Numerical Methods for PDEs: Exploring Finite Element and Discontinuous Galerkin Approaches

Numerical Methods for PDEs: Exploring Finite Element and Discontinuous…

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https://www.youtube.com/embed/hdsVpGj4XtY  + 723
강연자 신동욱
소속 아주대학교
Various natural and physical phenomena can be described by partial differential equations (PDEs). To solve these equations, numerical methods based on the Finite Element (FE) and Discontinuous Galerkin (DG) approaches have been widely used and developed. In this talk, we focus on FE and DG methods for solving second-order elliptic equations. We discuss their theoretical foundations, numerical implementation, and key differences, particularly in terms of accuracy, stability, and conservation properties. While these methods allow for high-order accuracy, their convergence behavior is significantly influenced by the regularity of the solution. When the solution is not sufficiently smooth, adaptive algorithms play a crucial role in improving efficiency and accuracy. Here, we introduce reliable and fully computable a posteriori error estimators, which guide adaptive mesh refinement to enhance solution accuracy while reducing computational cost. Finally, numerical results will be presented to validate the theoretical findings and demonstrate the effectiveness of adaptive algorithms.
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Real spectra of large asymmetric Gaussian random matrices

강연자 : 이용우 | 소속 : 서울대학교
Gaussian random matrices play a central role in random matrix theory, serving as canonical models whose spectral properties often persist across much broader classes of ensembles. Many observables in non-Gaussian settings—including the limiting spectrum, local correlation functions, and counting statistics—exhibit the same asymptotic behavior as in the Gaussian case. In this talk, we briefly review the importance of Gaussian random matrices. We then focus on the real eigenvalues of asymmetric Gaussian random matrices, and discuss recent progress on their statistical properties.

Regularity theory of very weak solutions

강연자 : 임민규 | 소속 : KAIST
A very weak solution refers to a solution of a partial differential equation that has the lowest regularity and is defined in terms of distribution. Although the solutions best suited for establishing regularity theory are weak solutions, there are also regularity results for very weak solutions under certain conditions. This presentation will introduce the historical background as well as recent developments in the regularity theory of very weak solutions.

A brief introduction of noncommutative harmonic analysis

강연자 : Zhang Hao | 소속 : 서울대학교
Noncommutative harmonic analysis has been developing rapidly in recent years and has attracted considerable attention. It is deeply connected with classical Banach space theory, operator spaces, operator theory, geometric group theory, noncommutative geometry, Boolean analysis, quantum probability, and quantum information, among others, and has found wide-ranging applications across these fields. In this talk, I will present a brief overview of several important applications of noncommutative harmonic analysis. If time permits, I will also introduce my recent work on Schatten class membership of commutators.

Analytic subalgebras of Beurling-Fourier algebras and complexification of Lie groups

강연자 : 이헌 | 소속 : 하얼빈공업대학교
In this talk, we focus on how we can interpret the actions of the elements in the Gelfand spectrum of a Beurling-Fourier algebra on connected Lie groups. They can be viewed as evaluations on specific points of the complexification of the underlying Lie group. For this purpose, we need to introduce a particular dense subalgebra of the Beurling-Fourier algebra, which we call an analytic subalgebra. We first introduce an analytic subalgebra allowing a ``local" solution for general connected Lie groups. We will demonstrate that a ``global" solution is also possible for simply-connected nilpotent Lie groups.

Finding Spacecraft Orbits around Moons and Planets

강연자 : 정찬규 | 소속 : 서울대학교
The three-body problem asks a simple question: how do three objects move under mutual gravity? Unlike the two-body problem, which has general closed-form solutions, there is no single unified method for finding periodic solutions, even under the restricted assumption that the mass of the third body is negligible. In this talk, I will give a tour of some of the mathematical ideas behind finding periodic solutions, from classical ideas of Poincar\’e, to modern computer-assisted proofs. I will highlight how these ideas connect to developments in symplectic dynamics and also to applications in space missions.

Hamiltonian Dynamics and Three-body Problem

강연자 : 이동호 | 소속 : QSMS

Hamiltonian dynamics, originated in classical mechanics, lies at the root of symplectic geometry and provides a wealth of examples. One of the central topics is the study of periodic orbits, and over the years many tools have been developed to investigate them. Among the most interesting and historically important examples is the three-body problem. Despite centuries of research, however, a complete picture of its periodic orbits is still lacking, with only partial results available. In this talk, I will present the basic ideas and techniques of Hamiltonian dynamics and symplectic geometry including Floer theory, and introduce the three-body problem and related systems. I will also briefly describe some of my recent results related with the rotating Kepler problem and three-body problem.

Khintchine’s theorem on Diophantine approximation

강연자 : 김성민 | 소속 : 서울대학교

Diophantine approximation is the study of approximating real numbers by rational numbers. For example, one can ask whether a real number x is ψ-approximable; that is, whether there are infinitely many rationals p/q satisfying |x-p/q|<ψ(q)/q for a given monotonic function ψ. A century ago, Khintchine discovered a remarkable dichotomy for the Lebesgue measure of the set of ψ-approximable numbers. Since then, Khintchine’s theorem has been extended in various directions, including inhomogeneous approximation and higher-dimensional generalizations. In this talk, I will introduce the Allen-Ramírez conjecture on removing the monotonicity condition from the inhomogeneous Khintchine-Groshev theorem and discuss a recent proof of the conjecture in the case (n,m)=(2,1).

음향문제로 강연 후반 음성이 출력되지 않습니다

Various function filed extensioins over a finite field

강연자 : 유진주 | 소속 : 서울대학교

An algebraic function field is an algebraic extension of finite degree over the rational function field. This field was first introduced by Dedekind, Kronecker, and Weber in the nineteenth century and was further developed by Artin, Hasse, Schmidt, and Weil in the first half of the twentieth century. In this talk, we discuss various function field extensions over a finite field and examine important invariants such as the number of rational places and the class numbers.

음향문제로 강연 시작 4분 후부터 음성이 출력됩니다.

Bochner-Riesz means and spectral projection for Hermite expansions

강연자 : 유재현 | 소속 : 이화여자대학교

Hermite functions play an important role in many areas, including quantum mechanics, partial differential equations, and probability theory. In this talk, I will introduce two problems about Hermite functions that have recently attracted much attention in harmonic analysis. To help the audience follow along, I will begin with some background.

Moduli space of vector bundles on curves

강연자 : 임우남 | 소속 : 연세대학교

The moduli spaces of vector bundles on curves lie at the crossroads of geometry, topology, and representation theory. In this talk, I will introduce these moduli spaces with a focus on their topological aspects. In particular, I will present formulas for their Betti numbers and describe their intersection theory, along with some applications. If time permits, I will also discuss some recent developments about these classical moduli spaces.