A Multilevel Correction Method for Interior Transmission Eigenvalue Problem

In this paper, we give a numerical analysis for the transmission eigenvalue problem by the finite element method. A type of multilevel correction method is proposed to solve the transmission eigenvalue problem. The multilevel correction method can transform the transmission eigenvalue solving in the finest finite element space to a sequence of linear problems and some transmission eigenvalue solving in a very low dimensional spaces. Since the main computational work is to solve the sequence of linear problems, the multilevel correction method improves the overfull efficiency of the transmission eigenvalue solving. Some numerical examples are provided to validate the theoretical results and the efficiency of the proposed numerical scheme.

[1]  Lucas Chesnel,et al.  On the use of T-coercivity to study the interior transmission eigenvalue problem , 2011 .

[2]  Hehu Xie.  A type of multi-level correction scheme for eigenvalue problems by nonconforming finite element methods , 2015 .

[3]  D. Colton,et al.  The interior transmission problem , 2007 .

[4]  Fioralba Cakoni,et al.  Transmission Eigenvalues , 2021, Applied Mathematical Sciences.

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

[6]  Xia Ji,et al.  A Multigrid Method for Helmholtz Transmission Eigenvalue Problems , 2014, J. Sci. Comput..

[7]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

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

[9]  Wenbin Chen,et al.  Error Estimates of the Finite Element Method for Interior Transmission Problems , 2013, J. Sci. Comput..

[10]  Jiguang Sun,et al.  A coupled BEM and FEM for the interior transmission problem in acoustics , 2011, J. Comput. Appl. Math..

[11]  Jiguang Sun.  Iterative Methods for Transmission Eigenvalues , 2011, SIAM J. Numer. Anal..

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

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

[14]  Q. Lin,et al.  A MULTILEVEL CORRECTION TYPE OF ADAPTIVE FINITE ELEMENT METHOD FOR STEKLOV EIGENVALUE PROBLEMS , 2012 .

[15]  Xia Ji,et al.  Algorithm 922: A Mixed Finite Element Method for Helmholtz Transmission Eigenvalues , 2012, TOMS.

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

[17]  A. Kirsch.  On the existence of transmission eigenvalues , 2009 .

[18]  Patrick Ciarlet,et al.  T-coercivity: Application to the discretization of Helmholtz-like problems , 2012, Comput. Math. Appl..

[19]  V. V. Shaidurov,et al.  Multigrid Methods for Finite Elements , 1995 .

[20]  D. Colton,et al.  The inverse electromagnetic scattering problem for anisotropic media , 2010 .

[21]  Lucas Chesnel,et al.  T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients , 2013, Numerische Mathematik.

[22]  Lucas Chesnel,et al.  T-COERCIVITY FOR SCALAR INTERFACE PROBLEMS BETWEEN DIELECTRICS AND METAMATERIALS , 2011 .

[23]  D. Colton,et al.  Analytical and computational methods for transmission eigenvalues , 2010 .

[24]  Xia Ji,et al.  A multi-level method for transmission eigenvalues of anisotropic media , 2013, J. Comput. Phys..

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

[26]  Patrick Ciarlet,et al.  Time harmonic wave diffraction problems in materials with sign-shifting coefficients , 2010, J. Comput. Appl. Math..

[27]  D. Colton,et al.  Transmission eigenvalues and the nondestructive testing of dielectrics , 2008 .

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

[29]  H. Haddar,et al.  Transmission Eigenvalues in Inverse Scattering Theory , 2012 .

[30]  Jiguang Sun,et al.  Estimation of transmission eigenvalues and the index of refraction from Cauchy data , 2010 .

[31]  Fioralba Cakoni,et al.  The Existence of an Infinite Discrete Set of Transmission Eigenvalues , 2010, SIAM J. Math. Anal..

[32]  Jie Shen,et al.  A Spectral-Element Method for Transmission Eigenvalue Problems , 2013, J. Sci. Comput..

[33]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .