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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.

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 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 *** 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 ***,we evaluate numerical method. for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a *** 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 *** 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 *** 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.

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.

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.

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.

Two-dimensional finite element model to study the effect of water flux on the nitrate dynamics in the rhizosphere of a maize root

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International Journal of Biomathematics 2017年第4期10卷 333-351页

作者： Ratnamala Badge Neeru Adlakha Department of Mathematics Maulana Azad National Institute of Technology Bhopal Madhya Pradesh 462003 India Department of MathematicsSardar Vallabhai National Institute of Technology Surat Gujarat India

The growth of plants and the quality and quantity of crop production depend on the uptake dynamics of water and mineral salts by the roots from the rhizosphere. This water and nutrient dynamics in the rhizosphere is still not well understood. In view of above a mathematical model is proposed to study the effect of water flux on the nitrate dynamics in the rhizosphere of a single maize root. The model incorporates the important mechanisms like diffusion and mass flow of solutes in the rhizosphere. The parameters like soil porosity, soil buffer power, radius of the root have incorporated in the model. Appropriate boundary and initial conditions have been framed based on the physical conditions of the problem. Coaxial circular sector elements have been employed to dis- cretize the circular cross-section of the root. Galerkin finite element method.has been employed to obtain the solution. The numerical results have been computed and used to study the effect of water flux on nitrate dynamics at the surface of the root and in the rhizosphere. The change percentage in nitrate concentration at the surface of the root in the rhizosphere of maize root due to change in water flux 10-7-10-6cms-1, 10-6-10-5cms-1 and 10-7-10-5cms-1 is observed to be 0.28-1.21%, 2.91 13.73% and 3.20-15.11% in μ mol cm-3 respectively which is quite significant.

关键词： Water flux nitrate finite element method.

Two-dimensional finite element model to study calcium distribution in astrocytes in presence of excess buffer

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International Journal of Biomathematics 2014年第3期7卷 137-147页

作者： Brajesh Kumar Jha Neeru Adlakha M. N. Mehta Department of Mathematics and Computer Science School of Technology Pandit Deendayal Petroleum University Raisen Gandhinagar Gujarat 382007 India Department of MathematicsM. A. National Institute of Technology Bhopal India Department of Applied MathematicsS. V. National Institute of Technology Surat India

In this paper a finite element model is developed to study cytosolic calcium concen- tration distribution in astrocytes for a two-dimensional steady-state case in presence of excess buffer. The mathematical model of calcium diffusion in astrocytes leads to a boundary value problem involving elliptical partial differential equation. The model con- sists of reaction-diffusion phenomena, association and dissociation rates and buffer. A point source of calcium is incorporated in the model. Appropriate boundary conditions have been framed. finite element method.is employed to solve the problem. A MATLAB program has been developed for the entire problem and simulated to compute the numer- ical results. The numerical results have been used to plot calcium concentration profiles in astrocytes. The effect of ECTA, BAPTA and aCa influx on calcium concentration distribution in astrocytes is studied with the help of numerical results.

关键词： Ca2+ profile buffer finite element method.