Critical Review of Preconditioners for Helmholtz Equation and their Spectral Analysis

Shaikh, A.G.; Sheikh, A. H.; Ali, Asif Iqbal; Zeb, Salman

doi:10.17485/ijst/2019/v12i20/144368

Cited by 2 publications

(2 citation statements)

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“…Finite difference, Finite element and Finite volume are amongst them. A detail of discretization can be found in [8,13,20]. Discretizing of the Helmholtz problem Eq.…”

Section: Model Problemmentioning

confidence: 99%

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

Rajput,

Shiekh,

Amur

2022

VFAST trans. math.

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A Long time deflation preconditioner is used to speed up the convergence of the Krylov subspace method. The discretization of Helmholtz equation with Dirichlet boundary condition by finite difference method obtained any linear system. Resolving a large wavenumber requires a larger number of Grid points, i.e. large linear systems. Thus due to the large linear system, many (sparse) direct methods have taken more memory, So we have used the (iterative technique) Krylov subspace method. One of the problems of the Krylov subspace method is the required preconditioner for better convergence. We use (CSLP) as a preconditioner and drive eigenvalues of (CSLP). However, with increasing wavenumber CSLP shows slow convergence behavior. Then we use another projection-type preconditioner as a deflation preconditioner. A rigorous Fourier analysis (RFA) is a separate research idea to examine the con- vergence of the iterative method included in this article. We analyze the deflation preconditioner with a complex shifted Laplace preconditioner (CSLP) which exhibition spectral behavior of the preconditioner, which is favorable to the Krylov method.

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“…Finite difference, Finite element and Finite volume are amongst them. A detail of discretization can be found in [8,13,20]. Discretizing of the Helmholtz problem Eq.…”

Section: Model Problemmentioning

confidence: 99%

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

Rajput,

Shiekh,

Amur

2022

VFAST trans. math.

View full text Add to dashboard Cite

show abstract

“…I n The finite difference approximation method [1], [2], changes differential equations, whether an ordinary differential or partial differential equation into, a linear system of equations, and it does not give a symbolic solution. Many problems related to steady state oscillations (mechanical, acoustical, and thermal), and wave scattering [3] is modelled by the Helmholtz equation [4], [5], [6], [7]. Wave scattering has many applications in physics, engineering, and science.…”

Section: Introductionmentioning

confidence: 99%

An Impact of the Small and Large Grid Sizes on Differential Equations

Shaikh

2021

QUESTRJ

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The finite difference technique is oldest numerical method to solve differential equations. Like many differential equations, Helmholtz differential equation which is used to describe many physical phenomena, has long been solved using finite difference method. can be described by Helmholtz Differential equations. The solution of the Helmholtz type differential equations is very important. The information that it belongs together because it tells one coherent story just knowing a little bit about finite differences through to how to solve differential equations an especial technique is used, how to implement finite difference method and the tool which is used as generic enough that will immediately be given a whole new differential equation. The analysis of small to moderate sized presented with the help of a few examples. The improved finite difference method is presented with examples, the method is simple, clear, and short the MatLab code is available, the improved finite difference method is suitable and easy to implement, manually as well as computationally.

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Critical Review of Preconditioners for Helmholtz Equation and their Spectral Analysis

Cited by 2 publications

References 0 publications

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

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

An Impact of the Small and Large Grid Sizes on Differential Equations

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