Yidu Yang

Guizhou Normal University

In this paper, we adopt the discontinuous Galerkin finite element method and the enriched Crouzeix‐Raviart finite element method to study the magnetohydrodynamic (MHD) Stokes eigenvalue problem describing the flow of a viscous and electrically conducting fluid in a duct under the influence of a uniform magnetic field. We give the convergence and er...

The elastic transmission eigenvalue problem, arising from the inverse scattering theory, plays a critical role in the qualitative reconstruction methods for elastic media. In this paper, we discuss the finite element method for solving the elastic transmission eigenvalue problem with different elastic tensors and different mass densities. The probl...

In this paper, we study the discontinuous Galerkin finite element method for the Steklov eigenvalue problem arising in inverse scattering. We present a complete error estimates including the a refined priori error estimate and the a posteriori error estimate, and prove the reliability and efficiency of the a posteriori error estimators for eigenfun...

We discuss the a posteriori error estimates of the H\begin{document}$ ^{2} $\end{document}-conforming finite element for the elastic transmission eigenvalue problem, which is of fourth order and non-selfadjoint, with vector-valued eigenfunctions. We first introduce the error indicators for primal eigenfunction, dual eigenfunction and eigenvalue, re...

In this paper, we study the nonconforming Crouzeix-Raviart (CR) element approximation for the fluid-solid vibration problem. We present the a priori error estimate and the a posteriori error estimate, and prove the reliability and efficiency of the a posteriori error indicator up to higher order terms. Moreover, we carry out the numerical experimen...

In this paper, we extend the work of Brenner and Sung [Math. Comp. 59, 321--338 (1992)] and present a regularity estimate for the elastic equations in concave domains. Based on the regularity estimate we prove that the constants in the error estimates of the nonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue...

In this paper, we use the non-conforming Crouzeix-Raviart element method to solve a Stekloff eigenvalue problem arising in inverse scattering. The weak formulation corresponding to this problem is non-self-adjoint and indefinite, and its Crouzeix-Raviart element discretization does not meet the condition of the Strang lemma. We use the standard dua...

The elastic transmission eigenvalue problem is quadratic in the eigenvalue parameter, nonselfadjoint, and of fourth order. In this paper, we apply the Ciarlet–Raviart mixed method to this problem, give a mixed variational form, and establish two mixed methods using the classical Lagrange finite element and the spectral element, respectively. We ded...

In this study, for the first time, we discuss the posteriori error estimates and adaptive algorithm for the non-self-adjoint Steklov eigenvalue problem in inverse scattering. The differential operator corresponding to this problem is non-self-adjoint and the associated weak formulation is not H 1-elliptic. Based on the study of Armentano et al. [Ap...

In this paper, using new correction to the Crouzeix-Raviart finite element eigenvalue approximations, we obtain lower eigenvalue bounds for the Steklov eigenvalue problem with variable coefficients on d-dimensional domains (d = 2,3). In addition, we prove that the corrected eigenvalues asymptotically converge to the exact ones from below whether th...

In this paper, for the second‐order elliptic and Stokes eigenvalue problems with variable coefficients, we propose a correction method to nonconforming eigenvalue approximations and prove that the corrected eigenvalues converge to the exact ones asymptotically from below. In particular, the asymptotic lower bound property of corrected eigenvalues i...

We propose a multigrid correction scheme to solve a new Steklov eigenvalue problem in inverse scattering. With this scheme, solving an eigenvalue problem in a fine finite element space is reduced to solve a series of boundary value problems in fine finite element spaces and a series of eigenvalue problems in the coarsest finite element space. And t...

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correc...

Based on the work of Xu and Zhou (Math Comp 69:881–909, 2000), we establish local and parallel algorithms for the Helmholtz transmission eigenvalue problem. For the \(H^2\)-conforming finite element and the spectral element approximations, we prove the local error estimates and the efficiency of local and parallel algorithms. Numerical experiments...

