(Institute of Computer Science, ASCR):
Moments, Krylov subspace methods and model reduction with application in ellipsometry
Abstract: Joint work with Dr. Petr Tichy.

Reduced order modeling plays a central role in approximation of large-scale dynamical systems, and techniques based on Krylov subspaces are used in that area for decades, see, e.g. [2] and the recent monograph [1]. Krylov subspace methods can not only be viewed as tools for model reduction. Many of them by their nature represent model reductions based on matching moments. In order to explain the deep link between matching moments and Krylov subspace methods, we start with a modification of the classical Stieltjes moment problem, and show that its solution is given:

  • in the language of orthogonal polynomials by the Gauss-Christoffel quadrature;
  • in the matrix form by the conjugate gradient method (see [4]).

    In order to allow straightforward generalizations, we use the Vorobyev vector moment problem [3], and present some basic Krylov subspace methods from the moment matching model reduction point of view [5]. The second part of the talk will be devoted to application in numerical modeling of the diffraction of light on periodic media [6].
    [1] A. C. Antoulas, Approximation of large-scale dynamical systems, vol. 6 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.
    [2] K. Gallivan, E. Grimme, and P. Van Dooren, Asymptotic wave-form evaluation via a Lanczos method, Appl. Math. Lett., 7 (1994), pp. 7580.
    [3] Y. V. Vorobyev, Methods of moments in applied mathematics, Translated from the Russian by Bernard Seckler, Gordon and Breach Science Publishers, New York, 1965.
    [4] Meurant, G. and Strakos, Z., The Lanczos and conjugate gradient algorithms in finite precision arithmetic, Acta Numerica, 15 (2006), pp. 471542.
    [5] Strakos, Z., Model reduction using the Vorobyev moment problem, Numerical Algorithms, 51, (2008), pp. 363376.
    [6] Hench, J. and Strakos, Z., The RCWA method - a Case study with open questions and perspectives of algebraic computations, Electronic Transactions on Numerical Analysis, 31 (2009), pp. 331357.

  • 11.1.2010
    Mgr. Tomas Ligursky
    (Mathematical Institute of the Charles University, Praha):
    Theoretical analysis of discrete contact problems with Coulomb friction
    Abstract: A discrete model of a two-dimensional Signorini problem with Coulomb friction and a coefficient of friction $mathcal{F}$ depending on the spatial variable will be analysed. It will be shown that a solution exists for any $mathcal{F}$ and is globally unique if $mathcal{F}$ is sufficiently small. The Lipschitz continuity of this unique solution as a function of $mathcal{F}$ as well as as a function of the load vector $boldsymbol{f}$ will be obtained. Furthermore, local uniqueness of solutions for arbitrary $mathcal{F} > 0$ will be studied. The question of existence of locally Lipschitz-continuous branches of solutions with respect to the coefficient $mathcal{F}$ will be converted to the question of existence of locally Lipschitz-continuous branches of solutions with respect to the load vector $boldsymbol{f}$. A condition guaranteeing the existence of locally Lipschitz-continuous branches of solutions in the latter case and results for determining their directional derivatives will be given. Finally, the general approach will be illustrated on an elementary example, whose solutions are known exactly.
    Prof. Hans-Goerg Roos
    (TU Dresden):
    Solution-decomposition and classification of meshes
    Abstract: First lecture of the mini-course Finite element methods for convection-diffusion problems II: Layer-adapted meshes

