arXiv subjects

Yidu Yang

Publications and source records attributed to Yidu Yang.

At least 19 recordsLinked to original sources

arXiv2026

A high order stabilization-free virtual element method for general second-order elliptic eigenvalue problem

In this paper, we discuss a novel higher-order stabilization-free virtual element method for general second-order elliptic eigenvalue problems. Optimal a priori error estimates are derived for both the approximate eigenspace and eigenvalues. Numerical experiments are conducted on regular convex polygonal meshes, convex-concave polygonal meshes, and concave polygonal meshes. The numerical results validate the effectiveness of the proposed method.

math.NA↗

arXiv2026

Adaptive FEM with optimal convergence rate for non-self-adjoint eigenvalue problems

In this paper, we first discuss the optimal convergence of the adaptive finite element methods for non-self-adjoint eigenvalue problems. We present new theoretical error estimators and computable error estimators for multiple and clustered eigenvalues with the help of the error estimators of finite element solutions for the corresponding source problems, and prove the equivalence between these two estimators. We propose an adaptive algorithm for the eigenvalue cluster and demonstrate that it achieves the optimal convergence rate.We also provide numerical experiments to support our theoretical findings.

math.NA↗

arXiv2025

The discontinuous Galerkin method for the Oseen eigenvalue problem

In this paper, we focus on investigating symmetric and nonsymmetric discontinuous Galerkin (DG) methods for solving the Oseen eigenvalue problem based on the velocity-pressure formulation in $\mathbb{R}^{d}(d=2,3)$. We derive the a priori and a posteriori error estimates for the approximate eigenpairs for each method. We establish an adjoint-consistent symmetric DG method and derive optimal a priori error estimates, and prove the reliability and effectiveness of the error estimators for approximate eigenfunctions, as well as the reliability of the estimator for approximate eigenvalues. Numerical experiments confirm our theoretical analysis and demonstrate that the symmetric DG method achieves the optimal order of convergence, and that the nonsymmetric DG methods produce fewer spurious eigenvalues than the symmetric DG method for a fixed small penalty parameter $\gamma$.

math.NA↗

arXiv2022

The a posteriori error estimates and an adaptive algorithm of the FEM for transmission eigenvalues for anisotropic media

The transmission eigenvalue problem arising from the inverse scattering theory is of great importance in the theory of qualitative methods and in the practical applications. In this paper, we study the transmission eigenvalue problem for anisotropic inhomogeneous media in $Ω\subset \mathbb{R}^d$,(d=2,3). Using the T-coercivity and the spectral approximation theory, we derive an a posteriori estimator of residual type and prove its effectiveness and reliability for eigenfunctions. In addition, we also prove the reliability of the estimator for transmission eigenvalues. The numerical experiments indicate our method is efficient and can reach the optimal order $DoF^{-2m/d}$ by using piecewise polynomials of degree $m$ for real eigenvalues.

math.NA↗

arXiv2021

The nonconforming Crouzeix-Raviart element approximation and two-grid discretizations for the elastic eigenvalue problem

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 problem are independent of the Lame constant, which means the nonconforming Crouzeix-Raviart element approximations are locking-free. We also establish two kinds of two-grid discretization schemes for the elastic eigenvalue problem and analyze that when the mesh sizes of the coarse grid and fine grid satisfy some relationship, the resulting solutions can achieve optimal accuracy. Numerical examples are provided to show the efficiency of two-grid schemes for the elastic eigenvalue problem.

math.NA↗

arXiv2019

Lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients

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 the eigenfunctions are singular or smooth and whether the eigenvalues are large enough or not. Further, we prove that the corrected eigenvalues still maintain the same convergence order as that of uncorrected eigenvalues. Finally, numerical experiments validate our theoretical results.

math.NA↗

arXiv2018

Non-conforming Crouzeix-Raviar element approximation for Stekloff eigenvalues in inverse scattering

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 use the standard duality techniques to prove an extension of Strang lemma. And we prove the convergence and error estimate of discrete eigenvalues and eigenfunctions using the spectral perturbation theory for compact operators. Finally, we present some numerical examples not only on uniform meshes but also in an adaptive refined meshes to show that the Crouzeix-Raviart method is efficient for computing real and complex eigenvalues as expected.

math.NA↗

arXiv2018

A multigrid correction scheme for a new Steklov eigenvalue problem in inverse scattering

