角形域上Neumann边值问题小波自然边界元法研究
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摘要
小波自然边界元法是近几年发展起来的一种新型用于求解偏微分方程的数值计算方法。目前关于它的研究结果还相对较少，但这种方法从一开始就展现了它独特的优点和强劲的生命力。论文主要是研究小波基函数在自然边界元方法中，应用于求角形区域上调和方程Neumann边值问题的数值解。目的是解决自然边界元方法中，计算奇异积分时存在的困难，从而减小计算量，提高计算精度。其基本思想是首先引入保角映射得到角形区域上的自然积分方程，将微分方程经过自然边界归化得到与之等价的变分问题，再利用Galerkin-wavelet方法或小波插值法将其离散，得到具有独特优点的刚度矩阵，这样就能大大减小计算量。同时对数值解的误差估计作了进一步的分析。给出的算例表明了各种小波自然边界元方法的可行性和有效性。
首先，综述了边界元方法的发展历史、研究现状，然后对边界归化方法、小波分析及保角映射的基本理论进行了概述，并分析了该课题的研究意义。同时介绍了调和方程在典型域上通过自然边界归化所得的自然边界积分方程和Poisson积分公式，以及一般单连通区域上的自然边界归化及其对角形域，扇形域与矩形域的应用。
其次，论文针对角形域上调和方程的Neumann边值问题在求解过程中遇到的强奇异积分的困难，论文充分利用Shannon小波在频域上有限带宽的优良性质，采用Shannon小波边界元方法求解，使问题简化且得到的数值解比较精确。同时论文又充分利用了拟小波基在时域中光滑性高、快速衰减且具有插值性的特点，使其与自然边界元方法相结合，再引入了保角映射，有效解决了角形域上调和方程边值问题时存在的奇异积分困难。
最后，论文采用Hermite三次样条多小波边界元方法对角形域上调和方程的Neumann边值问题做了进一步的探讨，并通过算例验证了该方法的有效性。
In recent years, wavelet boundary element method has developed a new numericalmethod. Although there is few works about this method, it still shows unique advantagesand strong vitality at the beginning. To overcome the shortcoming of singular integraldifficulty existed in the natural boundary element method, in this paper, we mainly solvethe Neumann boundary value problem of Laplace equation in the angle domain to studythe application of the wavelet base function in the natural boundary element method, it notonly simplify the calculation process, but also improve the computation accuracy. Thebasic idea is that introducing conformal mapping to obtain the natural integral equation,transforming the differential equations into its equivalent variational problem by thenatural boundary naturalization, and then discreting it by the Galerkin-wavelet method orwavelet interpolation to get the corresponding stiffness matrix with unique advantage, itsubstantially reducing computation. Meanwhile we make the further analysis to the errorestimate of the numerical solution. The given example indicates the validity of each kindof wavelet boundary element method.
Firstly, we retrospect the development history and research situation of the boundaryelement method, the boundary naturalization theory, conformal mapping and the waveletanalysis theory, and analyze the significance of the research projects. At the same time, weintroduce the natural boundary integral equation of Laplace equation and the Poissonintegral formula by the natural boundary naturalization on the typical domain, and thenatural boundary naturalization theory in the simply-connected domain and the applicationof the above theory in the angle domain, sector domain and rectangle domain are alsogiven.
Secondly, since there is difficulty in solving the Neumann boundary value problem ofthe Laplace equation in the angle domain encounters hypersingular integral. In order tosimplify the problem and get more precise numerical solution, we take full advantages ofthe Shannon wavelet limited bandwidth in frequency domain and use Shannon waveletboundary element method to solve the problem. In the meantime we combine the properties of smooth, fast weaken and the interpolation of the wavelet base with thenatural boundary element method, it introduces the conformal mapping. And it effectivelyresolves the trouble of the singular integral when solving the boundary value problem ofLaplace equation in angle domain.
Lastly, we apply Hermite Cubic Spline Multi-wavelet Natural Boundary ElementMethod to the Neumann boundary value problem of the Laplace equation in the angledomain and make further research. Moreover, the given example proves that the method iseffective and flexible.
