拟周期小波边界元法求解拉普拉斯方程
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摘要
边界元法(BEM)是求解偏微分方程的一种有效的数值计算方法,边界元法将求解区域内的微分方程边值问题归化到边界上,然后在边界上离散求解。边界元法主要优点是降维,从而使问题所需的方程少,数据少,求解工作大为简化,这在处理高维问题时具有优势。
小波作为一个新兴的数学分支,应起始于S. Mallat和Y. Meyer在八十年代中后期所做的工作,即构造小波基的通用方法,此后小波得到了迅猛的发展。小波变换克服了传统Fourier变换的不足,是继Fourier分析之后的一个突破性进展。小波在时域和频域都具有良好的局部化特性,在应用领域更是掀起了一股应用小波的热潮,它具有丰富的理论,应用十分广泛,如信号处理、图像分析等,是工程应用中强有力的方法和工具,给许多相关领域带来了崭新的思想,并使其被越来越多的数学研究工作者所关注,由于小波兼有光滑性和局部紧支撑性质,能更好的处理积分和微分方程的数值求解问题。
小波BEM自上世纪90年代提出以来,一直是国内外学者研究的热点。小波BEM具有迭代效率高、矩阵预条件简单等优点,因此成为众多学者关注的一种BEM快速求解方法。
全文分为四章,第一章主要介绍了论文选题背景,小波与边界元研究的历史和现状,以及本文要做的工作。第二章介绍了小波分析的基本理论,包括小波和小波变换的定义、性质,多分辨分析、尺度函数的定义、性质以及周期拟小波的基本理论。第三章介绍了边界元基本理论,加权余量法、变分法概述,以及边界积分方程的推导过程。第四章是本文的主要工作,主要研究了利用拟周期小波边界元方法讨论二维拉普拉斯方程的数值求解问题。首先用边界元方法将待求方程化为边界积分方程,然后用拟周期小波作为基底,将积分方程在边界上展开,化为对应的代数方程组,求解系数,得到方程的近似解。在对方程组求解过程中,系数矩阵作小波矩阵变换,利用多尺度方法,求解新的代数方程组,减少计算量,并分析了此算法的收敛性、算法复杂度及误差分析。
Boundary element method is an effective method to solve partial differential equation. Boundary element method naturalizes the differential equation in solve region to boundary and then disperses boundary to solve. The main virtue of boundary element method is the reduce of dimensions, thus equations and dates the question needed is decreased so the solving work is simplified, that is advantage to solve high dimension equations.
Wavelet theory as a new mathematical branche starts from the study of S. Mallat and Y. Meyer in 1980s, that is the construction of wavelet basis and has developed rapidly ever since. Wavelet transform overcomes defects of traditional Fourier transform,which is a breakthrough progress after Fourier analysis. Wavelet has well localized property both in time and frequency domains, creates an wavelet upsurge in application fields. It possess abundant theory and is applied widely, such as signal processing, image analysis. It is a powerful method and tools in application fields. It gives new idea to related fields and is concerned by more and more math research workers, because of its own smoothness and local compact support properties, can solve numerical computing integral and differential equations better.
Since was introduced in 90s last century, wavelet boundary element method is always hot spots studied by scholars home and aboard. Wavelet BEM possess high iteration efficiency and preconditioning matrix simple, thus concerned by numerous scholars as a fast BEM solving method.
The paper is divided to four chapters. The first chapter mainly relates background of selected subjects, history and present situations of wavelet and boundary element method researched and study work of this paper. The second chapter relates basic wavelet analysis theory including definition and property of wavelet, wavelet transform, multi resolution analysis and scale function as well as periodic quasi wavelet theory. The third chapter introduces basis theory of boundary element, summary of method of weight residual and variational method and deduction process of boundary integral equation. The fourth chapter is the main work of this paper, its mainly studied numerical computing of two dimensions Laplace equation using quasi periodic wavelet boundary element method. First using boundary element method coverts equation to be solved into boundary integral equation, then spread boundary integral equations by quasi wavelet basis turns to corresponding algebraic equations and solve it getting approximately solutions. In the process of solving equations do matrix transform using wavelet matrix and use multi scale method solving new algebraic equations reduce calculated amount, discuss convergence, complexity, error analysis of this arithmetic.
