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ERROR ESTIMATES OF THE finite element method.WITH WEIGHTED BASIS FUNCTIONS FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATION

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Journal of Computational Mathematics 2011年第2期29卷 227-242页

作者： Xianggui Li Xijun Yu Guangnan Chen School of Applied Science Beijing Information Science and Technology University Beijing 100101 China Institute of Applied Physics and Computational Mathematics Beijing 100088 China

In this paper, we establish a convergence theory for a finite element method.with weighted basis functions for solving singularly perturbed convection-diffusion equations. The stability of this finite element method.is proved and an upper bound O（h｜lnε｜3/2） for errors in the approximate solutions in the energy norm is obtained on the triangular Bakhvalov-type mesh. Numerical results are presented to verify the stability and the convergent rate of this finite element method.

关键词： Convergence Singular perturbation Convection-diffusion equation finite element method.

A finite element method.WITH RECTANGULAR PERFECTLY MATCHED LAYERS FOR THE SCATTERING FROM CAVITIES

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Journal of Computational Mathematics 2009年第6期27卷 812-834页

作者： Deyue Zhang Fuming Ma Heping Dong School of Mathematics Jilin University Changchun 130012 China

We develop a finite element method.with rectangular perfectly matched layers （PMLs） for the wave scattering from two-dimensional cavities. The unbounded computational domain is truncated to a bounded one by using of a rectangular perfectly matched layer at the open aperture. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. Numerical experiments are carried out to illustrate the competitive behavior of the proposed method.

关键词： Cavity Perfectly matched layers finite element method.

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A Stabilized finite element method.for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore

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Communications in Computational Physics 2014年第1期15卷 93-125页

作者： Jehanzeb Hameed Chaudhry Jeffrey Comer Aleksei Aksimentiev Luke N.Olson Department for Mathematics Colorado State UniversityFort CollinsCO 80523USA. Department for Physics University of Illinois at Urbana-ChampaignUrbanaIL 61801USA. Department for Computer Science University of Illinois at Urbana-ChampaignUrbanaIL 61801USA

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly.This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials,in particular,near highly charged surfaces.A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations.Here,we evaluate numerical method. for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore.We describe a novel,robust finite element solver that combines the applications of the Newton’s method.to the nonlinear Galerkin form of the equations,augmented with stabilization terms to appropriately handle the drift-diffusion processes.To make direct comparison with particle-based simulations possible,our method.is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain.We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates,calculations of the ionic current though a nanopore subject to an external electric field,and modeling the effect of a DNA molecule on the ion concentration and nanopore current.

关键词： Steric effects nucleic acids ionic current continuum transport theory ion channels PNP SUPG finite element method.

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

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Journal of Computational Mathematics 2003年第4期21卷 435-450页

作者： Hou-de Han Xin Wen (Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China) Department of Mathematical Sciences Tsinghua University 北京 100084

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the upstream artificial boundary and two vertical sides are introduced as the downstream artificial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method. we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.

关键词： Ship wave Potential flow Global artificial boundary condition finite element method.

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EXPLICIT ERROR ESTIMATES FOR COURANT, CROUZEIX-RAVIART AND RAVIART-THOMAS finite element method.

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Journal of Computational Mathematics 2012年第4期30卷 337-353页

作者： Carsten Carstensen Joscha Gedicke Donsub Rim Institut fur Mathematik Humboldt-Universitiit zu Berlin Unter den Linden 6 10099 Berlin Germany Department of Computational Science and Engineering Yonsei University 120-749 Seoul Korea Institut fur Mathematik Humboldt-Universitat zu Berlin Unter den Linden 6 10099 Berlin Germany Yonsei School of Business and Department of Computational Science and Engineering Yonsei University 120-749 Seoul Korea

The elementary analysis of this paper presents explicit expressions of the constants in the a priori error estimates for the lowest-order Courant, Crouzeix-Raviart nonconforming and P^viart-Thomas mixed finite element method. in the Poisson model problem. The three constants and their dependences on some maximal angle in the triangulation are indeed all comparable and allow accurate a priori error control.

