高精度特征解及其外推求解位势方程

High Accuracy Eigensolution and Its Extrapolation for Potential Equations

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,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和低的计算复杂度.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,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.

By the potential theorem,fundamental boundary eigenproblems were converted into boundary integral equations（BIE） with logarithmic singularity.Mechanical quadrature methods （MQMs） were presented to obtain eigensolutions 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 h^3-Richardson extrapolation algorithm（EA）,the accuracy order of the approximation can be greatly improved,and a posterior error estimate can be obtained as the self-adaptive algorithms.The efficiency of the algorithm is illustrated by examples.