly interesting for the discretization of convectiondominated problems. This scheme admits a simple variational formulation, it iseasy to analyze and it is also suitable for problems with a relatively smoothflux variable.