A two-grid discretization scheme of non-conforming finite elements for transmission eigenvalues

Abstract In this paper, for the Helmholtz transmission eigenvalue problem, we propose a two-grid discretization scheme of non-conforming finite elements. With this scheme, the solution of the transmission eigenvalue problem on a fine grid π h is reduced to the solution of the primal and dual eigenvalue problem on a much coarser grid π H and the solutions of two linear algebraic systems with the same positive definite Hermitian and block diagonal coefficient matrix on the fine grid π h . We prove the resulting solution still maintains an asymptotically optimal accuracy, and we report some numerical examples in two dimension and three dimension on the modified-Zienkiewicz element to validate the efficiency of our approach for solving transmission eigenvalues.

[1]  X. Hu,et al.  Two-Grid Methods for Maxwell Eigenvalue Problems , 2014, SIAM J. Numer. Anal..

[2]  Jinchao Xu.  A new class of iterative methods for nonselfadjoint or indefinite problems , 1992 .

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

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

[5]  Hai Bi,et al.  Non-conforming finite element methods for transmission eigenvalue problem ☆ , 2016, 1601.01068.

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

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

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

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

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

[11]  Jiguang Sun,et al.  Finite Element Methods for Maxwell's Transmission Eigenvalues , 2012, SIAM J. Sci. Comput..

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

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

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

[15]  Jie Shen,et al.  Spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem , 2015, Comput. Math. Appl..

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

[17]  Rong An,et al.  Optimal Error Estimates of Linearized Crank–Nicolson Galerkin Method for Landau–Lifshitz Equation , 2016, J. Sci. Comput..

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

[19]  Liwei Xu,et al.  Computation of Maxwell’s transmission eigenvalues and its applications in inverse medium problems , 2013 .

[20]  Hai Bi,et al.  Error estimates and a two grid scheme for approximating transmission eigenvalues , 2015, 1506.06486.

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

[22]  Hai Bi,et al.  Mixed Methods for the Helmholtz Transmission Eigenvalues , 2016, SIAM J. Sci. Comput..

[23]  Jiguang Sun,et al.  Error Analysis for the Finite Element Approximation of Transmission Eigenvalues , 2014, Comput. Methods Appl. Math..

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

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

[26]  Jiayu Han,et al.  An Adaptive Finite Element Method for the Transmission Eigenvalue Problem , 2016, Journal of Scientific Computing.

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

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

[29]  Liwei Xu,et al.  A spectral projection method for transmission eigenvalues , 2016, 1605.00727.

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

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

[32]  Ming Wang,et al.  A new class of Zienkiewicz-type non-conforming element in any dimensions , 2007, Numerische Mathematik.