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Partition of unity method on nonmatching grids for the Stokes problem

Journal of Numerical mathematics 2005年第3期13卷 157-169页

作者： Bacuta, C. Chen, J. Huang, Y. Xu, J. Zikatanov, L. Mathematical Sciences University of Delaware Newark DE 19716 School of Mathematics and Computer Science Nanjing Normal University Nanjing 210097 China Institute for Computational and Applied Mathematics Department of Mathematics Xiangtan University 411105 China Department of Mathematics Pennsylvania State University University Park PA 16802

We consider the Stokes problem on a plane polygonal domain Ω ⊂ 2. We propose a finite element method for overlapping or nonmatching grids for the Stokes problem based on the partition of unity method. We prove that the discrete inf-sup condition holds with a constant independent of the overlapping size of the subdomains. The results are valid for multiple subdomains and any spatial dimension. © VSP 2005.

关键词： Problem solving

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A combination of energy method and spectral analysis for study of equations of gas motion

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中国高等学校学术文摘·数学 2009年第2期4卷 253-282页

作者： Renjun DUAN Seiji UKAI Tong YANG Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenbergerstrasse 69 A-4040 Linz Austria Liu Bie Ju Centre for Mathematical Sciences City University of Hong Kong Hong Kong China Department of Mathematics City University of Hong Kong Hong Kong China

There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic profiles. In this paper, we will first illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier-Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative systems in fluid dynamics and kinetic models such as the model system of radiating gas and the Vlasov-Poisson-Boltzmann system.

关键词： Energy method linearization asymptotic stability optimal rate spectral analysis

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C^0DISCONTINUOUS GALERKIN METHODS FOR A PLATE FRICTIONAL CONTACT PROBLEM

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Journal of computational mathematics 2019年第2期37卷 184-200页

作者： Fei Wang Tianyi Zhang Weimin Han School of Mathematics and Statistics Xi 'an Jiaotong University Xi'an 710049 China Program in Applied Mathematical and Computational Sciences University of Iowa Iowa City IA 52242 USA School of Mathematics and Statistics Xi 'an Jiaotong University Xi'an 710049 China Departinent of Mathematics University of Iowa Iowa City. IA 52242 USA

Numerous C^0 discontinuous Galerkin (DG) schemes for the Kirchhoff plate bending problem are extended to solve a plate frictional contact problem, which is a fourth-order elliptic variational inequality of the second kind. This variational inequality contains a nondifferentiable term due to the frictional contact. We prove that these C^0 DG methods are consis tent and st able, and derive optimal order error estima tes for the quadratic element. A numerical example is presented to show the performance of the C^0 DG methods;and the numerical convergence orders confirm the theoretical prediction.

关键词： Variational inequality of fourth-order Discontinuous Galerkin method Plate frictional contact problem Optimal order error estimate

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Exact Periodic Solitary Solutions to the Shallow Water Wave Equation

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Chinese Physics Letters 2009年第5期26卷 151-153页

作者：李栋龙赵俊霄 Department of Information and Computation Science Guangxi University of Technology Liuzhou 545006 Institute of Applied Physics and Computational Mathematics Beijing 100088 School of Mathematical Sciences Graduate University of Chinese Academy of Sciences Beijing 100049

Exact solutions to the shallow wave equation are studied based on the idea of the extended homoclinic test and bilinear method. Some explicit solutions, such as the one soliton solution, the doubly-periodic wave solution and the periodic solitary wave solutions, are obtained. In addition, the properties of the solutions are investigated.

关键词： gamma-ray bursts gamma-rays relativity

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Theory of differential approximations of radiative transfer equation 1

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作者： Han, Weimin Eichholz, Joseph A. Sheng, Qiwei Department of Mathematics and Program in Applied Mathematical and Computational Sciences University of Iowa Iowa City IA 52242 United States Department of Mathematics Rose-Hulman Institute of Technology Terre Haute IN 47803 United States

ISBN: (数字)9781461463931

ISBN: (纸本)9781461463924

The radiative transfer equation (RTE) arises in a variety of applications. The equation is challenging to solve numerically for a couple of reasons: high dimensionality, integro-differential form, highly forward-peaked scattering in application. In the literature, various approximations of RTE have been proposed in the literature. In an earlier publication, we explored a family of differential approximations to RTE, to be called RT/DA equations. In this paper, we study the RT/DA equations and investigate numerically the closeness of solutions of the RT/DA equations to that of the RTE. © Springer Science+Business Media New York 2013.