    Next lectures:
    March 4 (Seminar on Numerical mathematics, room K3, 2nd floor), 14:00: Galerkin-FEM on layer-adapted meshes
    March 8, 17:20: Nitsche-Mortaring and stabilization on layer-adapted meshes
    March 11 (SNM, K3), 14:00 Solution recovery and non-stationary problems
    Dr. Adrian Hirn
    (University of Heidelberg):
    Approximation of the p-Stokes equations with equal-order finite elements
    Abstract: Non-Newtonian fluid motions are often modeled by the p-Stokes equations where the extra stress tensor exhibits p-structure. In this talk we study the discretization of the p-Stokes equations with equal-order finite elements. Here, the stabilization of the pressure-gradient, which is essential due to the violation of the discrete inf-sup condition, is based on the local projection stabilization method. In this talk, a priori error estimates are derived and numerical simulations are shown.
    Dr. Tomas Bodnar
    (Faculty of Mechanical Engineering, Czech Technical University, Prague):
    On the implementation of a new viscoelastic shear-thinning model for blood flow simulations
    Abstract: The talk will summarize the first experience with the implementation of a new viscoelastic model specially designed for blood flow simulations. This model is based on the thermodynamic framework approach developed by Rajagopal & Srinivasa and further extended for blood flow simulations by Anand & Rajagopal.

    K. Rajagopal, A. Srinivasa, A thermodynamic frame work for rate type fluid models, Journal of Non-Newtonian Fluid Mechanics 80 (2000) 207-227.

    M. Anand, K. R. Rajagopal, A shear-thinning viscoelastic fluid model for describing the flow of blood, International Journal of Cardiovascular Medicine and Science 4 (2) (2004) 59-68.

    Prof. Hans-Goerg Roos
    (TU Dresden):
    Nitsche-Mortaring and stabilization on layer-adapted meshes
    Abstract: Third lecture of the mini-course Finite element methods for convection-diffusion problems II: Layer-adapted meshes

    Last lecture:
    March 11 (Seminar on Numerical Mathematics, room K3, 2nd floor), 14:00 Solution recovery and non-stationary problems
    Prof. Jens Frehse
    (Hausdorff Center for Mathematics, University of Bonn):
    Improved Lp-estimates for the strain velocities in hardening problems
    Abstract: Problems of elastic plastic deformation with kinematic or isotropic hardening and von Mises flowrule are considered. It is shown that the velocities of the stresses, strains and hardening parameters satisfy an improved Lp-property that is σ˙, ξ˙, ∇˙u ∈ L(0,T;L2+2δ(Ω)) with some δ > 0. For dimension n = 2 this implies continuity of u˙ in spatial direction, furthermore it can be used as tool to prove boundary differentialbility σ,ξ ∈ L(L1+ϵ) and σ,ξ ∈ L(0,T;N1∕2+δ′,2), where 1∕2 + δ′ is the order of fractional Nichol’skii differentiability. For large values of the Hardening modulus we achieve δ = 1∕3.
    Dr. Riccarda Rossi
    (University of Brescia, Italy):
    Analysis of rate-independent model of adhesive contact with thermal effects
    Abstract: In this talk we present a joint work with T. Roubicek on the analysis of a model for adhesive contact, which encompasses both thermal effects and a rate-independent evolution for the adhesive parameter. After describing the model, we present some existence and approximation results.
    Prof. Dr. Torsten Linss
    (TU Dresden):
    Maximum-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems

    In the first part of the lectures we consider stationary reaction-diffusion equations of the type


    while the second part is concerned with its time-dependent analogue


    Both are equipped with homogeneous Dirichlet boundary conditions and, in case of the time-dependent problem, with appropriate initial conditions. The parameter ε >0 is small, while the reaction cofficient c is assumed to satisfy c ≥ γ2 with some positive constantγ.

    The efficiency of standard numerical methods deteriorates as the perturbation parameter ε approaches zero. This is because layers form. These are regions where the solution varies rapidely.

    In the present talk, bounds for the Greens function associated with the differential operators L and M are derived. These bounds are applied to obtain a posteriori error estimators in the maximum norm for difference schemes and for FEM. These estimators are robust with respect to the perturbation parameter. They can be applied to design adaptive mesh-movement algorithms that give numerical methods which converge uniformly with respect to the perturbation parameter ε.

    Numerical results will be presented to illustrate the theoretical findings.

    The lecture will be delivered also at the Seminar on Numerical mathematics on 1st April, 14:00, seminar room K3.

