A Nonnested Augmented Subspace Method for Eigenvalue Problems with Curved Interfaces

In this paper, we present a nonnested augmented subspace algorithm and its multilevel correction method for solving eigenvalue problems with curved interfaces. The augmented subspace algorithm and the corresponding multilevel correction method are designed based on a coarse finite element space which is not the subset of the finer finite element space. The nonnested augmented subspace method can transform the eigenvalue problem solving on the finest mesh to the solving linear equation on the same mesh and small scale eigenvalue problem on the low dimensional augmented subspace. The corresponding theoretical analysis and numerical experiments are provided to demonstrate the efficiency of the proposed algorithms.

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

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

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

[4]  Hehu Xie,et al.  A multilevel correction adaptive finite element method for Kohn-Sham equation , 2018, J. Comput. Phys..

[5]  Hehu Xie,et al.  Adaptive Multilevel Correction Method for Finite Element Approximations of Elliptic Optimal Control Problems , 2017, J. Sci. Comput..

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

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

[9]  Robert D. Falgout,et al.  hypre: A Library of High Performance Preconditioners , 2002, International Conference on Computational Science.

[10]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[11]  Hehu Xie,et al.  A Multilevel Correction Method for Interior Transmission Eigenvalue Problem , 2015, J. Sci. Comput..

[12]  Fei Xu,et al.  An Efficient Multigrid Method for Ground State Solution of Bose-Einstein Condensates , 2017, 1711.02519.

[13]  Hehu Xie,et al.  Local and Parallel Finite Element Algorithm Based On Multilevel Discretization for Eigenvalue Problem , 2014, 1401.4969.

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

[15]  Jens Markus Melenk,et al.  Optimal a priori estimates for higher order finite elements for elliptic interface problems , 2010 .

[16]  Hehu Xie,et al.  A multigrid method for the ground state solution of Bose–Einstein condensates based on Newton iteration , 2014, BIT Numerical Mathematics.

[17]  William Gropp,et al.  Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries , 1997, SciTools.

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

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

[20]  Hehu Xie,et al.  An Eigenwise Parallel Augmented Subspace Method for Eigenvalue Problems , 2019, ArXiv.

[21]  D. Sorensen Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[22]  Fei Xu,et al.  A full multigrid method for semilinear elliptic equation , 2016, Applications of Mathematics.

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

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

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

[26]  Hehu Xie,et al.  A full multigrid method for eigenvalue problems , 2016, J. Comput. Phys..

[27]  Hehu Xie,et al.  Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction , 2018, SIAM J. Numer. Anal..

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

[29]  Ivo Babuska,et al.  The finite element method for elliptic equations with discontinuous coefficients , 1970, Computing.

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

[31]  G. Burton Sobolev Spaces , 2013 .

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

[33]  Hehu Xie,et al.  A Cascadic Multigrid Method for Eigenvalue Problem , 2014, 1409.2923.

[34]  Pingwen Zhang,et al.  Moving mesh methods in multiple dimensions based on harmonic maps , 2001 .

[35]  Hehu Xie,et al.  A multigrid method for eigenvalue problems based on shifted-inverse power technique , 2015 .

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

[37]  Hehu Xie,et al.  A multilevel finite element method for Fredholm integral eigenvalue problems , 2015, J. Comput. Phys..

[38]  Hehu Xie,et al.  A cascadic multigrid method for nonsymmetric eigenvalue problem , 2019 .

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

[40]  Q. Lin,et al.  Multilevel correction adaptive finite element method for semilinear elliptic equation , 2015 .

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

[42]  Hehu Xie N A ] 8 J an 2 01 5 A Multigrid Method for Nonlinear Eigenvalue Problems : Version 2 ∗ , 2015 .

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

[44]  Xia Ji,et al.  A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation , 2018, J. Sci. Comput..

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

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

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

[48]  MOVING FINITE ELEMENT METHOD , 2018, Boundary Elements and other Mesh Reduction Methods XLI.

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

[50]  Yuanzhe Xi,et al.  A multi-level mixed element scheme of the two-dimensional Helmholtz transmission eigenvalue problem , 2017, IMA Journal of Numerical Analysis.

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

[52]  Hehu Xie,et al.  A Multilevel Correction Method for Optimal Controls of Elliptic Equations , 2014, SIAM J. Sci. Comput..

[53]  Hehu Xie,et al.  A full multigrid method for nonlinear eigenvalue problems , 2015, 1502.04657.