小波作为一个新兴的数学分支,应起始于S. Mallat和Y. Meyer在八十年代中后期所做的工作,即构造小波基的通用方法,此后小波得到了迅猛的发展。小波变换克服了传统Fourier变换的不足,是继Fourier分析之后的一个突破性进展。小波在时域和频域都具有良好的局部化特性,在应用领域更是掀起了一股应用小波的热潮,它具有丰富的理论,应用十分广泛,如信号处理、图像分析等,是工程应用中强有力的方法和工具,给许多相关领域带来了崭新的思想,并使其被越来越多的数学研究工作者所关注,由于小波兼有光滑性和局部紧支撑性质,能更好的处理积分和微分方程的数值求解问题。
小波BEM自上世纪90年代提出以来,一直是国内外学者研究的热点。小波BEM具有迭代效率高、矩阵预条件简单等优点,因此成为众多学者关注的一种BEM快速求解方法。
全文分为四章,第一章主要介绍了论文选题背景,小波与边界元研究的历史和现状,以及本文要做的工作。第二章介绍了小波分析的基本理论,包括小波和小波变换的定义、性质,多分辨分析、尺度函数的定义、性质以及周期拟小波的基本理论。第三章介绍了边界元基本理论,加权余量法、变分法概述,以及边界积分方程的推导过程。第四章是本文的主要工作,主要研究了利用拟周期小波边界元方法讨论二维拉普拉斯方程的数值求解问题。首先用边界元方法将待求方程化为边界积分方程,然后用拟周期小波作为基底,将积分方程在边界上展开,化为对应的代数方程组,求解系数,得到方程的近似解。在对方程组求解过程中,系数矩阵作小波矩阵变换,利用多尺度方法,求解新的代数方程组,减少计算量,并分析了此算法的收敛性、算法复杂度及误差分析。
Boundary element method is an effective method to solve partial differential equation. Boundary element method naturalizes the differential equation in solve region to boundary and then disperses boundary to solve. The main virtue of boundary element method is the reduce of dimensions, thus equations and dates the question needed is decreased so the solving work is simplified, that is advantage to solve high dimension equations.
Wavelet theory as a new mathematical branche starts from the study of S. Mallat and Y. Meyer in 1980s, that is the construction of wavelet basis and has developed rapidly ever since. Wavelet transform overcomes defects of traditional Fourier transform,which is a breakthrough progress after Fourier analysis. Wavelet has well localized property both in time and frequency domains, creates an wavelet upsurge in application fields. It possess abundant theory and is applied widely, such as signal processing, image analysis. It is a powerful method and tools in application fields. It gives new idea to related fields and is concerned by more and more math research workers, because of its own smoothness and local compact support properties, can solve numerical computing integral and differential equations better.
Since was introduced in 90s last century, wavelet boundary element method is always hot spots studied by scholars home and aboard. Wavelet BEM possess high iteration efficiency and preconditioning matrix simple, thus concerned by numerous scholars as a fast BEM solving method.
The paper is divided to four chapters. The first chapter mainly relates background of selected subjects, history and present situations of wavelet and boundary element method researched and study work of this paper. The second chapter relates basic wavelet analysis theory including definition and property of wavelet, wavelet transform, multi resolution analysis and scale function as well as periodic quasi wavelet theory. The third chapter introduces basis theory of boundary element, summary of method of weight residual and variational method and deduction process of boundary integral equation. The fourth chapter is the main work of this paper, its mainly studied numerical computing of two dimensions Laplace equation using quasi periodic wavelet boundary element method. First using boundary element method coverts equation to be solved into boundary integral equation, then spread boundary integral equations by quasi wavelet basis turns to corresponding algebraic equations and solve it getting approximately solutions. In the process of solving equations do matrix transform using wavelet matrix and use multi scale method solving new algebraic equations reduce calculated amount, discuss convergence, complexity, error analysis of this arithmetic.