    Prof. Dr. Soeren Bartels
    (Institute for Numerical Simulation, University of Bonn):
    Robust Approximation of Phase Field Models past Topological Changes
    Abstract: Phase field models are often used to describe the evolution of submanifolds, e.g., the Allen-Cahn equation approximates motion by mean curvature and more sophisticated phase field models provide regularizations of the Willmore flow and other geometric evolution problems. The models involve small regularization parameters and we discuss the dependence of a priori and a posteriori error estimates for the numerical solution of the regularized problems on this parameter. In particular, we address the question whether robust error estimation is possible past topological changes. We provide an affirmative answer for a priori error estimates assuming a logarithmic scaling law of the time averaged principal eigenvalue of the linearized Allen-Cahn or Ginzburg-Landau operator about the exact solution. This scaling law is confirmed by numerical experiments for generic topological changes. The averaged eigenvalue about the approximate solution enters a posteriori error estimates exponentially and therefore, critical scenarios are detected automatically by related adaptive finite element methods. The devised scheme extracts information about the stability of the evolution from the approximate solution and thereby allows for a rigorous a posteriori error analysis. This is joint work with Ruediger Mueller (HU Berlin) and Christoph Ortner (U Oxford).
    Prof. Endre Suli
    (University of Oxford):
    An adaptive finite element approximation of a variational model of brittle fracture
    Abstract: The energy of the Francfort--Marigo model of brittle fracture, posed as a free-discontinuity problem over the space of special functions of bounded variation, can be approximated, in the sense of $Gamma$-convergence, by the Ambrosio--Tortorelli functional. In this talk we formulate and analyze an adaptive finite element algorithm for the computation of local minimizers of the Ambrosio--Tortorelli regularization of the Francfort--Marigo model. We combine a Newton-type method with an adaptive mesh-refinement algorithm driven by an a posteriori error bound. We present theoretical results that demonstrate the convergence of our algorithm. The theoretical results are illustrated by numerical experiments. The lecture are based on joint work with Siobhan Burke and Christoph Ortner at the University of Oxford.
    Dr. Ewelina Kaminska
    (Warsaw University, Poland):
    Analysis of nonlocal model of compressible fluid in 1-D
    Abstract: The talk will be devoted to a nonlocal modification of the compressible Navier-Stokes equations in mono dimensional case with a boundary condition characteristic for the free boundary problem. From the formal point of view the system is an intermediate between the Euler and Navier-Stokes equations. Under certain assumptions, imposed on initial data and viscosity coefficient the local and global existence of solutions can be proved. Moreover, we have a uniform in time bound on the density of fluid.
    Incompressible fluids with pressure and shear-rate dependent viscosity

    15:40 Prof. Josef Malek (Faculty of Math. and Phys., Charles University in Prague)

    15:55 Mgr. Martin Lanzendorfer (Institute of Computer Science, AS CR)
    On the existence analysis of fluids whose viscosity depends on the pressure and the shear rate

    16:25 MSc. Adrian Hirn (IWR, University of Heidelberg)
    Finite element approximation of flows of fluids with shear rate and pressure dependent viscosity

    16:45 MSc. Stefan Knauf (IWR, University of Heidelberg)
    Numerical simulations for ball bearings containing fluids with pressure-dependent viscosity

