Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q1rot, EQ1rot and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to validate our theoretical results.

[1]  John E. Osborn,et al.  APPROXIMATION OF STEKLOV EIGENVALUES OF NON-SELFADJOINT SECOND ORDER ELLIPTIC OPERATORS , 1972 .

[2]  Dorin Bucur,et al.  Asymptotic analysis and scaling of friction parameters , 2006 .

[3]  Qun Lin,et al.  Extrapolation and superconvergence of the Steklov eigenvalue problem , 2010, Adv. Comput. Math..

[4]  Differential operators with spectral parameter incompletely in the boundary conditions , 1990 .

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

[6]  S. Bergman,et al.  Kernel Functions and Elliptic Differential Equations in Mathematical Physics , 2005 .

[7]  Jacob T. Schwartz,et al.  Linear operators. Part II. Spectral theory , 2003 .

[8]  H. Ahn.  Vibrations of a pendulum consisting of a bob suspended from a wire: the method of integral equations , 1981 .

[9]  Christopher Beattie,et al.  Methods for computing lower bounds to eigenvalues of self-adjoint operators , 1995 .

[10]  Shun Zhang,et al.  Discontinuous Galerkin Finite Element Methods for Interface Problems: A Priori and A Posteriori Error Estimations , 2011, SIAM J. Numer. Anal..

[11]  F. Goerisch,et al.  The convergence of a new method for calculating lower bounds to eigenvalues , 1986 .

[12]  Claudio Padra,et al.  A posteriori error estimates for the Steklov eigenvalue problem , 2008 .

[13]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[14]  Zhangxin Chen,et al.  On the implementation of mixed methods as nonconforming methods for second-order elliptic problems , 1995 .

[15]  Andrey B. Andreev,et al.  Isoparametric finite-element approximation of a Steklov eigenvalue problem , 2004 .

[16]  Yidu Yang,et al.  The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators , 2008 .

[17]  JinHuang,et al.  THE MECHANICAL QUADRATURE METHODS AND THEIR EXTRAPOLATION FOR SOLVING BIE OF STEKLOV EIGENVALUE PROBLEMS , 2004 .

[18]  Yidu Yang,et al.  Nonconforming finite element approximations of the Steklov eigenvalue problem , 2009 .

[19]  Alfredo Bermúdez,et al.  A finite element solution of an added mass formulation for coupled fluid-solid vibrations , 2000, Numerische Mathematik.

[20]  Hai Bi,et al.  Lower spectral bounds by Wilson's brick discretization , 2010 .

[21]  R. Rannacher,et al.  Simple nonconforming quadrilateral Stokes element , 1992 .

[22]  P. Henrici.  Review: S. Bergman and M. Schiffer, Kernel functions and elliptic differential equations in mathematical physics , 1955 .

[23]  Wei Chen,et al.  Two-sided bounds of the discretization error for finite elements , 2011 .

[24]  Yang,et al.  A POSTERIORI ERROR ESTIMATES IN ADINI FINITE ELEMENT FOR EIGENVALUE PROBLEMS , 2000 .

[25]  Q. Lin,et al.  Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation , 2005 .

[26]  Lin Qun,et al.  Stokes Eigenvalue Approximations from Below with Nonconforming Mixed Finite Element Methods , 2010 .

[27]  Frédéric Hecht,et al.  Error indicators for the mortar finite element discretization of the Laplace equation , 2002, Math. Comput..

[28]  Ana Alonso,et al.  Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods , 2009 .

[29]  Zhimin Zhang,et al.  Eigenvalue approximation from below using non-conforming finite elements , 2010 .

[30]  Friedrich Goerisch,et al.  The Determination of Guaranteed Bounds to Eigenvalues with the Use of Variational Methods I , 1990, Computer Arithmetic and Self-Validating Numerical Methods.

[31]  Jun Hu,et al.  The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods , 2011, 1112.1145.