L形区域特征值问题的九点差分近似的误差估计与高精度校正法

Error Estimates and High Accuracy Corrections of the Approximate Eigenvalues by the Nine-point Difference Scheme on the L-shaped Region

讨论了L型区域Laplace算子的各个特征函数在凹点邻域的性态,得到九点差分格式的各个近似特征值的误差估计,并证明了除第一特征值外,其余特征值皆可使用已有的校正法取得更高的精度.所得结果不仅更正了对L型区域第一特征值的渐进展开之猜想,而且解决了其数值解为何是振荡的这一难题.

Discussed the behavior at the reentrant corner of eigenfunctions for Laplace operator on the L-shaped region and presents error estimates and high accuracy corrections of the approximate eigenvalues with the nine-point difference scheme. Proved the methods of Lü Tao is correct, by which the authors gained higher accuracy, and are efficient except the first eigenvalue. Moreover, the result of this paper not only makes corrections to the hypotheseize of asympotic expansion of the first eigenvalue on L-shaped region in, but also explains the phenomena of vibrancy of approximation solution which have puzzled a lot of mathematicians for a long time.