    Dr. Tomas Furst
    (Palacky University, Olomouc, Czech Republic):
    Richards equation and finger-like solutions: an impossibility result
    Abstract: The understanding and description of water movement in unsaturated porous media rates among the most challenging (and still not fully resolved) problems with important applications in oil recovery, environment protection, nuclear waste deposition, etc. Traditionally, fluid motion in porous media has been described in the framework of continuum mechanics which has lead to various forms of the celebrated Richards equation (1931). However, this approach runs into several problems, which will be addressed in the seminar. Most thoroughly, the problem of finger flow (observed as early as 1960) will be addressed. Finger flow represents a widely observed (rather generic than exceptional) mode of water infiltration into an initially dry porous medium. Recently, there has been a considerable discussion about the possibility of obtaining finger-like solutions to the Richards equation. In the seminar, it will be demonstrated that Richards equation, in principle, cannot admit finger-like solutions for three-dimensional homogeneous unsaturated porous media flow, subject to monotone boundary conditions. This will be demonstrated for any reasonable type of homogeneous porous material; the result will not be dependent on any particular constitution assumptions. Moreover, it will be explained why hysteresis of the retention curve does not play any role in the proof. An alternative approach to finger flow modeling will be discussed which uses the ideas of cellular automata.
    Dr. Rostislav Vodak
    (Palacky University, Olomouc, Czech Republic):
    Two models of non-linear diffusion in leather processing
    Abstract: 1. Processing of raw hide into leather comprises more than thirty chemical, physical and mechanical processes which need to be performed optimally in order not to damage the valuable raw material. Before the transport to tanneries, hide is usually conserved by salt. Even the process of salination is subject to optimisation because its incorrect course may lead to a considerable economic loss. Although raw hide represents a complicated (water-saturated) porous medium, the process of salination can be modeled by molecular diffusion. However, the concentrations involved are so high (up to 10 %) that the infiltration of salt into the water-saturated hide induces a counter-flow of fresh water out of the sample. This so called self-induces convection makes the mathematical model far more interesting. In the seminar, a one-dimensional form of the model will be presented. Existence of a weak solution to the system will be proved by means of the Schaefer Fixed Point Theorem. Moreover, a minimum principle for salt concentration will be derived.

