高精度特征解方法和外推解位势方程

程攀; 黄晋; 曾光

doi:10.3879/j.issn.1000-0887.2010.12.005

高精度特征解方法和外推解位势方程

doi: 10.3879/j.issn.1000-0887.2010.12.005

程攀^1,2,
黄晋¹,
曾光¹

电子科技大学数学科学院,成都 611731;

基金项目: 国家自然科学基金资助项目(10871034)

详细信息

作者简介:
程攀(1976- ),男,重庆人,讲师,博士(联系人.E-mail:cheng_pass@sina.com).

中图分类号: O24;O39
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- 被引次数: 0
出版历程
- 收稿日期: 1900-01-01
- 修回日期: 2010-11-05
- 刊出日期: 2010-12-15

High Accuracy Eigensolution and Its Extrapolation for Potential Equations

School of Mathematical Science, University of Electronic Science & Technology of China, Chengdu 611731, P. R. China;
School of Science, Chongqing Jiaotong University, Chongqing 400074, P. R. China

摘要

摘要: 根据位势理论，基本边界特征值问题可转化为具有对数奇性的边界积分方程．利用机械求积方法求解特征值和特征向量λ(l),u(l)，以及利用这些特征解求解Laplace方程．特征解和Laplace方程的解具有高精度O(h3)和低的计算复杂度O(h-1)．利用Anselone聚紧和渐近紧理论，证明了方法的收敛性和稳定性．此外，还给出了误差的奇数阶渐近展开．利用h3-Richardson外推，不仅误差近似的精度阶大为提高O(h5)，而且，得到的后验误差估计可以构造自适应算法．具体的数值例子说明了算法的有效性．
- 位势方程 /
- 机械求积法 /
- Richardson外推 /
- 后验误差估计
Abstract: By the potential theorem,fundamental boundary eigenproblems were converted into boundary in tegral equations (BIE) with logarithmic singularity.Mechanical quadrature methods (MQMs) were presented to obtain eigen solutions which were used to solve Laplace's equations.And the MQMs possess high accuracies and low computing complexities.The convergence and stability were proved based on Anselone's collective compact and asymptotical compact theory.Furthermore,an asymptotic expansion with odd powers of the errors is presented.Using h³-Richardson extrapolation algorithm (EA),the accuracy order of the approximation can be greatly improved,and a posterior error estmiate can be obtained as the self-adaptive algorithms.The efficiency of the algorithm is illustrated by examples.
- potential equation /
- mechanical quadrature methods /
- Richard sonextrapolation /
- posteriori error estimate

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参考文献(14)

[1]	Courant R, Hilbert D. Methods of Mathematical Physics[M]. New York: John Wiley and Sons, 1953.
[2]	Lanczos C. Discourse on Fourier Series[M]. Edinburgh: Oliver and Boyd, 1966.
[3]	Hadjesfandiari A R, Dargush G F. Theory of boundary eigensolutions in engineering mechanics[J]. J Appl Mech, 2001,68(1): 101-108. doi: 10.1115/1.1331059
[4]	Banerjee P K. The Boundary Element Methods in Engineering[M]. London: McGraw-Hill, 1994.
[5]	Li Z C. Combinations of method of fundamental solutions for Laplace’s equation with singularities[J]. Engineering Analysis With Boundary Elements, 2008, 32(10): 856-869. doi: 10.1016/j.enganabound.2008.01.002
[6]	Amini S. Nixon S P. Preconditioned multiwavelet Galerkin boundary element solution of Laplace’s equation[J]. Engineering Analysis With Boundary Elements, 2006, 30(7): 523-530. doi: 10.1016/j.enganabound.2006.02.003
[7]	吕涛, 黄晋. 第二类弱奇异积分方程的高精度Nystrm方法与外推[J].计算物理，1997, 14(3): 349-355.
[8]	Liu C S. A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation[J]. Engineering Analysis With Boundary Elements, 2008, 32(9): 778-785. doi: 10.1016/j.enganabound.2007.12.002
[9]	Sidi A, Israrli M. Quadrature methods for periodic singular and weakly singular Fredholm integral equations[J]. J Sci Comput, 1988, 3(2): 201-231. doi: 10.1007/BF01061258
[10]	Huang J, Wang Z. Extrapolation algorithm for solving mixed boundary integral equations of the Helmholtz equation by mechanical quadrature methods[J]. SIAM J Sci Comput, 2009, 31(6): 4115-4129.
[11]	Sloan I H, Spence A. The Galerkin method for integral equations of the first kind with logarithmic kernel: theorm[J].IMA J Numer Anal,1988, 8(1):105-122. doi: 10.1093/imanum/8.1.105
[12]	Anselone P M. Collectively Compact Operator Approximation Theory[M]. New Jersey: Prentice-Hall, Englewood Cliffs, 1971.
[13]	Chatelin F. Spectral Approximation of Linear Operator[M]. Academic Press, 1983.
[14]	Lin C B, Lü T, Shih T M. The Splitting Extrapolation Method[M]. Singapore: World Scientific, 1995.