In this paper, we establish a new local and parallel finite element discrete scheme based on the shifted‐inverse power method for solving the biharmonic eigenvalue problem of plate vibration. We prove the local error estimation of finite element solution for the biharmonic equation/eigenvalue problem and prove the error estimation of approximate so...

In this paper, for biharmonic eigenvalue problems with clamped boundary condition in Rⁿ which include plate vibration problem and plate buckling problem, we primarily study the two-grid discretization based on the shifted-inverse iteration of Ciarlet–Raviart mixed method. With our scheme, the solution of a biharmonic eigenvalue problem on a fine me...

In this paper, for a new Stekloff eigenvalue problem which is non-selfadjoint and not H¹-elliptic, we establish and analyze two kinds of two-grid discretization scheme and a local finite element scheme. We present the error estimates of approximations of two-grid discretizations. We also prove a local error estimate which is suitable for the case t...

In this paper, we use the non-conforming Crouzeix-Raviart element method to solve a Stekloff eigenvalue problem arising in inverse scattering. The weak formulation corresponding to this problem is non-selfadjoint and does not satisfy $H^{1}$-elliptic condition, and its Crouzeix-Raviart element discretization does not meet the Strang lemma condition...

We propose a multigrid correction scheme to solve a new Steklov eigenvalue problem in inverse scattering. With this scheme, solving an eigenvalue problem in a fine finite element space is reduced to solve a series of boundary value problems in fine finite element spaces and a series of eigenvalue problems in the coarsest finite element space. And t...

In this paper, for a new Stekloff eigenvalue problem which is non-selfadjoint and not $H^1$-elliptic, we establish and analyze two kinds of two-grid discretization scheme and a local finite element scheme. We present the error estimates of approximations of two-grid discretizations. We also prove a local error estimate which is suitable for the cas...

The interior penalty methods using Lagrange elements ( IPG) developed in the last decade for the fourth order problems are an interesting topic in academia at present. In this paper, we apply the methods to the Helmholtz transmission eigenvalue problem which is quadratic and non self-adjoint and has important applications in the inverse scattering...

In this paper we make a further discussion on the finite elements approximation for the Steklov eigenvalue 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, which is different from the existing proof argument, and...

The classical weak formulation of the Helmholtz transmission eigenvalue problem can be linearized into an equivalent nonsymmetric eigenvalue problem. Based on this nonsymmetric eigenvalue problem, we first discuss the a posteriori error estimates and adaptive algorithm of conforming finite elements for the Helmholtz transmission eigenvalue problem....

Numerical methods for the transmission eigenvalue problems are hot topics in recent years. Based on the work of Lin and Xie [Math. Comp., 84(2015), pp. 71-88], we build a multigrid method to solve the problems. With our method, we only need to solve a series of primal and dual eigenvalue problems on a coarse mesh and the associated boundary value p...

This paper proposes and analyzes an a posteriori error estimator for the finite element multi-scale discretization approximation of the Steklov eigenvalue problem. Based on the a posteriori error estimates, an adaptive algorithm of shifted inverse iteration type is designed. Finally, numerical experiments comparing the performances of three kinds o...

The transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory. In this paper, for the Helmholtz transmission eigenvalue problem, we give a weak formulation which is a nonselfadjoint linear eigenvalue problem. Based on the weak formulation, we first discuss the non-conforming finite element approx...

The Ciarlet-Raviart mixed finite element method is popular for the biharmonic equations. In this paper, we apply this method to the Helmholtz transmission eigenvalue problem which is quadratic and non-self-adjoint, give a mixed variational form and a mixed method using the Lagrange elements, and prove the a priori error estimates of the discrete ei...

In this paper we develop an $H^m$-conforming ($m\ge1$) spectral element method on multi-dimensional domain associated with the partition into multi-dimensional rectangles. We construct a set of basis functions on the interval $[-1,1]$ that is made up of the generalized Jacobi polynomials (GJPs) and the nodal basis functions. So the basis functions...

In this paper, for the Helmholtz transmission eigenvalue problem, we propose a new equivalent weak formulation which is actually a nonselfadjoint linear eigenvalue problem. Based on this weak formulation we give a finite element discretization with good algebraic structure and prove the error estimates of the numerical eigenvalues. We also apply th...

