Numerical Analysis Seminar

Seminar information archive ~03/28Next seminarFuture seminars 03/29~

Date, time & place Tuesday 16:30 - 18:00 002Room #002 (Graduate School of Math. Sci. Bldg.)
Organizer(s) Norikazu Saito, Takahito Kashiwabara

Seminar information archive

2019/12/09

16:50-18:20   Room #117 (Graduate School of Math. Sci. Bldg.)
Xuefeng Liu (Niigata University)
Point-wise error estimation for the finite element solution to Poisson's equation --- new approach based on Kato-Fujita's method (Japanese)
[ Abstract ]
In 1950s, H. Fujita proposed a method to provide the upper and lower bounds in boundary value problems, which is based on the T*T theory of T. Kato about differential equations. Such a method can be regarded a different formulation of the hypercircle method from Prage-Synge's theorem.
Recently, the speaker extended Kato-Fujita's method to the case of the finite element solution of Poisson's equation and proposed a guaranteed point-wise error estimation. The newly proposed error estimation can be applied to problems defined over domains of general shapes along with general boundary conditions.

2019/09/25

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Guanyu Zhou (Tokyo University of Science)
Finite volume method for the Keller-Segel system (Japanese)

2019/08/19

13:00-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Eric Chung (The Chinese University of Hong Kong) 13:00-14:00
Staggered hybridisation for discontinuous Galerkin methods (英語)
[ Abstract ]
In this talk, we present a new staggered hybridization technique for discontinuous Galerkin methods to discretize linear elastodynamic equations and nonlinear Stokes equations. The idea of hybridization is used extensively in many discontinuous Galerkin methods, but the idea of staggered hybridization is new. Our new approach offers several advantages, namely energy conservation, high-order optimal convergence, preservation of symmetry for the stress tensor, block diagonal mass matrices as well as low dispersion error. The key idea is to use two staggered hybrid variables to enforce the continuity of the velocity and the continuity of the normal component of the stress tensor on a staggered mesh. We prove the stability and the convergence of the proposed scheme in both the semi-discrete and the fully-discrete settings. Numerical results confirm the optimal rate of convergence and show that the method has a superconvergent property for dispersion.
Feifei Jing (Northwestern Polytechnical University) 14:30-15:30
DG and HDG methods for the variational inequality problems (英語)
[ Abstract ]
There exist many numerical methods for solving the fluid dynamics equations, the main difference between them lies in the partitions of geometric domain and the discrete forms of governing equations. Due to the discontinuous piecewise polynomial subspaces, DG and HDG methods can be easily implemented on highly unstructured meshes, e.g. general polygonal mesh, and volume integrals could be calculated on physical elements, without reference elements and mappings between physical and reference elements. In this talk, DG and HDG methods employed to a class of variational inequality problems arising in hydrodynamics are studied. Some theoretical results will be shown, as well as the implementations of these methods are also put into practice.
Issei Oikawa (Hitotsubashi University) 16:00-16:30
A new HDG method using a hybridized flux (英語)
[ Abstract ]
We propose a new hybridizable discontinuous Galerkin (HDG) method for steady-state diffusion problems. In our method, both the trace and flux of the exact solution are hybridized. The Lehrenfeld-Schöberl stabilization is implicitly included in the method, so that the orders of convergence in all variables are optimal without postprocessing and computation of any projection. Numerical results are present to show the validation of our method.
Takahito Kashiwabara (The University of Tokyo) 16:30-17:00
Numerical approximation of the Stokes–Darcy problem using discontinuous linear elements (英語)
[ Abstract ]
We consider the Stokes–Darcy interface problem supplemented with the Beavers– Joseph–Saffman condition on the interface separating two domains. This condition allows for discontinuity in the tangential velocities and in the pressures along the interface. To effectively express it, we propose to use discontinuous linear finite elements to approximate all of the velocities/pressures in the Stokes/Darcy regions. The continuity of velocity in the normal direction is weakly enforced by adopting either the penalty method or Nitsche’s method. We present stability and error estimates for the proposed scheme, taking into account the situation where a curved interface is approximated by a polygonal curve or polyhedral surface.