    2. Removing of the preserving salt is followed by a de-hairing operation, which is performed by sodium sulphide under strongly alkaline conditions of calcium hydroxide. After de-hairing, the white hide contains an excess of calcium hydroxide, which has to be removed again by decantation washing. In terms of mathematical modeling, decantation washing represents diffusion accompanied by desorption, which makes the model non-linear. In the seminar, existence and uniqueness of a weak solution to the model will be presented, and its stabilization for time converging to infinity will be shown.
    Prof. Endre Suli
    (University of Oxford):
    Existence of global weak solutions to kinetic models of dilute polymers
    Abstract: We report on recent progress concerning the existence of global-in-time weak solutions to a general class of microscopic-macroscopic bead-spring models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We establish the existence of global-in-time weak solutions to the model for a general class of spring-force potentials including, in particular, the widely used finitely extensible nonlinear elastic (FENE) potential. A key ingredient of the argument is a special testing procedure in the weak formulation of the Fokker-Planck equation, based on a convex entropy function. This is joint work with John W. Barrett (Imperial College London).
    (IWR, University of Heidelberg):
    Boundedness of solutions of a haptotaxis model
    Abstract: we prove existence of global solutions of the haptotaxis model of cancer invasion for arbitrary nonnegative initial conditions. Uniform boundedness of the solutions is shown using the method of bounded invariant rectangles applied to the reformulated system of reaction-diffusion equations in divergence form with a diagonal diffusion matrix. Moreover, the analysis of the model shows how the structure of kinetics of the model is related to the growth properties of the solutions and how this growth depends on the ratio of the sensitivity function (describing the size of haptotaxis) and the diffusion coefficient. One of the implications of our analysis is that in the haptotaxis model with a logistic growth term, cell density may exceed the carrying capacity, which is impossible in the classical logistic equation and its reaction-diffusion extension.
    (Basque Center for Applied Mathematics):
    Prof. Akif Ibragimov
    (Department of Mathematics & Statistics, Texas Tech University):
    Stability analyzes of non-linear flows in porous media and application
    Abstract: Note: The seminar will exceptionally take place in the lecture room K3.
    Prof. Gerard Meurant
    (Commissariat a l Energie Atomique (CEA), former Research Director):
    Matrices, moments and quadrature I
    Abstract: The aim of this series of lectures is to describe and explain the beautiful mathematical relationships between matrices, moments, orthogonal polynomials, quadrature rules and the Lanczos and conjugate gradient algorithms. The main topic is to obtain numerical methods to estimate or in some cases to bound quantities like I[f]=u^T f(A)v where u and v are given vectors, A is a symmetric nonsingular matrix and f is a smooth function. There are many instances in which one would like to compute bilinear forms like u^T f(A)v. A first application is the computation of some elements of the matrix f(A) when it is not desired or feasible to compute all of f(A). Computation of quadratic forms r^T A^{-i}r for i=1,2 is interesting to obtain estimates of error norms when one has an appro ximate solution of a linear system Ax=b and r is the residual vector b-Ax. Bilinear or quadratic forms arise naturally for the computation of parameters in problems like least squares, total least squares and regularization methods for solving ill--posed problems. We will describe the algorithms and give some examples of applications. Particular topics addresssed: Orthogonal polynomials and properties of tridiagonal matrices, the Lanczos and conjugate gradient (CG) algorithms and computation of Jacobi matrices, Gauss quadrature and bounds for bilinear forms u^T f(A)v, and applications (bounds for elements of $f(A)$, estimates of error norms in CG, least squares and total least squares, discrete ill--posed problems). The presentation is available at http://www.karlin.mff.cuni.cz/ncmm/texts/Meurant_Lecture1.pdf
    Prof. Giulio Schimperna
    (Universita di Pavia):
    On a class of fourth order degenerate parabolic equations
    Abstract: I will present some existence, regularity, and long-time behaviour results for a fourth order parabolic equation related to the evolution of thin films. Applications to the Cahn-Hilliard model with degenerate mobility are also discussed. More precisely, after showing existence of at least one (weak) solution by means of an approximation - a priori estimates - passage to the limit argument, I will analyze the generalized dynamical process associated to the equation and prove the existence of a weak trajectory attractor. In case a viscosity term is added and slightly more restrictive conditions on the nonlinearities are assumed, this trajectory attractor can be intended with respect to the strong phase-space topology. Finally, I will present sufficient conditions for having strict positivity of the solution, entailing in particular uniqueness. The result presented in the talk have been obtained in collaboration with Sergey Zelik (university of Surrey).
    Prof. Vladislav Mantic
    (Universidad de Sevilla):
    Debonding at the fibre-matrix interface under transversal biaxial loads: an application of Interfacial and Finite Fracture Mechanics to crack initiation and propagation at micro scale in composites