Based on the work of Xu and Zhou [Math Comput 69 (2000) 881–909], we propose and analyze in this article local and parallel finite element algorithms for the Steklov eigenvalue problem. We also prove a local error estimate which is suitable for the case that the locally refined region contains singular points lying on the boundary of domain, which...

Based on the work of Xu and Zhou (2000), this paper combines mixed finite element method and the local defect-correction technique to establish new local and parallel multilevel discretization schemes for the Stokes eigenvalue problem. Theoretical analysis and numerical experiments show that the computational approach proposed in this paper is simp...

We propose a multilevel correction method for the convection-diffusion eigenvalue problems which is suitable for not only simple but also multiple eigenvalues. And we prove that the accuracy of resulting eigenpair approximations can be improved after each correction step. The scheme is easy to realize with Matlab, and numerical results are satisfac...

In this article, we combine mixed finite element method, multiscale discretization, and Rayleigh quotient iteration to propose a new adaptive algorithm based on residual type a posterior error estimates for the Stokes eigenvalue problem. Both reliability and efficiency of the error indicator are proved. The efficiency of the algorithm is also inves...

The shifted-inverse iteration based on the multigrid discretizations developed in recent years is an efficient computation method for eigenvalue problems. In this paper, for general self-adjoint eigenvalue problems, including the Maxwell eigenvalue problem and integral operator eigenvalue problem, we establish the inverse iteration with fixed shift...

Based on the work of Xu and Zhou [Math.Comput., 69(2000), pp.881-909], we establish new three-level and multilevel finite element discretizations by local defect-correction technique. Theoretical analysis and numerical experiments show that the schemes are simple and easy to carry out, and can be used to solve singular nonsymmetric eigenvalue probl...

In this paper we study the residual type a posteriori error estimates for general elliptic (not necessarily symmetric) eigenvalue problems. We present estimates for approximations of semisimple eigenvalues and associated eigenvectors. In particular, we obtain the following new results: 1) An error representation formula which we use to reduce the a...

We establish Crouzeix-Raviart element adaptive algorithm based on Rayleigh quotient iteration and give its a priori/a posteriori error estimates. Our algorithm is performed under the package of Chen, and satisfactory numerical results are obtained.

In this paper we first discover and prove that on adaptive meshes the eigenvalues by the Crouzeix–Raviart element approximate the exact ones from below when the corresponding eigenfunctions are singular. In addition, we use conforming finite elements to do the interpolation postprocessing to get the upper bound of the eigenvalues. Using the upper a...

This paper discusses the extrapolation of numerical eigenvalues by finite elements for differential operators and obtains the following new results: (a) By extending a theorem of eigenvalue error estimate, which was established by Osborn, a new expansion of eigenvalue error is obtained. Many achievements, which are about the asymptotic expansions o...

This paper discusses conforming mixed finite element approximations for the Stokes eigenvalue problem. Firstly, several mixed finite element identities are proved. Based on these identities, the following new results are given: (1) It is proved that the numerical eigenvalues obtained by mini-element, P 1-P 1 element and Q 1-Q 1 element approximate...

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 \(\left\| \cdot \right\|_{1,\Omega _0 }\) norm for conforming elements eigenfunction, which has not been...

In this paper, adaptive finite element methods for differential operator eigenvalue problems are discussed. For multi-scale discretization schemes based on Rayleigh quotient iteration (see Scheme 3 in [Y. Yang, H. Bi, A two-grid discretization scheme based on shifted-inverse power method, SIAM J. Numer. Anal. 49 (2011) 1602–1624]), a reliable and e...

This paper discusses spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic eigenvalue problems. The special work of this paper is as follows. (1) We prove a priori and a posteriori error estimates for spectral and spectral element methods. (2) We compare between spectral methods, spectral eleme...