引文
1 N. Nishimura. Fast Multipole Accelerated Boundary Integral Equation Methods. Appl. Mech. Rev.. 2002, 55(4): 299~324
2 J. Xiao, L. H. Wen and J. Tausch. A Multilevel Galerkin Boundary Element Method. 8th World Congress on Computational Mechanics & 5th European Congress on Computational Methods in Applied Sciences and Engineering, Venice, Italy, 2008: 15~24
3 J. Xiao, L .H. Wen and J. Tausch. On Fast Matrix-Vector Multiplication in Wavelet Galerkin BEM. Submitted to Eng. Anal. Bound. Elem. 2008: 2~13
4 Y. Yan. A Fast Boundary Element Method for the Two Dimensional Helmholtz Equations. Comput. Method. Appl Mech Engrg. 2008: 6~16
5 W. Dahmen, H. Harbrecht and R. Schneider. Compression Techniques for Boundary Integral Equations-Optimal Complexity Estimates. Math. Computer. 2007, 256(36): 1243~1274
6 H. Harbrecht, R. Schneider. Wavelet Galerkin Schemes for Boundary Integral Equations Implementation and Quadrature. SIAM Journal on Scientific Computing. 2006, 27(4): 1347~1370
7 J. Tausch, J. White. Surface Independent Multiscale Bases for the Sparse Representational of Integral Operators on Complex Geometry. SIAM Journal on Scientific Computing. 2003, 24(5): 1610~1629
8 F. Khellat, S. A. Yousefi. The Linear Legendre Mother Wavelets Operational Matrix of Integration and its Application. Journal of the Franklin Institute. 2006, (43): 181~190
9 K. Eppler, H. Harbrecht. Exterior Electromagnetic Shaping Using Wavelet BEM. Math. Meth. Appl. Sci. 2005, (28): 387~405
10 C. K. Chui, A. Lian. A Study of Orthogonormal Multiwavelet, Applied Numer. Math.. 1996, 20(20): 273~298,
11李弼时,罗建书.小波分析及其应用.电子工业出版社, 2003: 53-97
12 K. Malekndjad, F. Mirzaee. Using Rationalized Haar Wavelet for Solving Linear Integral Equations. Appl. Math. Comput.. 2005, (16): 579~587
13 T. V. Petersdorff, C. Schwab, R. Schneider. Multiwavelet for Second Integral Equations. SIAM. Numerical Analysis. 1997, 34(6): 2212~2227
14 J. Ravnik, L. Skerget and M.Hribersek. The Wavelet Transform for BEM Computational Fluid Dynamics. Eng. Anal. Bound. Elem. 2004, (28): 1303~1314
15王元淳著.边界元法基础.上海交通大学出版社, 1988: 15~42
16杨德全,赵忠生著.边界元理论及应用.北京理工大学出版社, 2002: 1~35
17 M. Fratantonio, J. Rencis. Exact Boundary Element Integration for Two-Dimensional Laplace Equation. Engineering Analysis with Boundary Elements. 2005, (29): 325~342
18赵忠生,徐光远. FEM与BEM中的积分离散计算.第一届全国解析与数值结合法会议文集.湖南大学出版社,1990: 29~53
19 I. Daubechies. Ten Lectures on Wavelets. CBMS/NSF Series in Applied Mathematics, V. 61. SIAM. 1992: 16~72
20崔锦态著.小波分析导论.程正兴译.西安交通大学出版社, 1995: 2~83
21李登峰,彭思龙,陈翰麟.一类周期小波的局部性质.数学学报. 2001, 44(5): 947~960
22 J. Xiao, L. Wen. Solving Second Kind Fredholm Integral Equations by Periodic Wavelet Galerkin Method. Applied Mathematics and Computations. 2006, 175(1): 508~518
23成礼智,王红霞,罗永.小波的理论与应用.科学出版社, 2004: 103~128
24徐长发,李国宽.实用小波方法.华中科技大学出版社, 2001: 27~53
25程正兴.小波分析算法及应用.西安交通大学出版社, 1998: 49~68
26刘明才.小波分析中的若干数值计算问题.吉林大学博士学位论文. 1999: 12~38
27程正兴,张玲玲.多小波分析与应用.工程数学学报. 2001, 18(1): 99~107