首先，综述了边界元方法的发展历史、研究现状，然后对边界归化方法、小波分析及保角映射的基本理论进行了概述，并分析了该课题的研究意义。同时介绍了调和方程在典型域上通过自然边界归化所得的自然边界积分方程和Poisson积分公式，以及一般单连通区域上的自然边界归化及其对角形域，扇形域与矩形域的应用。
其次，论文针对角形域上调和方程的Neumann边值问题在求解过程中遇到的强奇异积分的困难，论文充分利用Shannon小波在频域上有限带宽的优良性质，采用Shannon小波边界元方法求解，使问题简化且得到的数值解比较精确。同时论文又充分利用了拟小波基在时域中光滑性高、快速衰减且具有插值性的特点，使其与自然边界元方法相结合，再引入了保角映射，有效解决了角形域上调和方程边值问题时存在的奇异积分困难。
最后，论文采用Hermite三次样条多小波边界元方法对角形域上调和方程的Neumann边值问题做了进一步的探讨，并通过算例验证了该方法的有效性。
In recent years, wavelet boundary element method has developed a new numericalmethod. Although there is few works about this method, it still shows unique advantagesand strong vitality at the beginning. To overcome the shortcoming of singular integraldifficulty existed in the natural boundary element method, in this paper, we mainly solvethe Neumann boundary value problem of Laplace equation in the angle domain to studythe application of the wavelet base function in the natural boundary element method, it notonly simplify the calculation process, but also improve the computation accuracy. Thebasic idea is that introducing conformal mapping to obtain the natural integral equation,transforming the differential equations into its equivalent variational problem by thenatural boundary naturalization, and then discreting it by the Galerkin-wavelet method orwavelet interpolation to get the corresponding stiffness matrix with unique advantage, itsubstantially reducing computation. Meanwhile we make the further analysis to the errorestimate of the numerical solution. The given example indicates the validity of each kindof wavelet boundary element method.
Firstly, we retrospect the development history and research situation of the boundaryelement method, the boundary naturalization theory, conformal mapping and the waveletanalysis theory, and analyze the significance of the research projects. At the same time, weintroduce the natural boundary integral equation of Laplace equation and the Poissonintegral formula by the natural boundary naturalization on the typical domain, and thenatural boundary naturalization theory in the simply-connected domain and the applicationof the above theory in the angle domain, sector domain and rectangle domain are alsogiven.
Secondly, since there is difficulty in solving the Neumann boundary value problem ofthe Laplace equation in the angle domain encounters hypersingular integral. In order tosimplify the problem and get more precise numerical solution, we take full advantages ofthe Shannon wavelet limited bandwidth in frequency domain and use Shannon waveletboundary element method to solve the problem. In the meantime we combine the properties of smooth, fast weaken and the interpolation of the wavelet base with thenatural boundary element method, it introduces the conformal mapping. And it effectivelyresolves the trouble of the singular integral when solving the boundary value problem ofLaplace equation in angle domain.
Lastly, we apply Hermite Cubic Spline Multi-wavelet Natural Boundary ElementMethod to the Neumann boundary value problem of the Laplace equation in the angledomain and make further research. Moreover, the given example proves that the method iseffective and flexible.
引文
[1]李景湧.有限元法[M].北京:邮电大学出版社,2004:51-87.
[2]王勖成,邵敏.有限单元法基本原理和数值方法(第2版)[M].北京:清华大学出版社,1997:1-2.
[3]谢贻权,何福保.弹性和塑性力学中的有限单元法[M].北京:机械工业出版社,1981:1-5.
[4]刘继军.现代数值计算方法[M].北京:科学出版社,2010,3:1-160.
[5]姚振汉,杜庆华.边界元法应用的若干近期研究及国际新发展[J].清华大学学报(自然科学版),2001,4:89-93.
[6]杜庆华,姚振汉.边界积分方程边界元法的基本理论及其在弹性力学方面的若干工程应用[J].固体力学学报,1982,3:1-22.
[7]姚振汉,王海涛.边界元法[M].北京:高等教育出版社,2010,4:1-73.
[8]徐长发,李红.偏微分方程数值解法[M].武汉:华中科技大学出版社,2000:123-314.
[9]余德浩,汤华中.微分方程数值解法[M].北京:科学出版社,2003:421-459.
[10]陈传璋.积分方程论及其应用[M].北京:科学技术文献出版社,1987:1-86.