This paper discusses spectral method with the tensor-product nodal basis at the Legendre-Gauss-Lobatto points for solving the Steklov eigenvalue problem. A priori error estimates of spectral method are discussed, and based on the work of Melenk and Wohlmuth (2001), a posterior error estimator of the residual type is given and analyzed. In addition,...

Based on the work of Xu and Zhou [Math. Comp., 69 (2000), pp. 881-909], this paper combines the local defect-correction technique and the shifted-inverse power method to establish new local and parallel finite element three-scale schemes for a class of eigenvalue problems. It is proved that with these schemes, the solution of an eigenvalue problem...

This paper studies highly accurate algorithm for membrane vibration problem. Based on the theory of theinterpolation post-processing, this paper establishes a scheme of bi-quadratic interpolation post-processing of bilinearrectangular element on a locally refined mesh with hanging nodes. And the resulting solution is verysatisfactory.

This paper discusses highly efficient discretization schemes for the Maxwell eigenvalue problem. Two kinds of 2-grid discretization schemes (see Math Comp 70(2001)pp17-25, SIAM J Numer Anal 49(4)pp.1602-1624) are applied to the finite element filtration method for the Maxwell eigenvalue problem. MATLAB numerical experiments are presented to support...

In this paper, we explore efficient discretization schemes—two kinds of two-grid discretization schemes based on the parameterized approach for the Maxwell eigenvalue problem. We give numerical experiments by using Matlab to write program, and show that this algorithm is highly effective.

This paper discusses finite-element highly efficient calculation schemes for solving eigenvalue problem of electric field. Multigrid discretization is extended to the filter approach for eigenvalue problem of electric field. With this scheme one solves an eigenvalue problem on a coarse grid just at the first step, and then always solves a linear al...

This paper discusses efficient numerical methods for the Steklov eigenvalue problem and establishes a new multiscale discretization scheme and an adaptive algorithm based on the Rayleigh quotient iterative method. The efficiency of these schemes is analyzed theoretically, and the constants appeared in the error estimates are also analyzed elaborate...

Rannacher discovered by numerical results that the Morley element eigenvalues could approximate the exact eigenvalues from below. This discovery is very important in engineering and mechanics computing. This note provides a theoretical proof for Rannacher’s observations. KeywordsMorley element–Plate vibration problem–Eigenvalue approximation–Lower...

This paper discusses highly finite element algorithms for the eigenvalue problem of electric field. Combining the mixed finite element method with the Rayleigh quotient iteration method, a new multi-grid discretization scheme and an adaptive algorithm are proposed and applied to the eigenvalue problem of electric field. Theoretical analysis and num...

This paper discusses highly efficient discretization schemes for mixed variational formulation of eigenvalue problems. A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems. With this scheme, the solution of an eigenv...

This paper discusses a high efficient scheme for the Steklov eigenvalue problem. A two-grid discretization scheme of nonconforming Crouzeix–Raviart element is established. With this scheme, the solution of a Steklov eigenvalue problem on a fine grid πh is reduced to the solution of the eigenvalue problem on a much coarser grid πH and the solution o...

In the paper, a two-grid discretization scheme is discussed for the Steklov eigenvalue problem. With the scheme, the solution of the Steklov eigenvalue problem on a fine grid is reduced to the solution of the Steklov eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. Using spectral approximatio...

This paper discusses highly efficient discretization schemes for solving self-adjoint elliptic differential operator eigenvalue problems. Several new two-grid discretization schemes, including the conforming and nonconforming finite element schemes, are proposed by combining the finite element method with the shifted-inverse power method for matrix...

This paper characterizes the spectrum of a fourth-order Steklov eigenvalue problem by using the spectral theory of completely continuous operator. The conforming finite element approximation for this problem is analyzed, and the error estimate is given. Finally, the bounds for Steklov eigenvalues on the square domain are provided by Bogner-Fox-Schm...

This paper discusses the Wilson element approximation for the eigenvalue problem of Laplace operator on n-dimensional polygonal domain (n=2,3), and the main results are as follows: (1) We establish the relationship between the interpolation weak estimate of the Wilson element and the interpolation weak estimate of n-linear element. (2) We prove tha...

