Guaranteed lower bounds for eigenvalues

This paper introduces fully computable two-sided bounds on the eigenvalues of the Laplace operator on arbitrarily coarse meshes based on some approximation of the corresponding eigenfunction in the nonconforming Crouzeix-Raviart finite element space plus some postprocessing. The efficiency of the guaranteed error bounds involves the global mesh-size and is proven for the large class of graded meshes. Numerical examples demonstrate the reliability of the guaranteed error control even with an inexact solve of the algebraic eigenvalue problem. This motivates an adaptive algorithm which monitors the discretisation error, the maximal mesh-size, and the algebraic eigenvalue error. The accuracy of the guaranteed eigenvalue bounds is surprisingly high with efficiency indices as small as 1.4.

[1]  Carsten Carstensen,et al.  Guaranteed lower eigenvalue bounds for the biharmonic equation , 2014, Numerische Mathematik.

[2]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[3]  Wolfgang Dahmen,et al.  Adaptive eigenvalue computation: complexity estimates , 2007, Numerische Mathematik.

[4]  R. Durán,et al.  ASYMPTOTIC LOWER BOUNDS FOR EIGENVALUES BY NONCONFORMING FINITE ELEMENT METHODS , 2004 .

[5]  Sophia Blau,et al.  Analysis Of The Finite Element Method , 2016 .

[6]  R. Durán,et al.  A posteriori error estimators for nonconforming finite element methods , 1996 .

[7]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[8]  Carsten Carstensen,et al.  An Adaptive Finite Element Eigenvalue Solver of Asymptotic Quasi-Optimal Computational Complexity , 2012, SIAM J. Numer. Anal..

[9]  Carsten Carstensen,et al.  Computational survey on a posteriori error estimators for nonconforming finite element methods for the Poisson problem , 2013, J. Comput. Appl. Math..

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

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

[12]  Susanne C. Brenner,et al.  Convergence of nonconforming multigrid methods without full elliptic regularity , 1999, Math. Comput..

[13]  George E. Forsythe,et al.  Asymptotic lower bounds for the fundamental frequency of convex membranes , 1955 .

[14]  Lloyd N. Trefethen,et al.  Computed eigenmodes of planar regions , 2005 .

[15]  R. Laugesen,et al.  Minimizing Neumann fundamental tones of triangles: An optimal Poincaré inequality , 2009, 0907.1552.

[16]  A. Knyazev,et al.  A Geometric Theory for Preconditioned Inverse Iteration. III:A Short and Sharp Convergence Estimate for Generalized EigenvalueProblems. , 2001 .

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

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

[19]  Carsten Carstensen,et al.  An oscillation-free adaptive FEM for symmetric eigenvalue problems , 2011, Numerische Mathematik.

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

[21]  Carsten Carstensen,et al.  Explicit Error Estimates for Courant, Crouzeix-Raviart and Raviart-Thomas Finite Element Methods , 2012 .

[22]  Chao Yang,et al.  ARPACK users' guide - solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods , 1998, Software, environments, tools.

[23]  Eduardo M. Garau,et al.  Convergence of adaptive finite element methods for eigenvalue problems , 2008, 0803.0365.

[24]  R. Hoppe,et al.  A review of unified a posteriori finite element error control , 2012 .

[25]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

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

[27]  Ricardo G. Durán,et al.  A posteriori error estimates for non-conforming approximation of eigenvalue problems , 2012 .

[28]  Hans F. Weinberger,et al.  Upper and lower bounds for eigenvalues by finite difference methods , 1956 .