2019/07/08

16:50-18:20   Room #056 (Graduate School of Math. Sci. Bldg.)
Takeru Matsuda (University of Tokyo)
Parameter estimation and discretization errors for ordinary differential models (Japanese)

2019/07/01

16:50-18:20   Room #117 (Graduate School of Math. Sci. Bldg.)
Hideo Kawarada (AMSOK)
The effect of preventing scale formation by ceramic balls and its effect on the human body (Japanese)

2019/05/13

16:50-18:20   Room #056 (Graduate School of Math. Sci. Bldg.)
Kensuke Aishima (Hosei University)
Iterative refinement for symmetric eigenvalue problems (Japanese)

2019/04/22

16:50-18:20   Room #056 (Graduate School of Math. Sci. Bldg.)
Issei Oikawa (Hitotsubashi University )
Superconvergence of the HDG method (Japanese)

2019/04/08

16:50-18:20   Room #056 (Graduate School of Math. Sci. Bldg.)
Takeshi Takaishi (Musashino University)
Crack growth model of viscoelastic material with the phase field approach (Japanese)

2018/11/05

16:50-18:20   Room #002 (Graduate School of Math. Sci. Bldg.)
Hisashi Okamoto (Gakushuin University)
Tosio Kato as an applied mathematician (Japanese)
[ Abstract ]
Tosio Kato (1917-1999) is nowadays considered to be a rigorous analyst or theorist. Many people consider his contributions in quantum mechanics to be epoch-making, his work on nonlinear partial differential equations elegant and inspiring. However, around the time when he visited USA for the first time in 1954, he was studying problems of applied mathematics, too, notably numerical computation of eigenvalues. I wish to shed light on the historical background of his study of applied mathematics. This is a joint work with Prof. Hiroshi Fujita.

2018/10/22

16:50-18:20   Room #002 (Graduate School of Math. Sci. Bldg.)
Kensuke Aihara (Tokyo City University)
Residual smoothing technique for short-recurrence Krylov subspace methods (Japanese)

2018/10/15

16:50-18:20   Room #002 (Graduate School of Math. Sci. Bldg.)
Takeyuki Nagasawa (Saitama University)
Möbius invariant discretizations and decomposition of the Möbius energy (Japanese)

2018/07/31

14:00-15:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Jichun Li (University of Nevada Las Vegas)
Recent advances on numerical analysis and simulation of invisibility cloaks with metamaterials (English)
[ Abstract ]
In the June 23, 2006's issue of Science magazine, Pendry et al. and Leonhardt independently published their seminar papers on electromagnetic cloaking. Since then, there is a growing interest in using metamaterials to design invisibility cloaks. In this talk, I will first give a brief introduction to invisibility cloaks with metamaterials, then I will focus on some time-domain cloaking models we studied in the last few years. Well-posedness study and time-domain finite element method for these models will be presented. I will conclude the talk with some open issues.

2018/07/26

16:00-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Takahito Kashiwabara (University of Tokyo)
$L^\infty$ error estimates of the finite element method for elliptic and parabolic equations with the Neumann boundary condition in smooth domains (日本語)

2018/07/10

16:50-18:20   Room #126 (Graduate School of Math. Sci. Bldg.)
Junichi Matsumoto (AIST)
Free surface flow using orthogonal basis bubble function finite element method (Japanese)

2018/06/19

16:50-18:20   Room #002 (Graduate School of Math. Sci. Bldg.)
Shuji Yoshikawa (Oita University)
Small data global existence for the semi-discrete scheme of a model system of hyperbolic balance laws (Japanese)

2018/05/31

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Olivier Pironneau (Sorbonne University and Academy of Sciences)
Parallel Computing Methods for Quantitative Finance: the Parareal Algorithm for American Options (English)
[ Abstract ]
With parallelism in mind we investigate the parareal method for American contracts both theoretically and numerically. Least-Square Monte-Carlo (LSMC) and parareal time decomposition with two or more levels are used, leading to an efficient parallel implementation which scales linearly with the number of processors and is appropriate to any multiprocessor-memory architecture in its multilevel version. We prove $L^2$ superlinear convergence for an LSMC backward in time computation of American contracts, when the conditional expectations are known, i.e. before Monte-Carlo discretization. In all cases the computing time is increased only by a constant factor, compared to the sequential algorithm on the finest grid, and speed-up is guaranteed when the number of processors is larger than that constant. A numerical implementation will be shown to confirm the theoretical error estimates.