[11]冯康,余德浩.自然边界归化与区域分解[M].冯康文集,国防工业出版社,1994:367-371.
[12] Feng K, Yu D H. Canonical Intergral Equation of Elliptic Boundary Value Problem and TheirNumerical Solutions[J]. Proc. of China France Symp.on FEM(1982, Beijing), Science Press,Beijing,1983,211-252.
[13]冯康.有限元方法与自然边界归化[D].国际数学家大会论文集,华沙:1983:1439-1453.
[14]余德浩.自然边界元的数学理论[M].北京:科学出版社,1993:86-108.
[15]祝家麟,袁政强.边界元分析[M].北京:科学出版社,2009:45-55.
[16]冯康,余德浩.椭圆边值问题的正则积分方程与其数值解[D].中法有限元讨论会文集,北京:科学出版社,1993:211-252.
[17] Mallat S. Multiresolution Approximation and Wavelets[J]. Trans of American Math Society,1989,31(5):69-88.
[18] Meyer Y. Wavelets Algorithms and Application[J]. SIAM,1993:16-27.
[19]程正兴,杨守志,冯晓霞.小波分析的理论、算法、进展和应用[M].北京:国防工业出版社,2007,7:1-2.
[20]唐玲艳.双曲型方程数值解的小波方法研究[D].长沙:国防科学技术大学博士论文,2007,10:2-4.
[21]崔锦泰著,程正兴译.小波分析导论[M].西安:西安交通大学出版社,1995:41-46.
[22]徐长发,李国宽.实用小波方法[M].武汉:华中科技大学出版社,2001:92-139.
[23] Wei G W. Quasi Wavelets and Quasi Interpolating Wavelets[]J. Chem Phys Lett,1998,296(3-4):215-222.
[24] Wei G W. Discrete Singular Convolution for the Solution of the Fokker-Planck Equations[J]. JChem Phy,1999,110:8930-8942.
[25] Wei G W. A Unified Approach for the Solution of the Fokker-Planck Equations[J]. J Phys A.2000,33:4935-4953.
[26] Wei G W, Zhang D S, Althorpe S C. Wavelet Distributed Approximating Method for Solving theNavier Stokes Equation[J]. Computer Physics Communications,1998,115:18-24.
[27] Wan D C, Wei G W. The Study of Quasi Wavelets Based Numerical Method Applied toBurgers’Equations[J]. Appl Math Mech Engl,2000,21:1099-1110.
[28]成礼智,王红霞,罗永.小波的理论与应用[M].北京:科学出版社,2004:370-395.
[29] Well R O, Jr, Zhou X. Wavelet Solutions for the Dirichlet Problem[J]. Numerische Mathematik,1995,70:379-396.
[30] Cohen A, Dahmen W, Devore R. Adaptive Wavelet Methods for Elliptic Operator EquationsConvergence Rates[J]. Mathematics of Computation,2001,70(233):27-75.
[31] Cohen A, Masson R. Wavelet Methods for Second Order Elliptic Problems Preconditioning andAdaptivity[J]. SIAM Journal on Scientific Computing,1999,21(3):1006-1026.
[32] Alpha Mamadou Bah. Conformal Mapping of a Symmetrically Horizontal Slit Rectangle onto aRectangle with a Square Removed[J]. Nonlinear Analysis,2009,71:2427-2430.
[33]拉夫连季耶夫M A等著,施祥林等译.复变函数论方法[M].北京:高等教育出版社,2006:183-185.
[34]陈亚军,杜其奎.椭圆外无穷扇形区域边值问题的自然边界元法[J].南京师大学报(自然科学版),2009,32(2):6-12.
[35]张敏,杜其奎.椭圆外区域上Helmholtz问题的自然边界元法[J].计算数学,2008,30(1):75-88.
[36]陈亚浙,吴兰成.数学物理方程讲义[M].北京:高等教育出版社,2000:72-83.
[37]陈文胜,林伟. Hadamard奇异积分方程和Hermite小波方法[D].第5届国际有限或无限元复分析会议文集,北京:1997:13-22.
[38]徐耀群,孙明. Shannon小波混沌神经网络及其TSP问题的求解[J].控制理论与应用,2008,25(3):574-577.
