Superconvergence and a posteriori error estimates for the Stokes eigenvalue problems

In this paper we consider the finite element approximation of the Stokes eigenvalue problems based on projection method, and derive some superconvergence results and the related recovery type a posteriori error estimators. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares strategy. The results are based on some regularity assumptions for the Stokes equations, and are applicable to the finite element approximations of the Stokes eigenvalue problems with general quasi-regular partitions. Numerical results are presented to verify the superconvergence results and the efficiency of the recovery type a posteriori error estimators.

[1]  Lars B. Wahlbin,et al.  Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part II: The piecewise linear case , 2004, Math. Comput..

[2]  Aihui Zhou,et al.  Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates , 2006, Adv. Comput. Math..

[3]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[4]  Hehu Xie,et al.  Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods , 2009 .

[5]  Rolf Rannacher,et al.  A posteriori error control for finite element approximations of elliptic eigenvalue problems , 2001, Adv. Comput. Math..

[6]  Hongsen Chen,et al.  An Interior Estimate of Superconvergence for Finite Element Solutions for Second-Order Elliptic Problems on Quasi-uniform Meshes by Local Projections , 2003, SIAM J. Numer. Anal..

[7]  Xiu Ye.  Superconvergence of nonconforming finite element method for the Stokes equations , 2002 .

[8]  O. C. Zienkiewicz,et al.  The superconvergent patch recovery (SPR) and adaptive finite element refinement , 1992 .

[9]  Rolf Stenberg,et al.  A posteriori estimates for the Stokes eigenvalue problem , 2009 .

[10]  Jianhua Pan,et al.  Global Superconvergence for the Bilinear-Constant Scheme for the Stokes Problem , 1997 .

[11]  Ningning Yan,et al.  Enhancing finite element approximation for eigenvalue problems by projection method , 2012 .

[12]  Zhimin Zhang,et al.  Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes? , 2007, SIAM J. Numer. Anal..

[13]  Wei Chen,et al.  Approximation of an Eigenvalue Problem Associated with the Stokes Problem by the Stream Function-Vorticity-Pressure Method , 2006 .

[14]  Ricardo G. Durán,et al.  A Posteriori Error Estimates for the Finite Element Approximation of Eigenvalue Problems , 2003 .

[15]  Ricardo H. Nochetto,et al.  Small data oscillation implies the saturation assumption , 2002, Numerische Mathematik.

[16]  Mats G. Larson,et al.  A Posteriori and a Priori Error Analysis for Finite Element Approximations of Self-Adjoint Elliptic Eigenvalue Problems , 2000, SIAM J. Numer. Anal..

[17]  C. Chien,et al.  Superconvergence of high order FEMs for eigenvalue problems with periodic boundary conditions , 2009 .

[18]  Junping Wang,et al.  Superconvergence of Finite Element Approximations for the Stokes Problem by Projection Methods , 2001, SIAM J. Numer. Anal..

[19]  Jinchao Xu,et al.  Numerische Mathematik Convergence and optimal complexity of adaptive finite element eigenvalue computations , 2022 .

[20]  Zhimin Zhang,et al.  Enhancing Eigenvalue Approximation by Gradient Recovery , 2006, SIAM J. Sci. Comput..

[21]  Jinchao Xu,et al.  Local and Parallel Finite Element Algorithms for Eigenvalue Problems , 2002 .

[22]  Stefano Giani,et al.  A Convergent Adaptive Method for Elliptic Eigenvalue Problems , 2009, SIAM J. Numer. Anal..

[23]  Ping Wang,et al.  On the Monotonicity of (k;g,h)-graphs , 2002 .

[24]  Zhimin Zhang,et al.  A Posteriori Error Estimates Based on the Polynomial Preserving Recovery , 2004, SIAM J. Numer. Anal..

[25]  Xue-Cheng Tai,et al.  Superconvergence for the Gradient of Finite Element Approximations by L2 Projections , 2002, SIAM J. Numer. Anal..

[26]  Juan Sun,et al.  Recovery type a posteriori estimates and superconvergence for nonconforming FEM of eigenvalue problems , 2009 .

[27]  Zhimin Zhang,et al.  A New Finite Element Gradient Recovery Method: Superconvergence Property , 2005, SIAM J. Sci. Comput..

[28]  Jinchao Xu,et al.  A two-grid discretization scheme for eigenvalue problems , 2001, Math. Comput..

[29]  Hehu Xie,et al.  Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems , 2009 .

[30]  Haijun Wu,et al.  Enhancing eigenvalue approximation by gradient recovery on adaptive meshes , 2009 .

[31]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[32]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[33]  B. Mercier,et al.  Eigenvalue approximation by mixed and hybrid methods , 1981 .

[34]  I. Babuska,et al.  Finite element-galerkin approximation of the eigenvalues and Eigenvectors of selfadjoint problems , 1989 .

[35]  Daniele Boffi,et al.  Finite element approximation of eigenvalue problems , 2010, Acta Numerica.

[36]  Gabriel Wittum,et al.  Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: A smooth problem and globally quasi-uniform meshes , 2001, Math. Comput..