Accelerated spectral approximation

Abstract

Abstract. A systematic development of higher order spectral analysis, introduced by Dellwo and Friedman, is undertaken in the framework of an appropriate product space. Accelerated analogues of Osborn's results about spectral approximation are presented. Numerical examples are given by considering an integral operator.

Accelerated spectral refinement Part I: simple eigenvalue

“…,< maxHence by our hypothesis (H), it follows that N q%n M q%n -*• 1 as n -> oo, uniformly in g.For obtaining the bound for ||S ? , n ||oo, note that k q ," -> k as n ->• oo, uniformly in q = 2,3,... by[6, Theorem 3.6]. Hence 1 ,.…”

Accelerated spectral refinement Part II: Cluster of eigenvalues

“…. (see [9,Proposition 3.2 (b)] and [10, Proposition 2.2 (a)]). Let P,, n and P£° be the spectral projections associated with T ?n and A 9 , n = cr(T, n ) n IntT, and T^n ) and A, respectively.…”

“…We now introduce some convenient notation, which is consistent with the notations used in [8] and [9].…”

“…In order to treat the case when min{|A| : X e A] > € and 0 < e < 1, we let a = 1/e, A = {aX : X € A} and consider the scaled operators T,, t , ", T q n) and the corresponding spectral projections P,, P,,", P^1', respectively as given in [9,Section 3] and [10,Section 3].…”

A spectral collocation method for eigenvalue problems of compact integral operators

“…The analysis of [3,17] covers many methods and provides a basis for the convergence analysis of our method. In [13], Dellwo and Friedman proposed a new approach by solving a polynomial eigenvalue problem of a higher degree, based upon which Alam et al [1] obtained an accelerated spectral approximation for eigenelements. Kulkarni [16] introduced another method by involving a new approximation operator T n and obtained a high-order convergence rate.…”