Error Estimates And High Accuracy Corrections Of The Approximate Eigenvalues By The Nine-point Difference Scheme On The L-shaped Region

This paper discusses 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. We prove that correction methods in [5], by which we may gain higher accuracy, are efficient except the first eigenvalue. Moreover, the result of this paper not only makes corrections to the hypothesize of asymptotic expansion of the first eigenvalue on L-shaped region in [1], but also explains the phenomena of vibrancy of approximation solution which have puzzled a lot of mathematicians for a long time.