# Karlstad Applied Analysis Seminar (KAAS)

## Next seminars:

**When: **December 9, Monday 10:15**What: **An extended Maxwell viscoelastic model: Derivation and FEM**Who: **Hirofumi Notsu**Where: **21E415A

**Abstract: **An extended Maxwell viscoelastic model is studied from mathematical and numerical points of view. It is shown that the model is derived from a conceptual diagram in 1D, and that the model has a gradient flow structure with respect to a viscoelastic energy. A P1/P0 finite element scheme is presented and its stability in the sense of energy is proved by using a corresponding discrete gradient flow structure. The talk is given based on the following paper: [M. Kimura, H. Notsu, Y. Tanaka and H. Yamamoto. The gradient flow structure of an extended Maxwell viscoelastic model and a structure-preserving finite element scheme. Journal of Scientific Computing, Vol.78(2019), pp.1111--1131.]

**When:** December 12, Thursday 14:15**What**: Introduction to modelling and simulation on spreading epidemics** ****Who**: Boonkrong Pichit, Rangsit University, Pathum Thani, Thailand**Where:** 21E415A

**Abstract**: This seminar intentionally presents how a mathematical model and a social complex network can describe epidemiological phenomena. It is organised into two main parts including the analysis of epidemic models and the study of disease transmission on social complex networks.

In stability analysis, the existence of stability in each compartmental epidemic model is adjudicated through the basic reproduction number. For application, the epidemic network model is developed to study the spreading behavior of the disease. The intervention strategies including vaccination and quarantine are introduced for the epidemic control.

The simulation results have shown that the developed epidemic network model can capture the important features of the disease transmission in human society. Network hubs play a crucial role in disease transmission. Therefore, the intervention against the spreading epidemics should be executed through the network hubs.

**When: **December 12, Thursday 15:15**What: **Numerical analysis of the stabilized Lagrange-Galerkin method**Who: **Hirofumi Notsu, Kanazawa University, Japan**Where: **21E415A

**Abstract: **The Lagrange-Galerkin (LG) method is a finite element method (FEM) combined with the method of characteristics, and has the following advantages: (i) It is robust for convection-dominated flow problems. (ii) The resulting matrix is symmetric and the system of linear equations is solved by one of iterative linear solvers for symmetric matrices, e.g., CG and MINRES methods. In this talk, we introduce the stabilized LG method for the Navier-Stokes equations, which additionally has an advantage, a small number of degrees of freedom especially in 3D, since it employs the P1/P1 finite element with Brezzi-Pitkaranta's pressure-stabilization. After reviewing the basic idea of the method of characteristics and the stabilization technique, we present theoretical and numerical results.

Refs.

[1] H. Notsu and M. Tabata. Error estimates of a pressure-stabilized characteristics finite element scheme for the Oseen equations. Journal of Scientific Computing, Vol.65 (2015), pp.940-955.

[2] H. Notsu and M. Tabata. Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations. ESAIM: M2AN, Vol.50 (2016), pp.361-380. [3] P.-Y. Hsu, H. Notsu and T. Yoneda. A local analysis of the axisymmetric Navier-Stokes flow near a saddle point and no-slip flat boundary. Journal of Fluid Mechanics, Vol.794 (2016), pp.444-459.

## Future seminars:

**When: **T.B.A.**What: **Topological and Interfacial Effects on the Glass Transition in Confined Polymers**Who: **Alexey V. Lyulin, Technische Universiteit Eindhoven, The Netherlands**Where: **T.B.A.