2018/05/08

16:50-18:20   Room #002 (Graduate School of Math. Sci. Bldg.)
Norikazu Saito (University of Tokyo)
Various aspects of numerical analysis (Japanese)

2018/04/17

16:50-18:20   Room #002 (Graduate School of Math. Sci. Bldg.)
Yoshiki Sugitani (Tohoku University)
Introduction to Machine learning and its application to Medical diagnosis (Japanese)

2018/02/19

15:00-16:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Michael Plum (Karlsruhe Insitute of Technology)
Existence, multiplicity, and orbital stability for travelling waves in a nonlinearly supported beam (English)
[ Abstract ]
For a nonlinear beam equation with exponential nonlinearity, we prove existence of at least 36 travelling wave solutions for the specific wave speed c=1.3. Our proof makes heavy use of computer assistance: starting from numerical approximations, we use a fixed point argument to prove existence of solutions "close to" the approximate ones. Furthermore we investigate the orbital stability of these solutions by making use of both analytical and computer-assisted techniques.

2018/02/19

16:15-17:15   Room #056 (Graduate School of Math. Sci. Bldg.)
Kaori Nagatou (Karlsruhe Insitute of Technology)
An approach to computer-assisted existence proofs for nonlinear space-time fractional parabolic problems (English)
[ Abstract ]
We consider an initial boundary value problem for a space-time fractional parabolic equation, which includes the fractional Laplacian, i.e. a nonlocal operator. We treat a corresponding local problem which is obtained by the Caffarelli-Silvestre extension technique, and show how to enclose a solution of the extended problem by computer-assisted means.

2017/12/19

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)

2017/11/28

16:50-18:20   Room #117 (Graduate School of Math. Sci. Bldg.)
Daisuke Koyama (The University of Electro-Communications)
Hybrid discontinuous Galerkin methods for nearly incompressible elasticity problems
(Japanese)
[ Abstract ]
A Hybrid discontinuous Galerkin (HDG) method for linear elasticity problems has been introduced by Kikuchi et al. [Theor. Appl. Mech. Japan, vol.57, 395--404 (2009)], [RIMS Kokyuroku, vol.1971, 28--46 (2015)]. We consider to seek numerical solutions of the plane strain problem by the HDG method, especially in the case when materials are nearly incompressible, that is, when the first Lam\'e parameter $\lambda$ is large. In this talk, we consider two cases when the HDG method uses a lifting term and does not use it. When the lifting term is used, the method can be free of volumetric locking. On the other hand, when the lifting term is not used, we have to take an interior penalty parameter of order $\lambda$ as $\lambda$ tends to infinity, in order to guarantee the coercivity of the bilinear form. Taking such an interior penalty parameter causes volumetric locking phenomena. We thus conclude that the lifting term is essential for avoiding the volumetric locking in the HDG method.

2017/11/14

16:50-18:20   Room #002 (Graduate School of Math. Sci. Bldg.)
Ai Ishikawa (Kobe University)
Energy-preserving numerical method based on the variational principle and application to unconstrained optimization problems (Japanese)

2017/10/23

16:00-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Christian Klingenberg (Wuerzburg University, Germany)
On the numerical discretization of the Euler equations with a gravitational force and applications in astrophysics (English)
[ Abstract ]
We consider astrophysical systems that are modeled by the multidimensional Euler equations with gravity.
First for the homogeneous Euler equations we look at flow in the low Mach number regime. Here for conventional finite volume discretizations one has excessive dissipation in this regime. We identify inconsistent scaling for low Mach numbers of the numerical fux function as the origin of this problem. Based on the Roe solver a technique that allows to correctly represent low Mach number flows with a discretization of the compressible Euler equations is proposed. We analyze properties of this scheme and demonstrate that its limit yields a discretization of the incompressible limit system.
Next for the Euler equations with gravity we seek well-balanced methods. We describe a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear PDE, whose solutions are called hydrostatic equilibria. We present well-balanced methods, for which we can ensure robustness, accuracy and stability, since it satisfies discrete entropy inequalities.
We will then present work in progress where we combine the two methods above.

2017/10/10

16:50-18:20   Room #002 (Graduate School of Math. Sci. Bldg.)
Yumiharu Nakano (Tokyo Institute of Technology)
Meshfree collocation methods for linear and fully nonlinear parabolic equations

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