A full multigrid method for eigenvalue problems

A multigrid method is proposed to solve the eigenvalue problem by the finite element method based on the combination of the multilevel correction scheme for the eigenvalue problem and the multigrid method for the boundary value problem. In this scheme, solving eigenvalue problem is transformed to a series of solutions of boundary value problems by the multigrid method on multilevel meshes and a series of solutions of eigenvalue problems on the coarsest finite element space. Besides the multigrid scheme, all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems. The total computational work of this scheme can reach the optimal order as same as solving the corresponding boundary value problem. Therefore, this type of iteration scheme improves the overall efficiency of the eigenvalue problem solving. Some numerical experiments are presented to validate the efficiency of the method.

[1]  Jinchao Xu,et al.  Lower bounds of the discretization error for piecewise polynomials , 2013, Math. Comput..

[2]  R. Martin,et al.  Electronic Structure: Basic Theory and Practical Methods , 2004 .

[3]  Tony F. Chan,et al.  Subspace correction multi-level methods for elliptic eigenvalue problems , 2002, Numer. Linear Algebra Appl..

[4]  G. Burton.  Sobolev Spaces , 2013 .

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

[6]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

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

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

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

[10]  Huajie Chen,et al.  Numerical approximations of a nonlinear eigenvalue problem and applications to a density functional model , 2010 .

[11]  Hamilton-Jacobi Equations,et al.  Multigrid Methods for , 2011 .

[12]  F. Chatelin.  Spectral approximation of linear operators , 2011 .

[13]  Hai Bi,et al.  Two-Grid Finite Element Discretization Schemes Based on Shifted-Inverse Power Method for Elliptic Eigenvalue Problems , 2011, SIAM J. Numer. Anal..

[14]  J. Pasciak,et al.  New convergence estimates for multigrid algorithms , 1987 .

[15]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[16]  H. Yserentant.  On the multi-level splitting of finite element spaces , 1986 .

[17]  S. McCormick,et al.  Multigrid Methods for Differential Eigenproblems , 1983 .

[18]  L. Ridgway Scott,et al.  Higher-dimensional nonnested multigrid methods , 1992 .

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

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

[21]  Jinchao Xu,et al.  A Novel Two-Grid Method for Semilinear Elliptic Equations , 1994, SIAM J. Sci. Comput..

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

[23]  Aihui Zhou,et al.  Numerical analysis of finite dimensional approximations of Kohn–Sham models , 2011, Adv. Comput. Math..

[24]  Jinchao Xu,et al.  Lower Bounds of the Discretization for Piecewise Polynomials , 2011, 1106.4395.

[25]  James H. Bramble,et al.  The analysis of multigrid methods , 2000 .

[26]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[27]  A. Zhou.  MULTI-LEVEL ADAPTIVE CORRECTIONS IN FINITE DIMENSIONAL APPROXIMATIONS , 2009 .

[28]  Randolph E. Bank,et al.  An optimal order process for solving finite element equations , 1981 .

[29]  Wolfgang Hackbusch,et al.  On the Computation of Approximate Eigenvalues and Eigenfunctions of Elliptic Operators by Means of a Multi-Grid Method , 1979 .

[30]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[31]  Jinchao Xu,et al.  Local and Parallel Finite Element Algorithms for Eigenvalue Problems , 2002 .

[32]  R. Parr.  Density-functional theory of atoms and molecules , 1989 .

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

[34]  Xiaozhe Hu,et al.  Corrigendum to: "Acceleration of a two-grid method for eigenvalue problems" , 2011, Math. Comput..