**Abstract: **Glasses in general, and polymer glasses in particular, are not, perhaps surprisingly, technically solid in a crystalized form, but are substances frozen in a liquidlike structure. Many fundamental questions remain as to exactly how glasses form, transitioning from flowing liquid-like state to solid polymer glass. A central factor materials scientists study is the temperature where this occurs, the glass-transition temperature Tg. After some introduction, I will discuss in more detail the recent results of the molecular-dynamics computer simulations of atactic polystyrene (PS), for the bulk and free-standing films, and for both linear and cyclic polymers. Simulated volumetric glass-transition temperatures ([1,2] show a strong dependence on the film thickness below 10 nm [3]. Our studies reveal that the fraction of the chain-end groups is larger in the interfacial layer with its outermost region approx. 1 nm below the surface than it is in the bulk. The enhanced population of the end groups is expected to result in a more mobile interfacial layer and the consequent dependence of Tg on the film thickness. In addition, the simulations show an enrichment of backbone aliphatic carbons and concomitant deficit of phenyl aromatic carbons in the interfacial film layer. This deficit would weaken the strong phenyl-phenyl aromatic interactions and, hence, lead to a lower film-averaged Tg in thin films, as compared to the bulk sample. To investigate the relative importance of the two possible mechanisms (increased chain ends at the surface or weakened p-p interactions in the interfacial region), the data for linear PS are compared with those for cyclic PS. For the cyclic PS the reduction of the glass-transition temperature is also significant in thin films, albeit not as much as for linear PS. Moreover, the deficit of phenyl carbons in the film interface is comparable to that observed for linear PS. Therefore, chain-end effects alone cannot explain the observed pronounced Tg dependence on the thickness of thin PS films; the weakened phenyl-phenyl interactions in the interfacial region seems to be an important cause as well [4]. I will also discuss the interface characteristics of polystyrene in free-standing thin films and on a graphite surface simulated employing an explicit all-atom force field [5]. References [1] Ediger, M.D.; Forrest, J.A. Macromolecules 2014, 47, 471-478. [2] Barrat, J.-L.; Baschnagel, J.; Lyulin, A.V. Soft Matter 2010, 6, 3430-3446. [3] Hudzinskyy, D.; Lyulin, A.V.; Baljon, A.R.C.; Balabaev, N.K.; Michels, M.A.J. Macromolecules 2011, 44, 2299-2310. [4] A. V. Lyulin, N. K. Balabaev, A. R.C. Baljon, G. Mendoza, C. W. Frank and Do Y. Yoon, J. Chem. Phys. 2017, 146, 203314. [5] S. Lee, A. V. Lyulin, C. W. Frank, Do Y. Yoon, Polymer, 2017, 116, 540-548.

**When:** T.B.A.**What**: On robotics, falling cat, parallel parking and stock trading ** ****Who**: Maria Ulan, Linnaeus University**Where:** T.B.A.

**Abstract**: Configuration spaces of many real mechanical systems appear to be manifolds with singularity. A singularity often indicates that geometry of motion may change at the singular point of configuration space. During the talk, I will present one possible solution by considering a certain algebra instead of the configuration space, with a structure completely determined by the geometry of the singularity. We will discussed how one could describe different types of complex motion. We will start with simple examples like linkages, manifolds with corners. Then we will discussed examples with given motion of subsystem. Finally, we will consider some classical reachability problems. At the end of the talk, I will show some applications in industry.

**When:** T.B.A.**What**: Some methods of approximation by spline functions ** ****Who**: Adrian Branga, Lucian Blaga University, Sibiu, Romania**Where:** T.B.A.

**Abstract**: We investigate some classes of spline functions, which can be used to approximate linear functionals defined on unidimensional Sobolev spaces, as well as to find the approximate solutions of some classes of ordinary differential equation and partial derivative equations. The existence and uniqueness of the approximate solution, together with convergence criteria for these methods and estimates of the approximation error are studied and some computational results are given.

**When:** T.B.A.**What**: Ideals generated by quadratic forms in the exterior algebra** ****Who**: Veronica Crispin Quinonez, Uppsala University**Where:** T.B.A.

**Abstract**: (joint work with S. LUNDQVIST and G. NENASHEV) There is a longstanding conjecture due to Fröberg about the minimal Hilbert series of C[x_1, ..., x_n]/(f_1,...,f_r), where f_i are homogeneous forms. We introduce the problem and give some results.

Now let E_n denote the Exterior algebra on n generators over C. It is natural to believe that the Hilbert series of E_n/(f_1,...,f_r) should be equal to the conjectured series in the commutative case, if the f_i's are generic forms of even degree. In 2002, Moreno and Snellman showed it to be true for only one generic form f. However, the same year Fröberg and Löfwall gave a counterexample for the case of two generic forms.

We use the structure theory of pairs of skew-symmetric matrices to study the Hilbert function of two generic quadratic forms f and g in E_n. Further, we use combinatorial methods to describe the Hilbert series of E_n/(f,g). Among our results, we have a conjecture for the minimal Hilbert series.