A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L^2 projection method, the superconvergence results for the v...A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L^2 projection method, the superconvergence results for the velocity and pressure are obtained with a penalty parameter larger than that of the classical penalty scheme. The numerical experiments are carried out to confirm the theoretical results.展开更多
In this paper, the Wilson nonconforming finite element is considered for solving elliptic eigenvalue problems. Based on an interpolation postprocessing, superconvergence estimates of both eigenfunction and eigenvalue...In this paper, the Wilson nonconforming finite element is considered for solving elliptic eigenvalue problems. Based on an interpolation postprocessing, superconvergence estimates of both eigenfunction and eigenvalue are obtained.展开更多
This paper is a generalization of some recent results concerned with the lower bound property of eigenvalues produced by both the enriched rotated Q_(1) and Crouzeix-Raviart elements of the Stokes eigenvalue problem.T...This paper is a generalization of some recent results concerned with the lower bound property of eigenvalues produced by both the enriched rotated Q_(1) and Crouzeix-Raviart elements of the Stokes eigenvalue problem.The main ingredient are a novel and sharp L^(2) error estimate of discrete eigenfunctions,and a new error analysis of nonconforming finite element methods.展开更多
We formulate a coupled vibration between plate and acoustic field in mathematically rigorous fashion. It leads to a non-standard eigenvalue problem. A finite element approximation is considered in an abstract way, and...We formulate a coupled vibration between plate and acoustic field in mathematically rigorous fashion. It leads to a non-standard eigenvalue problem. A finite element approximation is considered in an abstract way, and the approximate eigenvalue problem is written in an operator form by means of some Ritz projections. The order of convergence is proved based on the result of Babugka and Osborn. Some numerical example is shown for the problem for which the exact analytical solutions are calculated. The results shows that the convergence order is consistent with the one by the numerical analysis.展开更多
In this paper,we analyze a nonconforming finite element method for the computation of transmission eigenvalues and the corresponding eigenfunctions.The error estimates of the eigenvalue and eigenfunction approximation...In this paper,we analyze a nonconforming finite element method for the computation of transmission eigenvalues and the corresponding eigenfunctions.The error estimates of the eigenvalue and eigenfunction approximation are given,respectively.Finally,some numerical examples are provided to validate the theoretical results.展开更多
Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper.These methods are based on the weakly continuous P1 vector fields and the locally divergence-free(...Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper.These methods are based on the weakly continuous P1 vector fields and the locally divergence-free(LDF)finite elements,which respectively penalize local divergence and are discontinuous across edges.These methods have no penalty factors and avoid solving the saddle-point problems.The existence and uniqueness of the velocity solution are proved,and the optimal error estimates of the energy norms and L^(2)-norms are obtained.Moreover,we propose unified pressure recovery algorithms and prove the optimal error estimates of L^(2)-norm for pressure.We design a unified iterative method for numerical experiments to verify the correctness of the theoretical analysis.展开更多
In this paper, we discuss a posteriori error estimates of the eigenvalue λ[sub h] given by Adini nonconforming finite element. We give an assymptotically exact error estimator of the λ[sub h]. We prove that the orde...In this paper, we discuss a posteriori error estimates of the eigenvalue λ[sub h] given by Adini nonconforming finite element. We give an assymptotically exact error estimator of the λ[sub h]. We prove that the order of convergence of the λ[sub h] is just 2 and the converge from below for sufficiently small h. [ABSTRACT FROM AUTHOR]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
This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou （Math. Comput., 70 （2001）, pp.17-25） to the nonconforming finite elements for eigenvalue problems. In...This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou （Math. Comput., 70 （2001）, pp.17-25） to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly.展开更多
This paper extends the unifying theory for a posteriori error analysis of the nonconformingfinite element methods to the second order elliptic eigenvalue problem.In particular,the authorproposes the a posteriori error...This paper extends the unifying theory for a posteriori error analysis of the nonconformingfinite element methods to the second order elliptic eigenvalue problem.In particular,the authorproposes the a posteriori error estimator for nonconforming methods of the eigenvalue problems andprove its reliability and efficiency based on two assumptions concerning both the weak continuity andthe weak orthogonality of the nonconforming finite element spaces,respectively.In addition,the authorexamines these two assumptions for those nonconforming methods checked in literature for the Laplace,Stokes,and the linear elasticity problems.展开更多
在结构健康监测(Structural Health Monitoring,SHM)技术中,基于Lamb波的损伤监测方法在板状结构中显示出了巨大的潜力。提出了一种基于近似非凸鲁棒主成分分析(Approximate Non-Convex Robust Principal Component Analysis,ANC-RPCA)...在结构健康监测(Structural Health Monitoring,SHM)技术中,基于Lamb波的损伤监测方法在板状结构中显示出了巨大的潜力。提出了一种基于近似非凸鲁棒主成分分析(Approximate Non-Convex Robust Principal Component Analysis,ANC-RPCA)的异常值分析方法。该算法对于高维测量信号,能够在降维条件下实现有效的损伤诊断。通过使用秩近似函数逼近矩阵的秩,采用非凸惩罚函数逼近?_(0)范数,非凸惩罚函数在一定条件下可以保证稀疏解的唯一性。随着数据矩阵规模的扩大,传统的RPCA采用核范数近似时,奇异值分解的计算复杂度也会上升。新的近似方法能在使计算效率更高的情况下,针对波场图像能够在更低秩的水平下保留有效信息,识别出异常值。将该算法运用到基于Lamb波的波场图像中,通过仿真和实验数据验证其有效性,使用非精确增广拉格朗日乘子(Inexact Augmented Lagrange Multiplier,IALM)法求解,并与目前使用较多的主流RPCA算法进行了效果对比。实验结果表明ANC-RPCA算法在异常值识别中具有良好的性能,相较于其他算法,在计算效率和低秩性等方面具有巨大的优势,证明了所提算法的可靠性和完整性。展开更多
基金supported by the National Natural Science Foundation of China (Nos. 10971203 and 11271340)the Research Fund for the Doctoral Program of Higher Education of China (No. 20094101110006)
文摘A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L^2 projection method, the superconvergence results for the velocity and pressure are obtained with a penalty parameter larger than that of the classical penalty scheme. The numerical experiments are carried out to confirm the theoretical results.