关键词： Radiative transfer

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Estimation of the spectral density of a homogeneous random stable discrete time field

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SORT 2005年第1期29卷 101-118页

作者： Demesh, N.N. Chekhmenok, S.L. Department of Information and Program-Mathematical Software of Automated Productions Faculty of Applied Mathematics and Computer Sciences Belarus State University F. Skoriny ave.4 Minsk 220000 Belarus

In earlier papers, 2π-periodic spectral data windows have been used in spectral estimation of discrete-time random fields having finite second-order moments. In this paper, we show that 2π-periodic spectral windows can also be used to construct estimates of the spectral density of a homoge-neous symmetric α-stable discrete-time random field. These fields do not have second-order moments if 0 < α < 2. We construct an estimate of the spectrum, calculate the asymptotic mean and variance, and prove weak consistency of our estimate.

关键词： Homogeneous Stable Fields spectral density estimate

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The graph limit of the minimizer of the Onsager-Machlup functional and its computation

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Science China mathematics 2021年第2期64卷 239-280页

作者： Qiang Du Tiejun Li Xiaoguang Li Weiqing Ren Department of Applied Phgysics and Applied Mathematics Columbia UniversityNeu YorkNY 10027USA LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Beijing Computational Science Research Certer Beijing 100193China MOE-LCSM School of Mathematics and StatisticsHunan Normal UniversityChangsha 410081China Department of Mathematics National University of SingaporeSingapore 119076Singapore

The Onsager-Machlup(OM)functional is well known for characterizing the most probable transition path of a diffusion process with non-vanishing ***,it suffers from a notorious issue that the functional is unbounded below when the specified transition time T goes to *** hinders the interpretation of the results obtained by minimizing the OM *** provide a new perspective on this *** mild conditions,we show that although the infimum of the OM functional becomes unbounded when T goes to infinity,the sequence of minimizers does contain convergent subsequences on the space of *** graph limit of this minimizing subsequence is an extremal of the abbreviated action functional,which is related to the OM functional via the Maupertuis principle with an optimal *** further propose an energy-climbing geometric minimization algorithm(EGMA)which identifies the optimal energy and the graph limit of the transition path *** algorithm is successfully applied to several typical examples in rare event *** interesting comparisons with the Freidlin-Wentzell action functional are also made.

关键词： Onsager-Machlup functional Freidlin-Wentzell functional graph limit geometric minimization Maupertuis principle

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Cosmological phase-space analysis of f(G)-theories of gravity

arXiv

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arXiv 2025年

作者： Papagiannopoulos, Giannis Luongo, Orlando Leon, Genly Paliathanasis, Andronikos Department of Physics National & Kapodistrian University of Athens Zografou Campus GR 157 73 Athens Greece School of Science and Technology University of Camerino Via Madonna delle Carceri Camerino62032 Italy Department of Nanoscale Science and Engineering University at Albany-SUNY AlbanyNY12222 United States INAF - Osservatorio Astronomico di Brera Milano Italy Sezione di Perugia Perugia06123 Italy Al-Farabi Kazakh National University Al-Farabi av. 71 Almaty050040 Kazakhstan Departamento de Matemáticas Universidad Católica del Norte Avda. Angamos 0610 Casilla Antofagasta1280 Chile Institute of Systems Science Department of Mathematics Faculty of Applied Sciences Durban University of Technology Durban4000 South Africa School for Data Science and Computational Thinking Department of Mathematical Sciences Stellenbosch University Stellenbosch7602 South Africa South Africa