    Abstract: Under loads normal to the direction of the fibres, composites reinforced by long fibres suffer failures that are known as matrix or interfibre failures, typically involving interface cracks between matrix and fibres, the coalescence of which originates macrocracks in the composite. A micromechanical model, based on the Interfacial and Finite Fracture mechanics and also on a numerical Boundary Element model, is presented aiming to explain the mechanism of appearance and propagation of the damage. To this end, the plane strain problem of a single circular cylindrical inclusion embedded in an unbounded matrix subjected to a far field uniaxial and biaxial transverse load is studied. First, a theoretical model for the simultaneous prediction of the initial size of a crack originated at the inclusion-matrix interface and of the critical remote load required to originate this crack is presented. Isotropic and linear elastic behaviour of both materials, with the inclusion being stiffer than the matrix, is assumed. The interface is considered to be strong (providing continuity of displacements and tractions across the interface surface) and brittle. A new dimensionless structural parameter, depending on bimaterial and interface properties together with the inclusion radius, which plays a key role in characterizing the interface crack onset, is introduced. A size effect inherent to this problem is predicted and analysed. Then, the crack growth along the inclusion-matrix interface and its subsequent kink towards the matrix is studied by means of a numerical model. Experiments show an excellent agreement between the predictions generated and the evolution of the damage in an actual composite.
    Prof. Gerard Meurant
    (Commissariat a l Energie Atomique (CEA), former Research Director):
    Matrices, moments and quadrature II
    Abstract: The presentation is available at http://www.karlin.mff.cuni.cz/ncmm/texts/Meurant_Lecture2.pdf
    Dr. Christos Panagiotopoulos
    (Universidad de Sevilla):
    Boundary Element Method for linear elasticity and implementation of an energetic approach to the delamination problem
    Abstract: Boundary Element Method (BEM) as a numerical tool for solving (initial)-boundary values problems with a focus on engineering mechanics will be introduced first. Some new trends in the BEM programming will be discussed. A novel application of the BEM to the delamination onset and growth by means of the energetic solution formulation will be presented together with some numerical results.
    (Institute of Information Theory and Automation, AS CR):
    Quasiconvexity at the boundary and weak lower semicontinuity of integral functionals
    Abstract: We show that the so-called quasiconvexity at the boundary, originally defined by J.M. Ball and J. Marsden to state necessary conditions for minimizers in nonlinear elasticity, plays a crucial role in the description of weak lower semicontinuity of integral functionals depending on gradients. As a consequence, we get higher integrability properties of some quasiaffine mappings.
    Prof. Gerard Meurant
    (Commissariat a l Energie Atomique (CEA), former Research Director):
    Matrices, moments and quadrature 3
    Abstract: The presentation is available at http://www.karlin.mff.cuni.cz/ncmm/texts/Meurant_Lecture3.pdf
    RNDr. Miroslav Bulicek, PhD.
    (Mathematical Institute of the Charles University, Prague):
    C^alpha-regularity for a class of non-diagonal elliptic systems with p-growth
    Abstract: We consider any weak solution to a system of nonlinear elliptic PDE s in $W^{1,p}$ setting that represents an Euler equation to a certain variational problem. Assuming that such a solution satisfies, in addition, the Noether equation, we identify new structural assumptions on the nonlinearity that guarantees Holder continuity of the solution. The new method is applicable also to a class of non-diagonal non-convex problems.
    Prof. Gerard Meurant
    (Commissariat a l Energie Atomique (CEA), former Research Director):
    Matrices, moments and quadrature 4
    Prof. Dr. Alexander Mielke
    (Weierstrass Inst. Berlin & Humboldt Universitat zu Berlin, Germany):
    Rate-independent plasticity as Gamma limit of a slow viscous gradient flow for wiggly energies
    Abstract: In a joint work with Lev Truskinovsky it is shown that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a visco-elastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishing-viscosity limit and the discrete to continuum limit. In the course of these limits a non-convex elastic problem transforms into a convex elastic problem while the quadratic rate-dependent dissipation of visco-elastic solid transforms into a singular rate-independent dissipation of an ideally plastic solid. In order to emphasize ideas we employ in our proofs the simplest prototypical system describing transformational plasticity of shape-memory alloys. The approach, however, is sufficiently general and can be used for similar reductions in the cases of more general plasticity and damage models.
    (Charles University, Prague & Academy of Sciences of the Czech Republic):
    Plasticity at small strains with or without hardening
    Abstract: Small-strain plasticity in its quasistatic formulation based on the energetic-solution concept will be presented, with the focus to the limit to the Prandt-Reuss elastic/perfectly plastic model when hardening parameters go to zero. Beside mere isothermal quasistatic evolution, also thermodynamical augmentation of these models will be discussed for isotropic materials undergoing thermal expansion. Numerical analysis will be outlined. The talk will be accompanied by sample computational experiments performed by S. Bartels (Univ. Bonn).
    RNDr. Miroslav Bulicek, Ph.D.
    (Charles University, Prague):
    On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions
    Abstract: We establish the large-data and long-time existence of a suitable weak solution to an initial and boundary value problem driven by a system of partial differential equations consisting of the Navier-Stokes equations with the viscosity $nu$ increasing with a scalar quantity $k$ that evolves according to an evolutionary convection diffusion equation with the right hand side nu(k)|D(v)|^2 that is merely L^1-integrable over space and time. We also formulate a conjecture concerning regularity of such a solution.