关键词： Error estimates Conforming Nonconforming Mixed finite element method.

GRADED MESHES FOR HIGHER ORDER FEM

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Journal of Computational Mathematics 2015年第1期33卷 1-16页

作者： Hans-Gorg Roos Ljiljana Teofanov Zorica Uzelac Institute of Numerical Mathematics Technical University of Dresden Dresden D-01062 Germany Department for Fundamental Disciplines Faculty of Technical Sciences University of Novi Sad Trg D. Obradovida 6 21000 Novi Sad Serbia

A singularly perturbed one-dimensional convection-diffusion problem is solved numeri- cMly by the finite element method.based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded （based on a mesh generating function） in the fine mesh region. Error estimates in the s-weighted energy norm are proved. We derive an ＇optimal＇ mesh generating function in order to min- imize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving er- ror estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element method.. Numerical experiments support all theoretical results.

关键词： Singular perturbation Boundary value problem Layer-adapted meshes Gradedmeshes finite element method.

THE RECONSTRUCTION OF OBSTACLES IN A WAVEGUIDE USING finite elementS

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Journal of Computational Mathematics 2018年第1期36卷 29-46页

作者： Ruming Zhang Jiguang Sun Department of Mathematical Sciences Michigan Technological University Michigan USA

This paper concerns the reconstruction of a penetrable obstacle embedded in a waveg- uide using the scattered data due to point sources, which is formulated as an optimization problem. We propose a fast reconstruction method.based on a carefully designed finite element scheme for the direct scattering problem. The method.has several merits： 1） the linear sampling method.is used to quickly obtain a good initial guess; 2） finite Fourier series are used to approximate the boundary of the obstacle, which is decoupled from the boundary used by the finite element method. and 3） the mesh is fixed and hence the stiff- ness matrix, mass matrix, and right hand side are assembled once and only minor changes are made at each iteration. The effectiveness of the proposed method.is demonstrated by numerical examples.

关键词： Inverse scattering problem Waveguides finite element method.

Cardiac nonrigid motion analysis from image sequences

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Progress in Natural Science:Materials International 2006年第10期16卷 1115-1118页

作者： LIU Huafeng (State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China) State Key Laboratory of Modern Optical Instrumentation Zhejiang University Hangzhou 310027 China

Noninvasive estimation of the soft tissue kinematics properties from medical image sequences has many important clinical and physiological implications, such as the diagnosis of heart diseases and the understanding of cardiac mechanics. In this paper, we present a biomechanics based strategy, framed as a priori constraints for the ill-posed motion recovery problems, to realize estimation of the cardiac motion and deformation parameters. By constructing the heart dynamics system equations from biomechanics principles, we use the finite element method.to generate smooth estimates of heart kinematics throughout the cardiac cycle. We present the application of the strategy to the estimation of displacements and strains from in vivo left ventricular magnetic resonance image sequence.

关键词： cardiac motion and deformation continnum mechanics model finite element method.

Sensitivity Analysis and Computations of the Time RelaxationModel

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Advances in Applied Mathematics and Mechanics 2015年第1期7卷 89-115页

作者： Monika Neda Fran Pahlevani Jiajia Waters Department of Mathematics University of NevadaLas VegasNV 89154USA Department ofMathematics Penn State University-Abington CollegePA 19001USA

This paper presents a numerical study of the sensitivity of a fluid model known as time relaxation model with respect to variations of the time relaxation coefficient c.The sensitivity analysis of this model is utilized by the sensitivity equation method.and uses the finite element method.along with Crank Nicolson method.in the fully discretization of the partial differential equations.We present a test case in support of the sensitivity convergence and also provide a numerical comparison between two different strategies of computing the sensitivity,sensitivity equation method.and forward finite differences.

关键词： Sensitivity analysis time relaxation model Navier-Stokes equations finite element method.