28 J. Gao, Y. L.Jiang. Trigonometric Hermite Wavelet Approximation for the Integral Equations of Second Kind with Weakly Singular Kernel. J. Comput. Appl. Math. 2007: 249~283
29 S. Yousefia, M. Razaaghib. Legendre Wavelets Method for the Nonlinear Volterra-Fredholm Integral Equations. Mathematics and Computers in Simulation. 2005, (70): 1~8
30 W. R. Madych, S. A. Nelson. Multivariate Interpolation and Conditionally Positive Definite Functions. Approximation Theory and its Appl. 1998, (4): 77~79
31陆金甫,关治著.偏微分方程数值解法.清华大学出版社, 1987: 116~132
32 E. Aksan. A Numerical Solution of Burgers Equation by Finite Element Constructed on the Method of Discretization in Time. Applied Mathematics and Computation. 2005, (170):895~904
33祝家麟著.椭圆边值问题的边界元分析.科学出版社, 1991: 68~93
34 B. Rathish. Wavelet-Taylor Galerkin Methods for the Burgers Equations. BitNumerical Mathematics. 2005, (45):543~560
35 S. Bertoluzza, G. Naldi. A Wavelet Collocation Method for the Numerical Solution of Partial Differential Equations. Appl. Comp. Harm. Anal. 1996, (3):83~97
36陈传璋等.积分方程理论及应用.上海科学技术出版社, 1987: 19~36
37王先标.积分方程数值解得小波方法.中山大学博士学位论文. 1997: 23~46
38高友兰.数值积分的Daubechies小波方法.大连理工大学硕士学位论文. 2006: 6~19
39梁学章,何甲兴,王新民,李强编著.小波分析.国防工业出版社, 2005: 72~112
40关履泰.有限区间截断B-样条小波及其消失矩性质.中山大学学报. 1996,(3): 28~33
41 H. L. Chen. Complex Harmonic Splines, Periodic Quasi-Wavelets. Theory and Applications. Kluwer Academic Publishers, 2000: 276~324
42 H. L. Chen, S. L. Peng. Localization of Dual Periodic Scaling and Wavelet Functions. Advances in Computational Mathematics. Special Issue on Pohang Conference. 2002: 383~426
43 L. Schumaker. Spline Functions, Basic Theory. New York, Wiley, 1981: 134~139
2 J. Xiao, L. H. Wen and J. Tausch. A Multilevel Galerkin Boundary Element Method. 8th World Congress on Computational Mechanics & 5th European Congress on Computational Methods in Applied Sciences and Engineering, Venice, Italy, 2008: 15~24
3 J. Xiao, L .H. Wen and J. Tausch. On Fast Matrix-Vector Multiplication in Wavelet Galerkin BEM. Submitted to Eng. Anal. Bound. Elem. 2008: 2~13
4 Y. Yan. A Fast Boundary Element Method for the Two Dimensional Helmholtz Equations. Comput. Method. Appl Mech Engrg. 2008: 6~16
5 W. Dahmen, H. Harbrecht and R. Schneider. Compression Techniques for Boundary Integral Equations-Optimal Complexity Estimates. Math. Computer. 2007, 256(36): 1243~1274
6 H. Harbrecht, R. Schneider. Wavelet Galerkin Schemes for Boundary Integral Equations Implementation and Quadrature. SIAM Journal on Scientific Computing. 2006, 27(4): 1347~1370
7 J. Tausch, J. White. Surface Independent Multiscale Bases for the Sparse Representational of Integral Operators on Complex Geometry. SIAM Journal on Scientific Computing. 2003, 24(5): 1610~1629
8 F. Khellat, S. A. Yousefi. The Linear Legendre Mother Wavelets Operational Matrix of Integration and its Application. Journal of the Franklin Institute. 2006, (43): 181~190
9 K. Eppler, H. Harbrecht. Exterior Electromagnetic Shaping Using Wavelet BEM. Math. Meth. Appl. Sci. 2005, (28): 387~405
10 C. K. Chui, A. Lian. A Study of Orthogonormal Multiwavelet, Applied Numer. Math.. 1996, 20(20): 273~298,