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 the coefficient matrices associated with those linear systems are constructed to be symmetric and positive definite. We prove error estimates of eigenvalues and eigenfunctions. Numerical results coincide in theoretical analysis and indicate our scheme is highly efficient in solving the eigenvalue problem.

math.NA↗

arXiv2018

Two-grid discretizations and a local finite element scheme for a non-selfadjoint Stekloff eigenvalue problem

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 case that the local refined region contains singular points lying on the boundary of domain. Numerical experiments are reported finally to show the efficiency of our schemes.

math.NA↗

arXiv2017

An adaptive C0IPG method for the Helmholtz transmission eigenvalue problem

The interior penalty methods using $C^0$ Lagrange elements ($C^0$IPG) developed in the last decade for the fourth order problems are an interesting topic in academia at present. In this paper, we discuss the adaptive fashion of $C^0$IPG method for the Helmholtz transmission eigenvalue problem.We give the a posteriori error indicators for primal and dual eigenfunctions, and prove their reliability and efficiency. We also give the a posteriori error indicator for eigenvalues and design a $C^0$IPG adaptive algorithm. Numerical experiments show that this algorithm is efficient and can get the optimal convergence rate.

math.NA↗

arXiv2017

New error estimates of linear triangle finite elements for the Steklov eigenvalue problem

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 prove a new and optimal error estimate in $\|\cdot\|_{0,\partial\Omega}$ for the eigenfunction of linear conforming finite element and the nonconforming Crouzeix-Raviart element, which is an improvement of the current results. Finally, we present some numerical experiments to support the theoretical analysis.

math.NA↗

arXiv2016

A mixed-element two-grid discretization for Helmholtz transmission eigenvalues

The Helmholtz transmission eigenvalue problem has received much concern in materials science, so it's significant to explore the efficient calculational method of the problem to mathematics and mechanics community. In this paper, based on a variational formulation proposed by Cakon, Monk and Sun, we introduce a mixed-element two-grid discretization and prove error estimates for this method theoretically. Some numerical results are presented to confirm the theoretical analysis and show that the method here is efficient.

math.NA↗

arXiv2016

A New Multigrid Finite Element Method for the Transmission Eigenvalue Problems

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 problems on the finer and finer meshes. Theoretical analysis and numerical results show that our method is simple and easy to implement and is efficient for computing real and complex transmission eigenvalues.

math.NA↗

arXiv2016

An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem

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 of different adaptive algorithms are provided, which illustrate the efficiency of the adaptive algorithm proposed here.

math.NA↗

arXiv2016

Non-conforming finite element methods for transmission eigenvalue problem

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 approximation, and prove the error estimates of the discrete eigenvalues obtained by the Adini element, Morley-Zienkiewicz element, modified-Zienkiewicz element et. al. And we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem.

math.NA↗

arXiv2015

An $H^m$-conforming spectral element method on multi-dimensional domain and its application to transmission eigenvalues

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 on multi-dimensional rectangles consist of the tensorial product of the basis functions on the interval $[-1,1]$. Then we construct the spectral element interpolation operator and prove the associated interpolation error estimates. Finally we apply the $H^2$-conforming spectral element method to the Helmholtz transmission eigenvalues that is a hot topic in the field engineering and mathematics.

math.NA↗

arXiv2015

The lower bound property of the Morley element eigenvalues

In this paper, we prove that the Morley element eigenvalues approximate the exact ones from below on regular meshes, including adaptive local refined meshes, for the fourth-order elliptic eigenvalue problems with the clamped boundary condition in any dimension. And we implement the adaptive computation to obtain lower bounds of the Morley element eigenvalues for the vibration problem of clamped plate under tension.

math.NA↗

arXiv2015

Error estimates and a two grid scheme for approximating transmission eigenvalues

In this paper, using the linearization technique we write the Helmholtz transmission eigenvalue problem as an equivalent nonselfadjoint linear eigenvalue problem whose left-hand side term is a selfadjoint, continuous and coercive sesquilinear form. To solve the resulting nonselfadjoint eigenvalue problem, we give an $H^{2}$ conforming finite element discretization and establish a two grid discretization scheme. We present a complete error analysis for both discretization schemes, and theoretical analysis and numerical experiments show that the methods presented in this paper can efficiently compute real and complex transmission eigenvalues.

math.NA↗