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Legendre-Fourier spectral approximation and error analysis for nonlinear eigenvalue problems in complex domains

JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2025年 1-28页

作者： Zheng, Jihui An, Jing Guizhou Normal Univ Sch Math Sci Guiyang 550025 Peoples R China

In this paper, we develop and analyze an efficient Legendre-Fourier spectral approximation for solving nonlinear eigenvalue problems in complex domains. The main idea is to employ the domain mapping method to convert the nonlinear eigenvalue problem on a complex domain into an equivalent form on a standard circular domain. Based on this, an effective Legendre-Fourier spectral method is implemented by utilizing Legendre polynomials and Fourier series approximations in the radial and tangential directions, respectively. As the initial step, we establish a priori error estimates for standard circular regions. Then, we define a new class of projection operators, demonstrate their approximation properties, and further prove the error estimates for approximating eigenvalues and their corresponding eigenfunctions. Subsequently, by employing region mapping techniques, we extend the algorithm to address nonlinear eigenvalue problems in two-dimensional complex domains, and validate its convergence and spectral accuracy through numerical examples.

关键词： nonlinear eigenvalue problems Legendre-fourier spectral method Error analysis Complex domains

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nonlinear eigenvalue problems and contour integrals

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2016年 292卷 526-540页

作者： Van Barel, Marc Kravanja, Peter Katholieke Univ Leuven Dept Comp Sci B-3001 Heverlee Belgium

In this paper Beyn's algorithm for solving nonlinear eigenvalue problems is given a new interpretation and a variant is designed in which the required information is extracted via the canonical polyadic decomposition of a Hankel tensor. A numerical example shows that the choice of the filter function is very important, particularly with respect to where it is positioned in the complex plane. (C) 2015 Elsevier B.V. All rights reserved.

关键词： nonlinear eigenvalue problems Contour integrals Keldysh' theorem Beyn's algorithm Canonical polyadic decomposition Filter function

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nonlinear eigenvalue problems of Schrodinger type admitting eigenfunctions with given spectral characteristics

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MATHEMATISCHE NACHRICHTEN 2002年第1期242卷 91-118页

作者： Heid, M Heinz, HP Weth, T Univ Mainz Fachbereich Math D-55099 Mainz Germany

The following work is an extension of our recent paper [10]. We still deal with nonlinear eigenvalue problems of the form (*)A(0)y + B(gamma)gamma = Agamma in a real Hilbert space H with a semi-bounded self-adjoint operator A(0), while for every y from a dense subspace X of W, B(gamma) is a symmetric operator. The left-hand side is assumed to be related to a certain auxiliary functional psi, and the associated linear problems (**)A(0)v + B(gamma)v = muv are supposed to have non-empty discrete spectrum (gamma is an element of X). We reformulate and generalize the topological method presented by the authors in [10] to construct solutions of (*) on a sphere S-R := {gamma is an element of X \ parallel togammaparallel to(H) = R} whose psi-value is the n-th Ljusternik-Schnirelman level of psi\s(R) and whose corresponding eigenvalue is the n-th eigenvalue of the associated linear problem (**), where R > 0 and n is an element of IN are given. In applications, the eigenfunctions thus found share any geometric property enjoyed by an n-th eigenfunction of a linear problem of the form (**). We discuss applications to elliptic partial differential equations with radial symmetry.

关键词： nonlinear eigenvalue problems Ljusternik-Schnirelman levels nonlinear Schrodinger equations nodal structure

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Convergence factors of Newton methods for nonlinear eigenvalue problems

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LINEAR ALGEBRA AND ITS APPLICATIONS 2012年第10期436卷 3943-3953页

作者： Jarlebring, Elias Katholieke Univ Leuven Dept Comp Sci B-3001 Louvain Belgium

Consider a complex sequence {lambda(k)}(k=0)(infinity) convergent to lambda(*) is an element of C with order p is an element of N. The convergence factor is typically defined as the fraction c(k) := (lambda(k+1) - lambda(*))/(lambda(k) - lambda(*))(p) in the limit k -> infinity. In this paper, we prove formulas characterizing ck in the limit k -> infinity for two different Newton-type methods for nonlinear eigenvalue problems. The formulas are expressed in terms of the left and right eigenvectors. The two treated methods are called the method of successive linear problems (MSLP) and augmented Newton and are widely used in the literature. We prove several explicit formulas for c(k) for both methods. Formulas for both methods are found for simple as well as double eigenvalues. In some cases, we observe in examples that the limit c(k) as k -> infinity does not exist. For cases where this limit does not appear to exist, we prove other limiting expressions such that a characterization of c(k) in the limit is still possible. (C) 2010 Elsevier Inc. All rights reserved.

关键词： nonlinear eigenvalue problems Newton's method Convergence factors

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Local bifurcation theory for multiparameter nonlinear eigenvalue problems

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nonlinear ANALYSIS-THEORY METHODS & APPLICATIONS 2002年第8期48卷 1077-1086页

作者： Bari, R Univ Dhaka Dept Math Dhaka 1000 Bangladesh

A local bifurcation theory for multiparameter nonlinear eigenvalue problems was analyzed. The problem was reduced to a finite-dimensional problem by using Lyapunov-Schmidt decomposition. The transversality condition was found to be generic. Analysis indicated the existence of an open set with no element satisfying the spanning condition existed.