11李弼时,罗建书.小波分析及其应用.电子工业出版社, 2003: 53-97
12 K. Malekndjad, F. Mirzaee. Using Rationalized Haar Wavelet for Solving Linear Integral Equations. Appl. Math. Comput.. 2005, (16): 579~587
13 T. V. Petersdorff, C. Schwab, R. Schneider. Multiwavelet for Second Integral Equations. SIAM. Numerical Analysis. 1997, 34(6): 2212~2227
14 J. Ravnik, L. Skerget and M.Hribersek. The Wavelet Transform for BEM Computational Fluid Dynamics. Eng. Anal. Bound. Elem. 2004, (28): 1303~1314
15王元淳著.边界元法基础.上海交通大学出版社, 1988: 15~42
16杨德全,赵忠生著.边界元理论及应用.北京理工大学出版社, 2002: 1~35
17 M. Fratantonio, J. Rencis. Exact Boundary Element Integration for Two-Dimensional Laplace Equation. Engineering Analysis with Boundary Elements. 2005, (29): 325~342
18赵忠生,徐光远. FEM与BEM中的积分离散计算.第一届全国解析与数值结合法会议文集.湖南大学出版社,1990: 29~53
19 I. Daubechies. Ten Lectures on Wavelets. CBMS/NSF Series in Applied Mathematics, V. 61. SIAM. 1992: 16~72
20崔锦态著.小波分析导论.程正兴译.西安交通大学出版社, 1995: 2~83
21李登峰,彭思龙,陈翰麟.一类周期小波的局部性质.数学学报. 2001, 44(5): 947~960
22 J. Xiao, L. Wen. Solving Second Kind Fredholm Integral Equations by Periodic Wavelet Galerkin Method. Applied Mathematics and Computations. 2006, 175(1): 508~518
23成礼智,王红霞,罗永.小波的理论与应用.科学出版社, 2004: 103~128
24徐长发,李国宽.实用小波方法.华中科技大学出版社, 2001: 27~53
25程正兴.小波分析算法及应用.西安交通大学出版社, 1998: 49~68
26刘明才.小波分析中的若干数值计算问题.吉林大学博士学位论文. 1999: 12~38
27程正兴,张玲玲.多小波分析与应用.工程数学学报. 2001, 18(1): 99~107
28 J. Gao, Y. L.Jiang. Trigonometric Hermite Wavelet Approximation for the Integral Equations of Second Kind with Weakly Singular Kernel. J. Comput. Appl. Math. 2007: 249~283
29 S. Yousefia, M. Razaaghib. Legendre Wavelets Method for the Nonlinear Volterra-Fredholm Integral Equations. Mathematics and Computers in Simulation. 2005, (70): 1~8
30 W. R. Madych, S. A. Nelson. Multivariate Interpolation and Conditionally Positive Definite Functions. Approximation Theory and its Appl. 1998, (4): 77~79
31陆金甫,关治著.偏微分方程数值解法.清华大学出版社, 1987: 116~132
32 E. Aksan. A Numerical Solution of Burgers Equation by Finite Element Constructed on the Method of Discretization in Time. Applied Mathematics and Computation. 2005, (170):895~904
33祝家麟著.椭圆边值问题的边界元分析.科学出版社, 1991: 68~93
34 B. Rathish. Wavelet-Taylor Galerkin Methods for the Burgers Equations. BitNumerical Mathematics. 2005, (45):543~560
35 S. Bertoluzza, G. Naldi. A Wavelet Collocation Method for the Numerical Solution of Partial Differential Equations. Appl. Comp. Harm. Anal. 1996, (3):83~97
36陈传璋等.积分方程理论及应用.上海科学技术出版社, 1987: 19~36
37王先标.积分方程数值解得小波方法.中山大学博士学位论文. 1997: 23~46
38高友兰.数值积分的Daubechies小波方法.大连理工大学硕士学位论文. 2006: 6~19
39梁学章,何甲兴,王新民,李强编著.小波分析.国防工业出版社, 2005: 72~112
40关履泰.有限区间截断B-样条小波及其消失矩性质.中山大学学报. 1996,(3): 28~33
41 H. L. Chen. Complex Harmonic Splines, Periodic Quasi-Wavelets. Theory and Applications. Kluwer Academic Publishers, 2000: 276~324
42 H. L. Chen, S. L. Peng. Localization of Dual Periodic Scaling and Wavelet Functions. Advances in Computational Mathematics. Special Issue on Pohang Conference. 2002: 383~426
43 L. Schumaker. Spline Functions, Basic Theory. New York, Wiley, 1981: 134~139