[39] Glowinski R, Lawton W, Ravachol M.一维空间上线性和非线性椭圆,双曲和抛物问题的小波解,应用科学和工程中的计算方法[J]. SIAM, Philadelphia,1990:55-120.(英文版)
[40]杜微微,邓彩霞,赵国良. Shannon小波变换像空间的描述[J].哈尔滨理工大学学报,2003,18(3):127-130.
[41] Tao R, Li B Z, Wang Y. Spectral Analysis and Reconstruction for Periodic Nonuniformly SampledSignals in Fractional Fourier Domain[J]. IEEE Trans Signal Process,2007,55(7):3541-3547.
[42]陈文胜,张永虎.调和方程自然边界元的Shannon小波方法[J].高等学校计算数学学报,2001,2:125-134.
[43]李俊贤,陈一鸣,管巍等. Neumann边值问题的小波边界元法[J].佳木斯大学学报(自然科学版),2008,26(6):794-2008.
[44] Mallat S, Multiresolution Approximation and Wavelet Orthonormal Bases of L2[J], Trans. of theAmerica Math. Society,1989,315:69-87.
[45] Wei G W, Zhao Y B, Xiang Y. Discrete Singular Convolution and Application to the Analysis ofPlates with Internal Support[J]. Part1: Theory and algorithm. Int.J. Numer.Methods.Engrg.2002,55:913-946.
[46]万德成,韦国伟.用拟小波方法数值求解Burgers方程[J].应用数学和力学,2000,21:991-1001.
[47]宋玲玲,缪守莉.测不准原理的分析[J].安徽农业技术师范学院学报,1999,3:91-92.
[48] Qian L W, Wei G W. A Note on Regularized Shannon’s Sampling Formulae[J]. Arxiv math.NA/0005003,2000,1:1-9.
[49]孙阳光.自然边界元的无网格方法和拟小波方法研究[D].武汉:华中科技大学硕士学位论文,2005,4:33-35.
[50] Quak E. Trigonometric Wavelets for Hermite Interpolation[J]. Math Comput,1996,65:683-722.
[51] Chen Xuefeng, Xiang Jiawei. Solving Diffusion Equation Using Wavelet Method[J]. AppliedMathematics and Computation,2011,217(13):6426–6432.
[52] Maleknejad K, Yousefi M. Numerical Solution of the Integral Equation of the Second Kind byUsing Wavelet Bases of Hermite Cubic Splines[J]. Applied Mathematics and Computation,2006,183(1):134-141.
[53] Dahmen W, Han B, Jia R Q, Kunoth A. Biorthogonal Multiwavelets on the Interval Cubic HermiteSplines[J]. Constr Approx,2000,16:221-259.
[54] Strela V, Strang G. Finte Element Multiwavelets. In Wavelet Applications in Signal and Image[J].Society of Photo-Optical Industrial Engineers. Bellingham WA,1994,(23):202-213.
[55]徐长发.对数型奇异核小波展开的递推算法[J].华中理工大学学报,1998,5:110-112.
[56] Zhu Jialin, Zhang Shougui. Galerkin Boundary Element Method for Solving the BoundaryIntegral Equation with Hypersingularity[J]. Journal of University of and Technclogy of China.2007,37(11):1357-1362.
[57]张玲玲,祝家麟,林鑫等.带超强奇异积分的Galerkin边界元法[J].重庆大学学报,2008,31(1):115-118.
[58] Cai Haotao. A Fast Solver for a Hypersingular Boundary Integral Equation[J]. Applied NumericalMathematics,2009,59:1960-1969.
[59] Boykov I V, Ventsel E S, Boykova A I. An Approximate Solution of Hypersingular IntegralEquations[J]. Applied Numerical Mathematics,2010,60:607-628.
[2]王勖成,邵敏.有限单元法基本原理和数值方法(第2版)[M].北京:清华大学出版社,1997:1-2.
[3]谢贻权,何福保.弹性和塑性力学中的有限单元法[M].北京:机械工业出版社,1981:1-5.
[4]刘继军.现代数值计算方法[M].北京:科学出版社,2010,3:1-160.
[5]姚振汉,杜庆华.边界元法应用的若干近期研究及国际新发展[J].清华大学学报(自然科学版),2001,4:89-93.
