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A MIXED FINITE ELEMENT METHOD ON A STAGGERED MESH FOR NAVIER-STOKES EQUATIONS

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Journal of Computational Mathematics 2008年第6期26卷 816-824页

作者： houde han Ming Yan Department of Mathematics University of Science and Technology of China Hefei 230026 China

In this paper, we introduce a mixed finite element method on a staggered mesh for the numerical solution of the steady state Navier-Stokes equations in which the two components of the velocity and the pressure are defined on three different meshes. This method is a conforming quadrilateral Q1 × Q1 - P0 element approximation for the Navier-Stokes equations. First-order error estimates are obtained for both the velocity and the pressure. Numerical examples are presented to illustrate the effectiveness of the proposed method.

关键词： Mixed finite element method Staggered mesh Navier-Stokes equations Error estimate.

A TAILORED FINITE POINT METHOD FOR THE HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS IN HETEROGENEOUS MEDIUM

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Journal of Computational Mathematics 2008年第5期26卷 728-739页

作者： houde han Zhongyi Huang Department of Mathematical Sciences Tsinghua University Beijing 100084 China

In this paper, we propose a tailored-finite-point method for the numerical simulation of the Helmholtz equation with high wave numbers in heterogeneous medium. Our finite point method has been tailored to some particular properties of the problem, which allows us to obtain approximate solutions with the same behaviors as that of the exact solution very naturally. Especially, when the coefficients are piecewise constant, we can get the exact solution with only one point in each subdomain. Our finite-point method has uniformly convergent rate with respect to wave number k in L^2-norm.

关键词： Tailored finite point method Helmholtz equation Inhomogeneous media High frequency wave.

POISSON PRECONDITIONING FOR SELF-ADJOINT ELLIPTIC PROBLEMS

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Journal of Computational Mathematics 2014年第5期32卷 560-578页

作者： houde han Chunxiong Zheng Department of Mathematical Sciences Tsinghua University Beijing 100084 China

In this paper, we formulate interface problem and Neumann elliptic boundary value problem into a form of linear operator equations with self-adjoint positive definite op- erators. We prove that in the discrete level the condition number of these operators is independent of the mesh size. Therefore, given a prescribed error tolerance, the classical conjugate gradient algorithm converges within a fixed number of iterations. The main computation task at each iteration is to solve a Dirichlet Poisson boundary value problem in a rectangular domain, which can be furnished with fast Poisson solver. The overall computational complexity is essentially of linear scaling.

关键词： Fast Poisson solver Interface problem Self-adjoint elliptic problem Conjugategradient method.

Split Local Artificial Boundary Conditions for the Two-Dimensional Sine-Gordon Equation on R^(2)

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Communications in Computational Physics 2011年第10期10卷 1161-1183页

作者： houde han Zhiwen Zhang Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China.

In this paper the numerical solution of the two-dimensional sine-Gordon equation is *** local artificial boundary conditions are obtained by the operator splitting *** the original problem is reduced to an initial boundary value problem on a bounded computational domain,which can be solved by the finite *** numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed method,and some interesting propagation and collision behaviors of the solitary wave solutions are observed.

关键词： Sine-Gordon equation operator splitting method artificial boundary condition soliton unbounded domain

A survey on artificial boundary method Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday

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Science China Mathematics 2013年第12期56卷 2439-2488页

作者： han houde WU XiaoNan Department of Mathematical Sciences Tsinghua University Department of Mathematics Hong Kong Baptist University

The artificial boundary method is one of the most popular and effective numerical methods tor solving partial differential equations on unbounded domains, with more than thirty years development. The artificiM boundary method has reached maturity in recent years. It has been applied to various problems in scientific and engineering computations, and the theoretical issues such as the convergence and error estimates of the artificial boundary method have been solved gradually. This paper reviews the development and discusses different forms of the artificial boundary method.

关键词： artificial boundary method global artificial boundary condition local artificial boundary condi-tion discrete artificial boundary condition implicit artificial boundary condition nonlinear artificial boundarycondition

Numerical Soliton Solutions for a Discrete Sine-Gordon System

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Communications in Computational Physics 2009年第9期6卷 903-918页

作者： houde han Jiwei Zhang Hermann Brunner Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong KongChina Department of Mathematics and Statistics Memorial University of NewfoundlandSt.John’sNLCanada A1C 5S7

In this paper we use an analytical-numerical approach to find,in a systematic way,new 1-soliton solutions for a discrete sine-Gordon system in one spatial *** the spatial domain is unbounded,the numerical scheme employed to generate these soliton solutions is based on the artificial boundary method.A large selection of numerical examples provides much insight into the possible shapes of these new 1-solitons.

关键词： Sine-Gordon equation soliton solutions numerical single solitons artificial boundary method

Numerical Simulation of Waves in Periodic Structures

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Communications in Computational Physics 2009年第5期5卷 849-870页

作者： Matthias Ehrhardt houde han Chunxiong Zheng Weierstraß-Institut fur Angewandte Analysis und Stochastik Mohrenstr.3910117 BerlinGermany Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China

In this work we improve and extend a technique named recursive doubling procedure developed by Yuan and Lu[*** Technology 25(2007),3649-3656]for solving periodic array *** turns out that when the periodic array contains an infinite number of periodic cells,our method gives a fast evaluation of the exact boundary Robin-to-Robin mapping if the wave number is complex,or real but in the stop *** technique is also used to solve the time-dependent Schr¨odinger equation in both one and two dimensions,when the periodic potential functions have some local defects.

关键词： Periodic media Helmholtz equation Schrodinger equation Dirichlet-to-Neumann maps Robin-to-Robin maps band structure Floquet-Bloch theory high-order finite elements

An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain

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Communications in Computational Physics 2011年第4期9卷 878-896页

作者： houde han Leevan Ling Tomoya Takeuchi Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong Department of Mathematics North Carolina State UniversityRaleighNorth Carolina 27695-8212USA.

Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the *** interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer *** Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary *** this work,we assume that the measurements are available on the whole outer boundary on an annulus *** imposing reasonable assumptions,the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization *** between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given.A novel numerical algorithm is proposed and numerical examples are given.

关键词： Inverse problem stability error bounds Tikhonov regularization method of fundamental solution

An Energy Regularization Method for the Backward Diffusion Problem and its Applications to Image Deblurring

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Communications in Computational Physics 2008年第6期4卷 177-194页

作者： houde han Ming Yan Chunlin Wu Department of Mathematics University of Science and Technology of ChinaHefeiAnhui 230026China.

For the backward diffusion equation,a stable discrete energy regularization algorithm is *** and uniqueness of the numerical solution are ***,the error between the solution of the given backward diffusion equation and the numerical solution via the regularization method can be *** numerical experiments illustrate the efficiency of the method,and its application in image deblurring.

关键词： Energy regularization method inverse problem heat equation backward diffusion equation image deblurring error estimate ill-posed well-posed.