A multigrid correction scheme for a new Steklov eigenvalue problem in inverse scattering

ABSTRACT We propose a multigrid correction scheme to solve a new Steklov eigenvalue problem in inverse scattering. With this scheme, solving an eigenvalue problem in a fine finite element space is reduced to solve a series of boundary value problems in fine finite element spaces and a series of eigenvalue problems in the coarsest finite element space. And the coefficient matrices associated with those linear systems are constructed to be symmetric and positive definite. We prove error estimates of eigenvalues and eigenfunctions. Numerical results coincide in theoretical analysis and indicate our scheme is highly efficient in solving the eigenvalue problem.

[1]  Hehu Xie,et al.  A Multilevel Correction Method for Steklov Eigenvalue Problem by Nonconforming Finite Element Methods , 2015 .

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

[3]  W. Gibbs,et al.  Finite element methods , 2017, Graduate Studies in Mathematics.

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

[5]  Jiguang Sun,et al.  Spectral Indicator Method for a Non-selfadjoint Steklov Eigenvalue Problem , 2018, Journal of Scientific Computing.

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

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

[8]  Tong Zhang,et al.  Two-Sided Arnoldi and Nonsymmetric Lanczos Algorithms , 2002, SIAM J. Matrix Anal. Appl..

[9]  Pedro Morin,et al.  Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems , 2011 .

[10]  Hehu Xie,et al.  A type of multilevel method for the Steklov eigenvalue problem , 2014 .

[11]  Hai Bi,et al.  The Shifted-Inverse Iteration Based on the Multigrid Discretizations for Eigenvalue Problems , 2015, SIAM J. Sci. Comput..

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

[13]  Hehu Xie,et al.  A Multilevel Correction Scheme for Nonsymmetric Eigenvalue Problems by Finite Element Methods , 2015, 1505.06288.

[14]  María G. Armentano,et al.  The effect of reduced integration in the Steklov eigenvalue problem , 2004 .

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

[16]  Antonmaria A. Minzoni,et al.  A discontinuous Steklov problem with an application to water waves , 1979 .

[17]  C. Conca,et al.  Fluids And Periodic Structures , 1995 .

[18]  Anahí Dello Russo,et al.  A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problems , 2011, Comput. Math. Appl..

[19]  Peter Monk,et al.  Stekloff Eigenvalues in Inverse Scattering , 2016, SIAM J. Appl. Math..

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

[21]  Yu Zhang,et al.  Two-grid discretizations and a local finite element scheme for a non-selfadjoint Stekloff eigenvalue problem , 2018, Comput. Math. Appl..

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

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

[24]  Hehu Xie,et al.  Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations , 2013, Applications of Mathematics.

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

[26]  Hehu Xie,et al.  A multi-level correction scheme for eigenvalue problems , 2011, Math. Comput..

[27]  Hehu Xie,et al.  A full multigrid method for eigenvalue problems , 2014, J. Comput. Phys..

[28]  M. Dauge.  Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions , 1988 .

[29]  Lei Zhang,et al.  Multiscale Asymptotic Method for Steklov Eigenvalue Equations in Composite Media , 2013, SIAM J. Numer. Anal..

[30]  Hai Bi,et al.  A Multilevel Correction Method for Convection-Diffusion Eigenvalue Problems , 2015 .

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

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

[33]  Vincent Cocquempot,et al.  Actuator Fault Diagnosis with Application to a Diesel Engine Testbed , 2015 .

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

[35]  Hai Bi,et al.  A new multigrid finite element method for the transmission eigenvalue problems , 2016, Appl. Math. Comput..