[6]杜庆华,姚振汉.边界积分方程边界元法的基本理论及其在弹性力学方面的若干工程应用[J].固体力学学报,1982,3:1-22.
[7]姚振汉,王海涛.边界元法[M].北京:高等教育出版社,2010,4:1-73.
[8]徐长发,李红.偏微分方程数值解法[M].武汉:华中科技大学出版社,2000:123-314.
[9]余德浩,汤华中.微分方程数值解法[M].北京:科学出版社,2003:421-459.
[10]陈传璋.积分方程论及其应用[M].北京:科学技术文献出版社,1987:1-86.
[11]冯康,余德浩.自然边界归化与区域分解[M].冯康文集,国防工业出版社,1994:367-371.
[12] Feng K, Yu D H. Canonical Intergral Equation of Elliptic Boundary Value Problem and TheirNumerical Solutions[J]. Proc. of China France Symp.on FEM(1982, Beijing), Science Press,Beijing,1983,211-252.
[13]冯康.有限元方法与自然边界归化[D].国际数学家大会论文集,华沙:1983:1439-1453.
[14]余德浩.自然边界元的数学理论[M].北京:科学出版社,1993:86-108.
[15]祝家麟,袁政强.边界元分析[M].北京:科学出版社,2009:45-55.
[16]冯康,余德浩.椭圆边值问题的正则积分方程与其数值解[D].中法有限元讨论会文集,北京:科学出版社,1993:211-252.
[17] Mallat S. Multiresolution Approximation and Wavelets[J]. Trans of American Math Society,1989,31(5):69-88.
[18] Meyer Y. Wavelets Algorithms and Application[J]. SIAM,1993:16-27.
[19]程正兴,杨守志,冯晓霞.小波分析的理论、算法、进展和应用[M].北京:国防工业出版社,2007,7:1-2.
[20]唐玲艳.双曲型方程数值解的小波方法研究[D].长沙:国防科学技术大学博士论文,2007,10:2-4.
[21]崔锦泰著,程正兴译.小波分析导论[M].西安:西安交通大学出版社,1995:41-46.
[22]徐长发,李国宽.实用小波方法[M].武汉:华中科技大学出版社,2001:92-139.
[23] Wei G W. Quasi Wavelets and Quasi Interpolating Wavelets[]J. Chem Phys Lett,1998,296(3-4):215-222.
[24] Wei G W. Discrete Singular Convolution for the Solution of the Fokker-Planck Equations[J]. JChem Phy,1999,110:8930-8942.
[25] Wei G W. A Unified Approach for the Solution of the Fokker-Planck Equations[J]. J Phys A.2000,33:4935-4953.
[26] Wei G W, Zhang D S, Althorpe S C. Wavelet Distributed Approximating Method for Solving theNavier Stokes Equation[J]. Computer Physics Communications,1998,115:18-24.
[27] Wan D C, Wei G W. The Study of Quasi Wavelets Based Numerical Method Applied toBurgers’Equations[J]. Appl Math Mech Engl,2000,21:1099-1110.
[28]成礼智,王红霞,罗永.小波的理论与应用[M].北京:科学出版社,2004:370-395.
[29] Well R O, Jr, Zhou X. Wavelet Solutions for the Dirichlet Problem[J]. Numerische Mathematik,1995,70:379-396.
[30] Cohen A, Dahmen W, Devore R. Adaptive Wavelet Methods for Elliptic Operator EquationsConvergence Rates[J]. Mathematics of Computation,2001,70(233):27-75.
[31] Cohen A, Masson R. Wavelet Methods for Second Order Elliptic Problems Preconditioning andAdaptivity[J]. SIAM Journal on Scientific Computing,1999,21(3):1006-1026.
[32] Alpha Mamadou Bah. Conformal Mapping of a Symmetrically Horizontal Slit Rectangle onto aRectangle with a Square Removed[J]. Nonlinear Analysis,2009,71:2427-2430.
[33]拉夫连季耶夫M A等著,施祥林等译.复变函数论方法[M].北京:高等教育出版社,2006:183-185.
[34]陈亚军,杜其奎.椭圆外无穷扇形区域边值问题的自然边界元法[J].南京师大学报(自然科学版),2009,32(2):6-12.