This is a survey article about using non-conforming finite elements in solving eigenvalue problems of elliptic operators, with emphasis on obtaining lower bounds. In addition, this article also contains some new materials for eigenvalue approximations of the Laplace operator, which include: 1) the proof of the fact that the non-conforming Crouzeix-...

This paper discusses the residual type a posteriori error estimate of EQ1rot element for second order elliptic boundary value problems. The a posterior error estimator is obtained by using the properties in nonconforming EQ1rot element space and the unified framework of Carstensen et al.(in SIAM. J.Numer.Anal.2007,45(1):68-82). The results of numer...

This paper extends the two-grid discretization scheme of the conforming finite elements proposed by J. C. Xu and A. H. Zhou [Math. Comput. 70, No. 233, 17–25 (2001; Zbl 0959.65119)] to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on the Rayleigh quotient technique are proposed....

This paper deals with nonconforming finite element approximations of the Steklov eigenvalue problem. For a class of nonconforming finite elements, it is shown that the j-th approximate eigenpair converges to the j-th exact eigenpair and error estimates for eigenvalues and eigenfunctions are derived. Furthermore, it is proved that the j-th eigenvalu...

This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfun...

We study the n-simplex nonconforming Crouzeix-Raviart element in approximating the n-dimensional second-order elliptic boundary value problems and the associated eigenvalue problems. By using the second Strang Lemma, optimal rate of convergence is established under the discrete energy norm. The error bound is also valid for the eigenfunction approx...

This paper discusses the order-preserving convergence for spectral approximation of the self-adjoint completely continuous operator T. Under the condition that the approximate operator T h converges to T in norm, it is proven that the k-th eigenvalue of T h converges to the k-th eigenvalue of T. (We sorted the positive eigenvalues in decreasing o...

We introduce a new extrapolation formula by combining Richardson extrapolation and Sloan iteration algorithms. Using this extrapolation formula, we obtain some asymptotic expansions of the Galerkin finite element method for semi-simple eigenvalue problems of Fredholm integral equations of the second kind and improve the accuracy of the numerical ap...

We consider the finite element approximation for the eigenvalue problem of the Laplace operator on a rectangular domain. We prove that the nonconforming Wilson element approximates eigenvalues from below, and thereby settle a long standing conjecture in the finite element method.

In this paper, we study numerical approximations of eigenvalues when using projection method for spectral approximations of completely con- tinuous operators. We improve the theory depending on the ascent of T ¡ " and provide a new approach for error estimate, which depends only on the ascent of Th ¡ "h. Applying this estimator to the integral oper...

Let (λ h ,σ h ,u h ) be the mixed finite element eigen-pair. Some error estimates for (λ h ,u h ) have been derived by Babuska and Osborn. This paper presents an abstract error estimate for σ h . The estimate is then applied to obtain new error estimates for two examples: Raviart-Thomas scheme for a second-order elliptic eigenvalue problem and Ciar...

In this paper, we discuss a posteriori error estimates of the eigenvalue lambda(h) given by Adini nonconforming finite element. We give an assymptotically exact error estimator of the lambda(h). We prove that the order of convergence of the lambda(h) is just 2 and the lambda(h) converge from below for sufficiently small h.

In this paper we introduce a Petrov-Galerkin approximation model to the solution of linear and semi-linear elliptic boundary value problems in which piecewise quadratic polynomial space and piecewise linear polynomial space are used as the shape function space and the test function space, respectively. We prove that the approximation order of the s...

This paper discusses computable bounds for the error between the linear finite element approximation eigenpair (λ h ,u h ) and the exact eigenpair (λ,u) of the second order selfadjoint elliptic eigenvalue problem Lu=λu, in G; u=0, on ∂G '' , and proves λ h -λ λ≤λ h MC 1 2 C 2 2 h 2 where M, C 1 and C 2 are constants satisfying the relations: ∫ G L...

For a second-order elliptic eigenvalue problem we prove that the Rayleigh quotient of the higher-order interpolation of a finite element eigenfunction has superconvergence.