Multilevel Preconditioner with Stable Coarse Grid Corrections for the Helmholtz Equation

In this paper we consider a class of robust multilevel preconditioners for the Helmholtz equation with high wave number. The key idea in this work is to use the continuous interior penalty finite element methods studied in [H. Wu, IMA J. Numer. Anal., 34 (2014), pp. 1266--1288; L. Zhu and H. Wu, SIAM J. Numer. Anal., 51 (2013), pp. 1828--1852] to construct the stable coarse grid correction problems. The multilevel methods, based on GMRES smoothing on coarse grids, are then served as a preconditioner in the outer GMRES iteration. In the one-dimensional case, the convergence property of the modified multilevel methods is analyzed by the local Fourier analysis. From our numerical results, we find that the proposed methods are efficient for a reasonable range of frequencies. The performance of the algorithms depends relatively mildly on wave number.

[1]  Haijun Wu,et al.  Preasymptotic Error Analysis of CIP-FEM and FEM for Helmholtz Equation with High Wave Number. Part II: hp Version , 2012, SIAM J. Numer. Anal..

[2]  Yogi A. Erlangga,et al.  Advances in Iterative Methods and Preconditioners for the Helmholtz Equation , 2008 .

[3]  F. Ihlenburg.  Finite Element Analysis of Acoustic Scattering , 1998 .

[4]  Lexing Ying,et al.  Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers , 2010, Multiscale Model. Simul..

[5]  Christiaan C. Stolk,et al.  A rapidly converging domain decomposition method for the Helmholtz equation , 2012, J. Comput. Phys..

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

[7]  Cornelis Vuik,et al.  Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian , 2007, SIAM J. Sci. Comput..

[8]  Joseph W. H. Liu,et al.  The Multifrontal Method for Sparse Matrix Solution: Theory and Practice , 1992, SIAM Rev..

[9]  Justin W. L. Wan,et al.  Practical Fourier analysis for multigrid methods , 2007, Math. Comput..

[10]  Haijun Wu,et al.  hp-Discontinuous Galerkin methods for the Helmholtz equation with large wave number , 2008, Math. Comput..

[11]  Haijun Wu,et al.  Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number , 2009, SIAM J. Numer. Anal..

[12]  Martin J. Gander,et al.  Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods , 2012 .

[13]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[14]  Wim Vanroose,et al.  Local Fourier analysis of the complex shifted Laplacian preconditioner for Helmholtz problems , 2011, Numer. Linear Algebra Appl..

[15]  Charbel Farhat,et al.  FETI-H: a scalable domain decomposition method for high frequency exterior Helmholtz problems , 1999 .

[16]  Soohyun Kim,et al.  Multigrid Simulation for High-Frequency Solutions of the Helmholtz Problem in Heterogeneous Media , 2002, SIAM J. Sci. Comput..

[17]  Martin J. Gander,et al.  Multigrid Methods for Helmholtz Problems: A Convergent Scheme in 1D Using Standard Components , 2013 .

[18]  Cornelis Vuik,et al.  A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems , 2005, SIAM J. Sci. Comput..

[19]  A. Brandt,et al.  WAVE-RAY MULTIGRID METHOD FOR STANDING WAVE EQUATIONS , 1997 .

[20]  Martin J. Gander,et al.  Optimized Schwarz Methods without Overlap for the Helmholtz Equation , 2002, SIAM J. Sci. Comput..

[21]  Panayot S. Vassilevski,et al.  An application of the abstract multilevel theory to nonconforming finite element methods , 1995 .

[22]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[23]  Achi Brandt,et al.  Accuracy Properties of the Wave-Ray Multigrid Algorithm for Helmholtz Equations , 2006, SIAM J. Sci. Comput..

[24]  Jens Markus Melenk,et al.  Direct and Inverse Problems in Wave Propagation and Applications , 2013 .

[25]  B. Engquist,et al.  Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation , 2010, 1007.4290.

[26]  C. Farhat,et al.  FETI-DPH: A DUAL-PRIMAL DOMAIN DECOMPOSITION METHOD FOR ACOUSTIC SCATTERING , 2005 .

[27]  Cornelis Vuik,et al.  On a Class of Preconditioners for Solving the Helmholtz Equation , 2003 .

[28]  F. Magoulès,et al.  An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation , 2007 .

[29]  Dianne P. O'Leary,et al.  A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations , 2001, SIAM J. Sci. Comput..

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