In this paper, we investigate a fourth-order three-point nonlinear eigenvalue problem. By the Krasnosel’skii’s fixed-point theorem in cone, some new results on the existence of positive solutions are obtained.
Steklov eigenvalue problem of Helmholtz equation is considered in the present paper. Steklov eigenvalue problem is reduced to a new variational formula on the boundary of a given domain, in which the self-adjoint prop...Steklov eigenvalue problem of Helmholtz equation is considered in the present paper. Steklov eigenvalue problem is reduced to a new variational formula on the boundary of a given domain, in which the self-adjoint property of the original differential operator is kept and the calculating of hyper-singular integral is avoided. A numerical example showing the efficiency of this method and an optimal error estimate are given.展开更多
We propose a numerical method for a non-selfadjoint Steklov eigenvalue problem of the Helmholtz equation.The problem is formulated using boundary integrals.The Nyström method is employed to discretize the integra...We propose a numerical method for a non-selfadjoint Steklov eigenvalue problem of the Helmholtz equation.The problem is formulated using boundary integrals.The Nyström method is employed to discretize the integral operators,which leads to a non-Hermitian generalized matrix eigenvalue problems.The spectral indicator method(SIM)is then applied to calculate the(complex)eigenvalues.The convergence is proved using the spectral approximation theory for(non-selfadjoint)compact operators.Numerical examples are presented for validation.展开更多
Based on the work of Xu and Zhou(2000),this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems,and proves a local a priori error estimate and a new local a pos...Based on the work of Xu and Zhou(2000),this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems,and proves a local a priori error estimate and a new local a posteriori error estimate in ||·||1,Ω0 norm for conforming elements eigenfunction,which has not been studied in existing literatures.展开更多
This paper is concerned with the finite elements approximation for the Steklov eigen- value problem on concave polygonal domain. We make full use of the regularity estimate and the characteristic of edge average inter...This paper is concerned with the finite elements approximation for the Steklov eigen- value problem on concave polygonal domain. We make full use of the regularity estimate and the characteristic of edge average interpolation operator of nonconforming Crouzeix- Raviart element, and prove a new and optimal error estimate in ｜｜ ｜｜o,δΩ for the eigenfunc- tion of linear conforming finite element and the nonconforming Crouzeix-Raviart element. Finally, we present some numerical results to support the theoretical analysis.展开更多
In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenp...In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine meshes and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.展开更多
In this paper, we study a class of fourth-order Neumann boundary value problem (NBVP for short). By virtue of fixed point index and the spectral theory of linear operators, the existence of positive solutions is obtai...In this paper, we study a class of fourth-order Neumann boundary value problem (NBVP for short). By virtue of fixed point index and the spectral theory of linear operators, the existence of positive solutions is obtained under the assumption that the nonlinearity satisfies sublinear or superlinear conditions, which are relevant to the first eigenvalue of the corresponding linear operator.展开更多
We establish the existence and multiplicity of solutions for Steklov problems under non- resonance or resonance conditions using variational methods. In our main theorems, we consider a weighted eigenvalue problem of ...We establish the existence and multiplicity of solutions for Steklov problems under non- resonance or resonance conditions using variational methods. In our main theorems, we consider a weighted eigenvalue problem of Steklov type.展开更多
基金the National Natural Science Foundation of China(11261053)The Natural Science Foundation of Gansu Province(1308RJZA125)
文摘In this paper, we investigate a fourth-order three-point nonlinear eigenvalue problem. By the Krasnosel’skii’s fixed-point theorem in cone, some new results on the existence of positive solutions are obtained.
文摘Steklov eigenvalue problem of Helmholtz equation is considered in the present paper. Steklov eigenvalue problem is reduced to a new variational formula on the boundary of a given domain, in which the self-adjoint property of the original differential operator is kept and the calculating of hyper-singular integral is avoided. A numerical example showing the efficiency of this method and an optimal error estimate are given.
文摘We propose a numerical method for a non-selfadjoint Steklov eigenvalue problem of the Helmholtz equation.The problem is formulated using boundary integrals.The Nyström method is employed to discretize the integral operators,which leads to a non-Hermitian generalized matrix eigenvalue problems.The spectral indicator method(SIM)is then applied to calculate the(complex)eigenvalues.The convergence is proved using the spectral approximation theory for(non-selfadjoint)compact operators.Numerical examples are presented for validation.
基金supported by National Natural Science Foundation of China(Grant Nos.11201093 and 11161012)
文摘Based on the work of Xu and Zhou(2000),this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems,and proves a local a priori error estimate and a new local a posteriori error estimate in ||·||1,Ω0 norm for conforming elements eigenfunction,which has not been studied in existing literatures.
文摘This paper is concerned with the finite elements approximation for the Steklov eigen- value problem on concave polygonal domain. We make full use of the regularity estimate and the characteristic of edge average interpolation operator of nonconforming Crouzeix- Raviart element, and prove a new and optimal error estimate in ｜｜ ｜｜o,δΩ for the eigenfunc- tion of linear conforming finite element and the nonconforming Crouzeix-Raviart element. Finally, we present some numerical results to support the theoretical analysis.
基金supported by National Natural Science Foundation of China(Grant Nos.11801021 and 11571027)Foundation for Fundamental Research of Beijing University of Technology(Grant No.006000546318504)International Research Cooperation Seed Fund of Beijing University of Technology(Grant No.2018B32)。
文摘In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine meshes and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.
基金Supported by NNSF of China (No.60665001) Educational Department of Jiangxi Province(No.GJJ08358+1 种基金 No.GJJ08359 No.JXJG07436)
文摘In this paper, we study a class of fourth-order Neumann boundary value problem (NBVP for short). By virtue of fixed point index and the spectral theory of linear operators, the existence of positive solutions is obtained under the assumption that the nonlinearity satisfies sublinear or superlinear conditions, which are relevant to the first eigenvalue of the corresponding linear operator.
基金Supported by Scientific Research Foundation of Hubei Provincial Department of Education(D20184301)the Hubei Key Laboratory of Applied Mathematics(Hubei University).
文摘We establish the existence and multiplicity of solutions for Steklov problems under non- resonance or resonance conditions using variational methods. In our main theorems, we consider a weighted eigenvalue problem of Steklov type.
