We consider an efficient iterative approach to the solution of the discrete Helmholtz equation with Dirichlet, Neumann, and Sommerfeld-like boundary conditions based on a compact sixth order approximation scheme and lower order preconditioned Krylov subspace methodology. The resulting systems of finite-difference equations are solved by different preconditioned Krylov subspace-based methods. In the analysis of the lower order preconditioning developed here, we introduce the term “kth order preconditioned matrix” in addition to the commonly used “an optimal preconditioner.” The necessity of the new criterion is justified by the fact that the condition number of the preconditioned matrix in some of our test problems improves with the decrease of the grid step size. In a simple 1D case, we are able to prove this analytically. This new parameter could serve as a guide in the construction of new preconditioners. The lower order direct preconditioner used in our algorithms is based on a combination of the separation of variables technique and fast Fourier transform (FFT) type methods. The resulting numerical methods allow efficient implementation on parallel computers. Numerical results confirm the high efficiency of the proposed iterative approach. 1. Introduction In recent years, the problem of increasing the resolution of existing numerical solvers has become an urgent task in many areas of science and engineering. Most of the existing efficient solvers for structured matrices were developed for lower-order approximations of partial differential equations. The need for improved accuracy of underlying algorithms leads to modified discretized systems and as a result to the modification of the numerical solvers (see, e.g., [1]). The use of a lower-order preconditioner for efficient implementation of high-order finite-difference and finite-element schemes has been under consideration for a long time (see, e.g., [2, 3]). In this paper, a compact sixth order approximation finite-difference scheme is developed, and a lower-order approximation direct solver as a preconditioner for an efficient implementation of this compact scheme for the Helmholtz equation in the Krylov subspace method framework is considered. This approach allows us to utilize the existing lower-order approximation solvers which significantly simplifies the implementation process of the higher resolution numerical methods. The model problem considered in the paper is the numerical solution of the Helmholtz equation with the Dirichlet, Neumann, and/or Sommerfeld-like (radiation) boundary
References
[1]
Y. Zhuang and X.-H. Sun, “A high-order fast direct solver for singular poisson equations,” Journal of Computational Physics, vol. 171, no. 1, pp. 79–94, 2001.
[2]
J. J. Heys, T. A. Manteuffel, S. F. McCormick, and L. N. Olson, “Algebraic multigrid for higher-order finite elements,” Journal of Computational Physics, vol. 204, no. 2, pp. 520–532, 2005.
[3]
S. Orszag, “Spectral methods for problems in complex geometries,” Journal of Computational Physics, vol. 37, pp. 70–92, 1980.
[4]
I. M. Babu？ka and S. A. Sauter, “Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?” SIAM Review, vol. 42, no. 3, pp. 451–484, 2000.
[5]
A. Bayliss, C. I. Goldstein, and E. Turkel, “On accuracy conditions for the numerical computation of waves,” Journal of Computational Physics, vol. 59, no. 3, pp. 396–404, 1985.
[6]
G. Sutmann, “Compact finite difference schemes of sixth order for the Helmholtz equation,” Journal of Computational and Applied Mathematics, vol. 203, no. 1, pp. 15–31, 2007.
[7]
M. Nabavi, M. H. K. Siddiqui, and J. Dargahi, “A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation,” Journal of Sound and Vibration, vol. 307, no. 3-5, pp. 972–982, 2007.
[8]
I. Singer and E. Turkel, “Sixth-order accurate finite difference schemes for the Helmholtz equation,” Journal of Computational Acoustics, vol. 14, no. 3, pp. 339–351, 2006.
[9]
S. Britt, S. Tsynkov, and E. Turkel, “Numerical simulation of time-harmonic waves in inhomogeneous media using compact high order schemes,” Communications in Computational Physics, vol. 9, no. 3, pp. 520–541, 2011.
[10]
A. Bayliss, C. I. Goldstein, and E. Turkel, “An iterative method for the Helmholtz equation,” Journal of Computational Physics, vol. 49, no. 3, pp. 443–457, 1983.
[11]
S. Kim, “Parallel multidomain iterative algorithms for the Helmholtz wave equation,” Applied Numerical Mathematics, vol. 17, no. 4, pp. 411–429, 1995.
[12]
J. Douglas Jr, J. L. Hensley, and J. E. Roberts, “Alternating-direction iteration method for Helmholtz problems,” Tech. Rep. 214, Mathematics Department, Purdue University, West Lafaette, Ind, USA, 1993.
[13]
N. Umetani, S. P. MacLachlan, and C. W. Oosterlee, “A multigrid-based shifted Laplacian preconditioner for a fourth-order Helmholtz discretization,” Numerical Linear Algebra with Applications, vol. 16, no. 8, pp. 603–626, 2009.
[14]
J. H. Bramble, J. E. Pasciak, and J. Xu, “The analysis of multigrid algorithms for nonsymmetric and indefinite problems,” Mathematics of Computation, vol. 51, pp. 389–414, 1988.
[15]
O. Ernst and G. H. Golub, “A domain decomposition approach to solving the Helmholtz equation with a radiation boundary condition,” in Domain Decomposition in Science and Engineering, A. Quarteroni, H. Periaux, Y. Kuznetsov, and O. Widdlund, Eds., pp. 177–192, American Mathematical Society, Providence, RI, USA, 1994.
[16]
H. C. Elman and D. P. O'Leary, “Efficient iterative solution of the three-dimensional Helmholtz equation,” Journal of Computational Physics, vol. 142, no. 1, pp. 163–181, 1998.
[17]
H. C. Elman and D. P. O'Leary, “Eigenanalysis of some preconditioned Helmholtz problems,” Numerische Mathematik, vol. 83, no. 2, pp. 231–257, 1999.
[18]
Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “GMRES computation of high frequency electrical field propagation in land mine detection,” Journal of Computational Physics, vol. 158, no. 1, pp. 98–115, 2000.
[19]
J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, Pa, USA, 1997.
[20]
A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, vol. 2, Birkh？user, Boston, Mass, USA, 1989.
[21]
S. H. Lui, Numerical Analysis of Partial Differential Equations, John Wiley & Sons, New York, NY, USA, 2011.
[22]
Y. A. Gryazin, “A Compact sixth order scheme combined with GMRES method for the 3D Helmholtz equation,” in Proceedings of the 10th International Conference on Mathematical and Numerical Aspects of Waves, pp. 339–442, Vancouver, Canada, July, 2011.
[23]
G. H. Golub and C. F. van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 2nd edition, 1989.
[24]
Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, Pa, USA, 2003.
[25]
E. Heikkola, T. Rossi, and J. Toivanen, “Fast direct solution of the Helmholtz equation with a perfectly matched layer or an absorbing boundary condition,” International Journal for Numerical Methods in Engineering, vol. 57, no. 14, pp. 2007–2025, 2003.