    Dr. Vit Prusa
    (Charles University, Prague):
    Flow of an electrorheological fluid in a journal bearing
    Abstract: Electrorheological fluids have numerous potential applications in vibration dampers, brakes, valves, clutches, exercise equipment, etc. The flows in such applications are complex three dimensional flows. Most models that have been developed to describe the flows of electrorheological fluids are one dimensional models. Here, we discuss the behaviour of two fully three dimensional models for electrorheological fluids. The models are such that they reduce, in the case of simple shear flows with the intensity of the electric field perpendicular to the streamlines, to the same constitutive relation, but they would not be identical in more complicated three dimensional settings. In order to show the difference between the two models we study the flow of these fluids between eccentrically placed rotating cylinders kept at different potentials, in the setting that corresponds to technologically relevant problem of flow of electrorheological fluid in journal bearing. Even though the two models have quite a different constitutive structure, due to the assumed forms for the velocity and pressure fields, the models lead to the same velocity field but to different pressure fields. This finding illustrates the need for considering the flows of fluids described by three dimensional constitutive models in complex geometries, and not restricting ourselves to flows of fluids described by one dimensional models or simple shear flows of fluids characterized by three dimensional models.
    Prof. Maria Lukacova
    (University of Mainz, Germany):
    Large time step finite volume schemes for shallow water flows
    Abstract: We present two new large time step methods within the framework of the well-balanced finite volume evolution Galerkin (FVEG) schemes. The methodology will be illustrated for low Froude number shallow water flows with source terms modelling the bottom topography and Coriolis forces, but results can be generalized to more complex systems of balance laws. The FVEG methods couple a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of multidimensional hyperbolic systems, such that all of the infinitely many directions of wave propagation are taken into account explicitly. We present two variants of large time step FVEG method: a semi-implicit time approximation and an explicit time approximation using several evolution steps along bicharacteristic cones. The behaviour of the methods will be discussed through a series of numerical experiments. This is a joint work with A. Hundertmark and F. Prill.
    Dr. Stefan Kroemer
    (Mathematisches Institut, Universitaet zu Koeln, Germany):
    Dimension reduction for functionals on solenoidal vector fields
    Abstract: We study integral functionals constrained to divergence-free vector fields in $L^p$ on a thin domain, in the limit as the thickness of the domain goes to zero. The Gamma-limit with respect to weak convergence in $L^p$ turns out to be given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting constraint can give rise to a nonlocal functional as illustrated in an example.
    Prof. Andreas Fischer
    (TU Dresden, Germany):
    Newton Methods for Generalized Nash Equilibrium Problems
    Abstract: In the generalized Nash equilibrium problem (GNEP) both the objective function and the feasible set of each player may depend on the other players strategies. In general, to find even one solution is a challenging task. Solutions of GNEPs can be characterized by necessary conditions. Based on this we provide two Newton methods, analyze their features and their range of applicability. In particular, we address the issue of the non isolated solutions that can cause severe difficulties within an application of standard methods. The talk is based on joint work with Francisco Facchinei and Veronica Piccialli.
    Prof. Maria Lukacova
    (University of Mainz, Germany):
    Efficient Fluid-Structure Algorithms with Application in Hemodynamics
    Abstract: The aim of this talk is to present recent results on numerical modelling of non-Newtonian flow in compliant stenosed vessels with application in hemodynamics. For the structure problem the generalized string equation for radial symmetric tubes is used and extended to a stenosed vessel. We consider two FSI algorithms: firstly, the global iterative approach with respect to the domain deformation and secondly, the kinematic splitting that is used to split a coupled fluid-structure problem in an efficient and stable way. Stability of the new FSI schemes will be studied theoretically as well as experimentally. At the end we present numerical experiments for some non-Newtonian models, comparisons with the Newtonian model and the results for hemodynamic wall parameters; the wall shear stress and the oscillatory shear index. This is a joint work with A. Hundertmark and G. Rusnakova
    Dr. Trygve Karper
    (Norwegian University of Science and Technology, Trondheim, Norway):
    Convergent finite element methods for compressible Stokes flow

    In the literature one can find a variety of numerical methods appropriate for viscous compressible flow. However, there are very few results with reference to the convergence properties of these methods. For instance, it is an open problem whether any numerical method for the compressible Navier-Stokes equations (in more than one dimension) converges as discretization parameters tend to zero.

    In this talk, I aim at discussing why proving convergence of numerical methods for viscous compressible flow is so hard. I will also present some recently developed methods that are provably convergent for compressible Stokes flow (neglecting convection). These methods solves half of the problems, but still seem inadequate for the problem with convection.

    Po semináři se podává opět čaj a káva.
    Všichni zájemci jsou srdečně zváni.