[35]张敏,杜其奎.椭圆外区域上Helmholtz问题的自然边界元法[J].计算数学,2008,30(1):75-88.
[36]陈亚浙,吴兰成.数学物理方程讲义[M].北京:高等教育出版社,2000:72-83.
[37]陈文胜,林伟. Hadamard奇异积分方程和Hermite小波方法[D].第5届国际有限或无限元复分析会议文集,北京:1997:13-22.
[38]徐耀群,孙明. Shannon小波混沌神经网络及其TSP问题的求解[J].控制理论与应用,2008,25(3):574-577.
[39] Glowinski R, Lawton W, Ravachol M.一维空间上线性和非线性椭圆,双曲和抛物问题的小波解,应用科学和工程中的计算方法[J]. SIAM, Philadelphia,1990:55-120.(英文版)
[40]杜微微,邓彩霞,赵国良. Shannon小波变换像空间的描述[J].哈尔滨理工大学学报,2003,18(3):127-130.
[41] Tao R, Li B Z, Wang Y. Spectral Analysis and Reconstruction for Periodic Nonuniformly SampledSignals in Fractional Fourier Domain[J]. IEEE Trans Signal Process,2007,55(7):3541-3547.
[42]陈文胜,张永虎.调和方程自然边界元的Shannon小波方法[J].高等学校计算数学学报,2001,2:125-134.
[43]李俊贤,陈一鸣,管巍等. Neumann边值问题的小波边界元法[J].佳木斯大学学报(自然科学版),2008,26(6):794-2008.
[44] Mallat S, Multiresolution Approximation and Wavelet Orthonormal Bases of L2[J], Trans. of theAmerica Math. Society,1989,315:69-87.
[45] Wei G W, Zhao Y B, Xiang Y. Discrete Singular Convolution and Application to the Analysis ofPlates with Internal Support[J]. Part1: Theory and algorithm. Int.J. Numer.Methods.Engrg.2002,55:913-946.
[46]万德成,韦国伟.用拟小波方法数值求解Burgers方程[J].应用数学和力学,2000,21:991-1001.
[47]宋玲玲,缪守莉.测不准原理的分析[J].安徽农业技术师范学院学报,1999,3:91-92.
[48] Qian L W, Wei G W. A Note on Regularized Shannon’s Sampling Formulae[J]. Arxiv math.NA/0005003,2000,1:1-9.
[49]孙阳光.自然边界元的无网格方法和拟小波方法研究[D].武汉:华中科技大学硕士学位论文,2005,4:33-35.
[50] Quak E. Trigonometric Wavelets for Hermite Interpolation[J]. Math Comput,1996,65:683-722.
[51] Chen Xuefeng, Xiang Jiawei. Solving Diffusion Equation Using Wavelet Method[J]. AppliedMathematics and Computation,2011,217(13):6426–6432.
[52] Maleknejad K, Yousefi M. Numerical Solution of the Integral Equation of the Second Kind byUsing Wavelet Bases of Hermite Cubic Splines[J]. Applied Mathematics and Computation,2006,183(1):134-141.
[53] Dahmen W, Han B, Jia R Q, Kunoth A. Biorthogonal Multiwavelets on the Interval Cubic HermiteSplines[J]. Constr Approx,2000,16:221-259.
[54] Strela V, Strang G. Finte Element Multiwavelets. In Wavelet Applications in Signal and Image[J].Society of Photo-Optical Industrial Engineers. Bellingham WA,1994,(23):202-213.
[55]徐长发.对数型奇异核小波展开的递推算法[J].华中理工大学学报,1998,5:110-112.
[56] Zhu Jialin, Zhang Shougui. Galerkin Boundary Element Method for Solving the BoundaryIntegral Equation with Hypersingularity[J]. Journal of University of and Technclogy of China.2007,37(11):1357-1362.
[57]张玲玲,祝家麟,林鑫等.带超强奇异积分的Galerkin边界元法[J].重庆大学学报,2008,31(1):115-118.
[58] Cai Haotao. A Fast Solver for a Hypersingular Boundary Integral Equation[J]. Applied NumericalMathematics,2009,59:1960-1969.
[59] Boykov I V, Ventsel E S, Boykova A I. An Approximate Solution of Hypersingular IntegralEquations[J]. Applied Numerical Mathematics,2010,60:607-628.