The impact of topological terms that modify the Hilbert-Einstein action is here explored by virtue of a further f(G) contribution. In particular, we investigate the phase-space stability and critical points of an equivalent scalar field representation that makes use of a massive field, whose potential is function of the topological correction. To do so, we introduce to the gravitational Action Integral a Lagrange multiplier and model the modified Friedmann equations by virtue of new non-dimensional variables. We single out dimensionless variables that permit a priori the Hubble rate change of sign, enabling regions in which the Hubble parameter either vanishes or becomes negative. In this respect, we thus analyze the various possibilities associated with a universe characterized by such topological contributions and find the attractors, saddle points and regions of stability, in general. The overall analysis is carried out considering the exponential potential first and then shifting to more complicated cases, where the underlying variables do not simplify. We compute the eigenvalues of our coupled differential equations and, accordingly, the stability of the system, in both a spatially-flat and non-flat universe. Quite remarkably, regardless of the spatial curvature, we show that a stable de Sitter-like phase that can model current time appears only a small fraction of the entire phase-space, suggesting that the model under exam is unlikely in describing the whole universe dynamics, i.e., the topological terms appear disfavored in framing the entire evolution of the universe. © 2025, CC BY.

关键词： Topology

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A new approximate model of nonlinearly elastic flexural shell and its numerical computation

A new approximate model of nonlinearly elastic flexural shel...

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作者： Shen, Xiaoqin Li, Kaitai Li, Can Department of Applied Mathematical School of Sciences Xi'An University of Technology Xi'an 710054 China Department of Computational Mathematical School of Mathematics and Statistics Xi'An Jiaotong University Xi'an 710049 China

In this article, we construct a two-dimensional model for the nonlinearly elastic flexural shell using differential geometry and tensor analysis under the assumption that flexural energy is dominant, that is, the metric of middle surface remains invariant. We conduct a numerical experiment for special shell - a portion of cylinder shell, which is applied to the forces along the opposite direction of the normal vector. The displacements distribution of all points in the middle surface when the shell deforms is obtained. Numerical experiment results are consistent with the theory, which proves the validity of the proposed model. We then compare the proposed model and Ciarlet's model with 3D model, which proves that the proposed model is more approximate to 3D model than Ciarlet's. © 2013 Wiley Periodicals, Inc.

关键词： Geometry

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A Partial RKDG Method for Solving the 2D Ideal MHD Equations Written in Semi-Lagrangian Formulation on Moving Meshes with Exactly Divergence-Free Magnetic Field

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Advances in applied mathematics and Mechanics 2023年第4期15卷 932-963页

作者： Shijun Zou Xijun Yu Zihuan Dai Fang Qing Xiaolong Zhao School of Mathematical Sciences Peking UniversityBeijing 100871China Institute of Applied Physics and Computational Mathematics Beijing 100088China Department of Mathematics Southern University of Science and TechnologyShenzhenGuangdong 518055China School of Mathematics and Statistics Zhengzhou UniversityZhengzhouHenan 450001China

A partial Runge-Kutta Discontinuous Galerkin(RKDG)method which preserves the exactly divergence-free property of the magnetic field is proposed in this paper to solve the two-dimensional ideal compressible magnetohydrodynamics(MHD)equations written in semi-Lagrangian formulation on moving quadrilateral *** this method,the fluid part of the ideal MHD equations along with zcomponent of the magnetic induction equation is discretized by the RKDG method as our previous paper[47].The numerical magnetic field in the remaining two directions(i.e.,x and y)are constructed by using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal *** the divergence of the magnetic field in 2D is independent of its z-direction component,an exactly divergence-free numerical magnetic field can be obtained by this *** propose a new nodal solver to improve the calculation accuracy of velocities of the moving meshes.A limiter is presented for the numerical solution of the fluid part of the MHD equations and it can avoid calculating the complex eigen-system of the MHD *** numerical examples are presented to demonstrate the accuracy,non-oscillatory property and preservation of the exactly divergence-free property of our method.

关键词： Ideal compressible MHD equations semi-Lagrangian formulation exactly divergence-free magnetic field Runge-Kutta discontinuous Galerkin method

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