A New Superconvergent Projection Method for Approximate Solutions of Eigenvalue Problems

Kulkarni, Rekha P.

doi:10.1081/nfa-120020246

Cited by 18 publications

(12 citation statements)

References 3 publications

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“…In [5] and [6], Kulkarni proposed to approximate a linear compact operator T by T K n = P n T P n + (I − P n )T P n + P n T (I − P n ). Since T is linear, T K n = P n T + T P n − P n T P n .…”

Section: A Projection Methodsmentioning

confidence: 99%

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A superconvergent projection method for nonlinear compact operator equations

Grammont¹,

Kulkarni

2005

Comptes Rendus. Mathématique

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Section: A Projection Methodsmentioning

confidence: 99%

“…In this Note, we adapt to the nonlinear case the method introduced by Kulkarni in [5] and [6] for a linear operator equation and the associated eigenvalue problem. In contrast to the iterated Galerkin solution, we show that the proposed solution is superconvergent to the exact solution without any additional conditions such as (4).…”

Section: Introductionmentioning

confidence: 99%

A superconvergent projection method for nonlinear compact operator equations

Grammont¹,

Kulkarni

2005

Comptes Rendus. Mathématique

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“…The approximations T P n , T S n , and T G n are known as the projection approximation, the Sloan approximation, and the Galerkin approximation of T , respectively. The approximation T K n was introduced by Kulkarni [9]. It may be called a modified projection approximation of T .…”

Section: − → T and Z ∈ ρ(T )mentioning

confidence: 99%

Operator Approximation

Limaye¹

2015

Semigroups, Algebras and Operator Theory

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We present an introduction to operator approximation theory. Let T be a bounded linear operator on a Banach space X over C. In order to find approximate solutions of (i) the operator equation z x − T x = y, where z ∈ C and y ∈ X are given, and (ii) the eigenvalue problem T ϕ = λϕ, where λ ∈ C and 0 = ϕ ∈ X , one approximates the operator T by a sequence (T n ) of bounded linear operators on X . We consider pointwise convergence, norm convergence, and nu convergence of (T n ) to T . We give several examples to illustrate possible scenarios. In most classical methods of approximation, each T n is of finite rank. We give a canonical procedure for reducing problems involving finite rank operators to problems involving matrix computations.

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“…In [8] we have considered the case of the orthogonal projections. We can also use the new operator T schemes for opertor equations as well as eigenvalue problems, multilevel methods, accelerated spectral approximation and extrapolation using this new operator.…”

Section: It Is Shown Thatmentioning

confidence: 99%

A new superconvergent collocation method for eigenvalue problems

Kulkarni

2006

Math. Comp.

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Abstract. Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ≤ r − 1, we show that the proposed method exhibits an error of the order of 4r for eigenvalue approximation and of the order of 3r for spectral subspace approximation. In the case of a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of the order 4r can be obtained. This improves upon the order 2r for eigenvalue approximation in the collocation/iterated collocation method and the orders r and 2r for spectral subspace approximation in the collocation method and the iterated collocation method, respectively. We illustrate this improvement in the order of convergence by numerical examples.

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A New Superconvergent Projection Method for Approximate Solutions of Eigenvalue Problems

Cited by 18 publications

References 3 publications

A superconvergent projection method for nonlinear compact operator equations

A superconvergent projection method for nonlinear compact operator equations

Operator Approximation

A new superconvergent collocation method for eigenvalue problems

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