文摘In this paper, the Wilson nonconforming finite element is considered for solving elliptic eigenvalue problems. Based on an interpolation postprocessing, superconvergence estimates of both eigenfunction and eigenvalue are obtained.
基金The author would like to thank Prof.Shangyou Zhang for helping the numerical experiments.The author was supported by the NSFC under Grants Nos.11571023 and 11401015.
文摘This paper is a generalization of some recent results concerned with the lower bound property of eigenvalues produced by both the enriched rotated Q_(1) and Crouzeix-Raviart elements of the Stokes eigenvalue problem.The main ingredient are a novel and sharp L^(2) error estimate of discrete eigenfunctions,and a new error analysis of nonconforming finite element methods.
文摘We formulate a coupled vibration between plate and acoustic field in mathematically rigorous fashion. It leads to a non-standard eigenvalue problem. A finite element approximation is considered in an abstract way, and the approximate eigenvalue problem is written in an operator form by means of some Ritz projections. The order of convergence is proved based on the result of Babugka and Osborn. Some numerical example is shown for the problem for which the exact analytical solutions are calculated. The results shows that the convergence order is consistent with the one by the numerical analysis.
基金Xia Ji is supported by the National Natural Science Foundation of China(No.11271018,No.91230203)the Special Funds for National Basic Research Program of China(973 Program 2012CB025904 and 863 Program 2012AA01A309)+1 种基金the national Center for Mathematics and Interdisciplinary Science,CAS.Hehu Xie is supported in part by the National Natural Science Foundations of China(NSFC 91330202,11001259,11371026,11031006,2011CB309703)the national Center for Mathematics and Interdisciplinary Science,CAS,the President Foundation of AMSS-CAS。
文摘In this paper,we analyze a nonconforming finite element method for the computation of transmission eigenvalues and the corresponding eigenfunctions.The error estimates of the eigenvalue and eigenfunction approximation are given,respectively.Finally,some numerical examples are provided to validate the theoretical results.
基金National Nature Science Foundation of China(No.11971337,No.11801387)。
文摘Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper.These methods are based on the weakly continuous P1 vector fields and the locally divergence-free(LDF)finite elements,which respectively penalize local divergence and are discontinuous across edges.These methods have no penalty factors and avoid solving the saddle-point problems.The existence and uniqueness of the velocity solution are proved,and the optimal error estimates of the energy norms and L^(2)-norms are obtained.Moreover,we propose unified pressure recovery algorithms and prove the optimal error estimates of L^(2)-norm for pressure.We design a unified iterative method for numerical experiments to verify the correctness of the theoretical analysis.
文摘In this paper, we discuss a posteriori error estimates of the eigenvalue λ[sub h] given by Adini nonconforming finite element. We give an assymptotically exact error estimator of the λ[sub h]. We prove that the order of convergence of the λ[sub h] is just 2 and the converge from below for sufficiently small h. [ABSTRACT FROM AUTHOR]
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 (No. 10761003)by the Foundation of Guizhou Province Scientific Research for Senior Personnel, China
文摘This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou （Math. Comput., 70 （2001）, pp.17-25） to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly.
文摘This paper extends the unifying theory for a posteriori error analysis of the nonconformingfinite element methods to the second order elliptic eigenvalue problem.In particular,the authorproposes the a posteriori error estimator for nonconforming methods of the eigenvalue problems andprove its reliability and efficiency based on two assumptions concerning both the weak continuity andthe weak orthogonality of the nonconforming finite element spaces,respectively.In addition,the authorexamines these two assumptions for those nonconforming methods checked in literature for the Laplace,Stokes,and the linear elasticity problems.
