### Analysis of Two-Level Complex Shifted Laplace Preconditioner and Deflation-Based Preconditioner for Helmholtz Equation

#### Abstract

#### Full Text:

PDF#### References

Bayliss, A., Goldstein, C. I. and Turkel, E. [1983], ‘An iterative method for the Helmholtz equation’,

Journal of Computational Physics 49, 443 – 457.

Davis, T. A. [2006], Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2), SIAM,

Philadelphia, PA, USA.

Erlangga, Y. A. [2005], A robust and effecient iterative method for numerical solution of Helmholtz

equation, PhD Thesis, DIAM, TU Delft.

Erlangga, Y. A. and Nabben, R. [2008], ‘Deﬂation and Balancing Preconditioners for Krylov Subspace

Methods Applied to Nonsymmetric Matrices’, SIAM J. Matrix Anal. Appl. 30, 684–699.

Erlangga, Y. A., Oosterlee, C. W. and Vuik, C. [2006], ‘A novel multigrid based preconditioner for heterogeneous

Helmholtz problems’, SIAM J. Sci. Comput 27, 1471–1492.

Erlangga, Y., Vuik, C. and Oosterlee, C. [2004], ‘On a class of preconditioners for solving the Helmholtz

equation’, Appl. Numer. Math. 50(3-4), 409–425.

Gijzen, M. B., Erlangga, Y. A. and Vuik, C. [2007], ‘Spectral Analysis of the Discrete Helmholtz Operator

Preconditioned with a Shifted Laplacian’, SIAM Journal on Scientiﬁc Computing 29, 1942–1958.

Humayoun, M., Burney, S. A., Sheikh, A. and Ghafoor, A. [2021], ‘Ritz Vectors-Based Deﬂation Preconditioner

for Linear System with Multiple Right-Hand Sides’, STATISTICS, COMPUTING AND INTERDISCIPLINARY

RESEARCH 3(2), 155–168.

Ihlenburg, F. and Babuska, I. [1995], ‘Finite element solution to the Helmholtz equation with high wave

numbers’, Computers and Mathematics with Applications 30, 9–37.

Mardoche, M. M. M. [2001], ‘Incomplete factorization-based preconditionings for solving the

Helmholtz equation’, International Journal for Numerical Methods in Engineering 50, 1077–1101.

Pillwein, V. and Takacs, S. [2014], ‘A local Fourier convergence analysis of a multigrid method using

symbolic computation’, Journal of Symbolic Computation 63, 1–20.

URL: https://linkinghub.elsevier.com/retrieve/pii/S0747717113001752

Saad, Y. [1996], Iterative Methods for Linear system, PWS Publishing Company.

Shaikh, A., Sheikh, A. H., Ali, A. and Zeb, S. [2019], ‘Critical Review of Preconditioners for Helmholtz

Equation and their Spectral Analysis’, Indian Journal of Science and Technology 12(20), 1–8.

URL: https://indjst.org/articles/critical-review-of-preconditioners-for-helmholtz-equation-and-theirspectral-analysis

Sheikh, A. H. [2014], Development Of The Helmholtz Solver Based On A Shifted Laplace Preconditioner

And A Multigrid Deﬂation Technique, PhD thesis, Delft University of Technology, The Netherlands.

URL: https://doi.org/10.4233/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1

Sheikh, A. H., Lahaye, D., Ramos, L. G., Nabben, R. and Vuik, C. [2016], ‘Accelerating the shifted Laplace

preconditioner for the Helmholtz equation by multilevel deﬂation’, Journal of Computational Physics

, 473–490. Publisher: Elsevier.

Sheikh, A. H., Lahaye, D. and Vuik, C. [2013], ‘On the convergence of shifted Laplace preconditioner

combined with multilevel deﬂation’, Numerical Linear Algebra with Applications 20, 645–662.

Sheikh, A. H., Vuik, C. and Lahaye, D. [2009], Fast iterative solution methods for the Helmholtz equation,

Technical Report 09-11, DIAM, TU Delft.

Sheikh, A. H., Vuik, C. and Lahaye, D. [2011], A scalable Helmholtz solver combining the shifted Laplace

preconditioner with Multigrid deﬂation, Technical Report 11-01, DIAM, TU Delft Netherlands.

Siyal, W. A., Sheikh, A. H., Mallah, M., Sandilo, S. H. and Shaikh, A. G. [2019], ‘Convergence Analysis of

Multigrid Method for Shifted Laplace at Various Levels Using Fourier Modes’, International Journal of

Computer Science and Network Security 19(9), 57–64.

URL: http://paper.ijcsns.org/07

book/201909/20190907.pdf

Siyal, W., Sheikh, A. H., Amur, K. B., Shaikh, G, A. and Malookani, R. A. [2020], ‘On the Eﬃciency of

Multigrid Solver for Shifted Laplace Equation in a Heterogeneous Medium’, International Journal of

Applied Mathematics and Statistics, 59(3), 102–114.

Stüben, K. [2001], An introduction to algebraic multigrid, in U. Trottenberg, C. Oosterlee and

A. Schüller, eds, ‘Multigrid’, Academic Press, San Diego, CA, pp. 413–528.

Vorst, H. A. v. d. [2003], Iterative Krylov Methods for Large Linear Systems, Cambridge University Press,

Cambridge.

Wienands, R. [2001], Extended local Fourier analysis for multigrid: Optimal smoothing, coarse grid

correction, and preconditioning, PhD Thesis, University of Cologne.

Wienands, R. and Joppich, W. [2005], Practical Fourier Analysis for Multigrid Methods, Numerical Insights,

Chapman & Hall/CRC, Boca Raton, Florida, USA.

DOI: http://dx.doi.org/10.21015/vtm.v10i2.1304

### Refbacks

- There are currently no refbacks.

This work is licensed under a Creative Commons Attribution 3.0 License.