关键词： bifurcation point nonlinear eigenvalue problems multiparameter operator Lyapunov-Schmidt method

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A new approach in studying one parameter nonlinear eigenvalue problems with constraints

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nonlinear ANALYSIS-THEORY METHODS & APPLICATIONS 2005年第3期60卷 443-463页

作者： Motreanu, D Univ Perpignan Dept Math F-66860 Perpignan France

The paper presents a new approach for obtaining existence and location of solutions to nonlinear eigenvalue problems depending on a parameter and subject to constraints. The location of eigensolutions and subsequent parameters is achieved by means of the graph of the derivative of a function used also to compensate the lack of coercivity. The applications concern one parameter families of semilinear elliptic eigenvalue problems with Dirichlet boundary conditions for which various qualitative properties of the solution sets are established. (C) 2004 Elsevier Ltd. All rights reserved.

关键词： nonlinear eigenvalue problems critical point theory semilinear elliptic equations location properties

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Solving large-scale finite element nonlinear eigenvalue problems by resolvent sampling based Rayleigh-Ritz method

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COMPUTATIONAL MECHANICS 2017年第2期59卷 317-334页

作者： Xiao, Jinyou Zhou, Hang Zhang, Chuanzeng Xu, Chao Northwestern Polytech Univ Sch Astronaut Xian 710072 Peoples R China Univ Siegen Dept Civil Engn D-57068 Siegen Germany

This paper focuses on the development and engineering applications of a new resolvent sampling based Rayleigh-Ritz method (RSRR) for solving large-scale nonlinear eigenvalue problems (NEPs) in finite element analysis. There are three contributions. First, to generate reliable eigenspaces the resolvent sampling scheme is derived from Keldysh's theorem for holomorphic matrix functions following a more concise and insightful algebraic framework. Second, based on the new derivation a two-stage solution strategy is proposed for solving large-scale NEPs, which can greatly enhance the computational cost and accuracy of the RSRR. The effects of the user-defined parameters are studied, which provides a useful guide for real applications. Finally, the RSRR and the two-stage scheme is applied to solve two NEPs in the FE analysis of viscoelastic damping structures with up to 1 million degrees of freedom. The method is versatile, robust and suitable for parallelization, and can be easily implemented into other packages.

关键词： Finite element methods Viscoelastic material structures nonlinear eigenvalue problems Rayleigh-Ritz projection

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The spectral-Galerkin approximation of nonlinear eigenvalue problems

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APPLIED NUMERICAL MATHEMATICS 2018年 131卷 1-15页

作者： An, Jing Shen, Jie Zhang, Zhimin Beijing Computat Sci Res Ctr Beijing 100193 Peoples R China Guizhou Normal Univ Sch Math Sci Guiyang 550025 Guizhou Peoples R China Xiamen Univ Fujian Prov Key Lab Math Modeling & High Performa Xiamen 361005 Peoples R China Xiamen Univ Sch Math Sci Xiamen 361005 Peoples R China Purdue Univ Dept Math W Lafayette IN 47907 USA Wayne State Univ Dept Math Detroit MI 48202 USA

In this paper we present and analyze a polynomial spectral-Galerkin method for nonlinear elliptic eigenvalue problems of the form -div(A del u) + Vu + f(u(2))u = lambda u, parallel to u parallel to(L2) = 1. We estimate errors of numerical eigenvalues and eigenfunctions. Spectral accuracy is proved under rectangular meshes and certain conditions of f. In addition, we establish optimal error estimation of eigenvalues in some hypothetical conditions. Then we propose a simple iteration scheme to solve the underlying an eigenvalue problem. Finally, we provide some numerical experiments to show the validity of the algorithm and the correctness of the theoretical results. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Spectral-Galerkin approximation Error estimation Iteration algorithm nonlinear eigenvalue problems

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An integral method for solving nonlinear eigenvalue problems

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LINEAR ALGEBRA AND ITS APPLICATIONS 2012年第10期436卷 3839-3863页

作者： Beyn, Wolf-Juergen Univ Bielefeld Dept Math D-33501 Bielefeld Germany

We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least k column vectors, where k is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension. We also give an extension of the method to the case where k is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour. (C) 2011 Elsevier Inc. All rights reserved.

关键词： nonlinear eigenvalue problems Numerical methods Contour integrals

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AN EXPLICIT FORMULA FOR THE SPLITTING OF MULTIPLE eigenvalueS FOR nonlinear eigenvalue problems AND CONNECTIONS WITH THE LINEARIZATION FOR THE DELAY eigenvalue PROBLEM

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2017年第2期38卷 599-620页

作者： Michiels, Wim Boussaada, Islam Niculescu, Silviu-Iulian Katholieke Univ Leuven Dept Comp Sci B-3001 Heverlee Belgium Univ Paris Sud IPSA Lab Signanx & Syst F-91192 Gif Sur Yvette France Univ Paris Sud CNRS Cent Supelec F-91192 Gif Sur Yvette France

We contribute to the perturbation theory of nonlinear eigenvalue problems in three ways. First, we extend the formula for the sensitivity of a simple eigenvalue with respect to a variation of a parameter to the case of multiple nonsemisimple eigenvalues, thereby providing an explicit expression for the leading coefficients of the Puiseux series of the emanating branches of eigenvalues. Second, for a broad class of delay eigenvalue problems, the connection between the finitedimensional nonlinear eigenvalue problem and an associated infinite-dimensional linear eigenvalue problem is emphasized in the developed perturbation theory. Finally, in contrast to existing work on analyzing multiple eigenvalues of delay systems, we develop all theory in a matrix framework, i.e., without reduction of a problem to the analysis of a scalar characteristic quasi-polynomial.

关键词： nonlinear eigenvalue problems systems of functional equations and inequalities perturbations of nonlinear operators asymptotic distribution of eigenvalues and eigenfunctions matrix and operator equations

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