The multilevel mixed finite element discretizations based on local defect-correction for the Stokes eigenvalue problem

Based on the work of Xu and Zhou (2000), we establish new three-level and multilevel finite element discretizations by local defect correction techniques. Theoretical analysis and numerical experiments show that the discretizations are simple and easy to implement, and can be used to solve nonsymmetric eigenvalue problems with non smooth eigenfunctions efficiently. We also discuss the local error estimates of finite element approximations; it is a new feature here that the estimates apply to the local domains containing corner points.

[1]  Yidu Yang,et al.  Generalized Rayleigh quotient and finite element two-grid discretization schemes , 2009 .

[2]  Carsten Carstensen,et al.  An adaptive homotopy approach for non-selfadjoint eigenvalue problems , 2011, Numerische Mathematik.

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

[4]  P. G. Ciarlet,et al.  Basic error estimates for elliptic problems , 1991 .

[5]  K. Kolman,et al.  A Two-Level Method for Nonsymmetric Eigenvalue Problems , 2005 .

[6]  Jinchao Xu.  Two-grid Discretization Techniques for Linear and Nonlinear PDEs , 1996 .

[7]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[8]  Tao Lü,et al.  Splitting extrapolation based on domain decomposition for finite element approximations , 1997 .

[9]  Hai Bi,et al.  A note on the residual type a posteriori error estimates for finite element eigenpairs of nonsymmetric elliptic eigenvalue problems , 2014 .

[10]  Carsten Carstensen,et al.  A posteriori error estimators for convection--diffusion eigenvalue problems , 2014 .

[11]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[12]  Hai Bi,et al.  The extrapolation of numerical eigenvalues by finite elements for differential operators , 2013 .

[13]  Jinchao Xu,et al.  Local and parallel finite element algorithms for the stokes problem , 2008, Numerische Mathematik.

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

[15]  Hao Li,et al.  Local and Parallel Finite Element Discretizations for Eigenvalue Problems , 2013, SIAM J. Sci. Comput..

[16]  L. Wahlbin,et al.  Local behavior in finite element methods , 1991 .

[17]  Jiayu Han,et al.  The multilevel mixed finite element discretizations based on local defect-correction for the Stokes eigenvalue problem , 2015 .

[18]  Zhimin Zhang,et al.  Function Value Recovery and Its Application in Eigenvalue Problems , 2012, SIAM J. Numer. Anal..

[19]  Roland Glowinski,et al.  An introduction to the mathematical theory of finite elements , 1976 .

[20]  Yinnian He,et al.  Newton Iterative Parallel Finite Element Algorithm for the Steady Navier-Stokes Equations , 2010, J. Sci. Comput..

[21]  Jinchao Xu,et al.  Local and parallel finite element algorithms based on two-grid discretizations , 2000, Math. Comput..

[22]  J. Bramble,et al.  Rate of convergence estimates for nonselfadjoint eigenvalue approximations , 1973 .

[23]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[24]  Jiayu Han,et al.  A Class of Spectral Element Methods and Its A Priori/A Posteriori Error Estimates for 2nd-Order Elliptic Eigenvalue Problems , 2013 .

[25]  Aihui Zhou,et al.  Three-Scale Finite Element Discretizations for Quantum Eigenvalue Problems , 2007, SIAM J. Numer. Anal..