A fast iterative solver for scattering by elastic objects in layered media

We developed a fast iterative solver for computing time-harmonic acoustic waves scattered by an elastic object in layered media. The discretization of the problem was performed using a finite element method with linear elements based on a locally body-fitted uniform triangulation. We used a domain decomposition preconditioner in the iterative solution of the resulting system of linear equations. The preconditioner was based on a cyclic reduction type fast direct solver. The solution procedure reduces GMRES iterates onto a sparse subspace which decreases the storage and computational requirements essentially. The numerical results demonstrate the effectiveness of the proposed approach for two-dimensional domains that are hundreds of wavelengths wide and require the solution of linear systems with several millions of unknowns.

[1]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[2]  Tuomo Rossi,et al.  A Domain Embedding Method for Scattering Problems with an Absorbing Boundary or a Perfectly Matched Layer , 2003 .

[3]  S. D. Chatterji.  Proceedings of the International Congress of Mathematicians , 1995 .

[4]  René-Édouard Plessix,et al.  Separation-of-variables as a preconditioner for an iterative Helmholtz solver , 2003 .

[5]  Elisabeth Larsson,et al.  A Domain Decomposition Method for the Helmholtz Equation in a Multilayer Domain , 1999, SIAM J. Sci. Comput..

[6]  William Gropp,et al.  Domain Decomposition: Parallel Mul-tilevel Methods for Elliptic PDEs , 1996 .

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

[8]  Kazufumi Ito,et al.  Material Surface Design to Counter Electromagnetic Interrogation of Targets , 2006, SIAM J. Appl. Math..

[9]  Kazufumi Ito,et al.  A Fast Helmholtz Solver for Scattering by a Sound-soft Target in Sediment , 2007 .

[10]  P. Swarztrauber.  THE METHODS OF CYCLIC REDUCTION, FOURIER ANALYSIS AND THE FACR ALGORITHM FOR THE DISCRETE SOLUTION OF POISSON'S EQUATION ON A RECTANGLE* , 1977 .

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

[12]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[13]  Tuomo Rossi,et al.  A Parallel Fast Direct Solver for Block Tridiagonal Systems with Separable Matrices of Arbitrary Dimension , 1999, SIAM J. Sci. Comput..

[14]  Kazufumi Ito,et al.  Preconditioned iterative methods on sparse subspaces , 2006, Appl. Math. Lett..

[15]  M. Guddati,et al.  Modified integration rules for reducing dispersion error in finite element methods , 2004 .

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

[17]  Semyon Tsynkov,et al.  Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions , 2005 .

[18]  Tuomo Rossi,et al.  Fast direct solution of the Helmholtz equation with a perfectly matched layer or an absorbing boundary condition , 2003 .

[19]  Yu. A. KUZNETSOV,et al.  On partial solution of systems of linear algebraic equations , 1989 .

[20]  Patrick Joly,et al.  Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem , 1990 .

[21]  G. Marchuk,et al.  Fictitious domain and domain decomposition methods , 1986 .

[22]  Christoph Börgers,et al.  A triangulation algorithm for fast elliptic solvers based on domain imbedding , 1990 .

[23]  Alexandra Banegas,et al.  Fast Poisson solvers for problems with sparsity , 1978 .

[24]  Elisabeth Larsson,et al.  Parallel Solution of the Helmholtz Equation in a Multilayer Domain , 2003 .

[25]  Tuomo Rossi,et al.  A Domain Decomposition Technique For Two-Dimensional Scattering Problems With Coated Obstacles , 2001 .

[26]  Y. Kuznetsov,et al.  3D Helmholtz wave equation by fictitious domain method , 1998 .

[27]  Yuri A. Kuznetsov,et al.  Fictitious Domain Methods for the Numerical Solution of Two-Dimensional Scattering Problems , 1998 .

[28]  Armand Wirgin,et al.  Marine Acoustics: Direct and Inverse Problems , 2004 .

[29]  Tuomo Rossi,et al.  A Nonstandard Cyclic Reduction Method, Its Variants and Stability , 1999, SIAM J. Matrix Anal. Appl..

[30]  Tuomo Rossi,et al.  A Parallel Fictitious Domain Method for the Three-Dimensional Helmholtz Equation , 2002, SIAM J. Sci. Comput..

[31]  J. Lopes,et al.  Subcritical detection of an elongated target buried under a rippled interface , 2004, Oceans '04 MTS/IEEE Techno-Ocean '04 (IEEE Cat. No.04CH37600).

[32]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .