# Shifted Laplace preconditioners for the Helmholtz equations

@article{Vuik2003ShiftedLP, title={Shifted Laplace preconditioners for the Helmholtz equations}, author={Cornelis Vuik and Yogi A. Erlangga}, journal={Reports of the Department of Applied Mathematical Analysis}, year={2003} }

In this paper, we present a numerical method to solve the time-harmonic wave equation in 2D heterogeneous media. The underlying equation governs wave propagations and scattering phenomena arising in acoustic and optical problems. In particular, we look for solutions of the Helmholtz equation discretized by a finite difference method. Since the number of gridpoints per wavelength should be sufficiently large to result in acceptable solutions, for very high wavenumbers the discrete problem… Expand

#### 7 Citations

A perfectly matched layer for the Helmholtz equation in a semi-infinite strip

- Mathematics
- 2004

The perfectly matched layer (PML) has become a widespread technique for preventing reflections from far field boundaries for wave propagation problems in both the time dependent and frequency… Expand

A PRECONDITIONED METHOD FOR THE SOLUTION OF THE ROBBINS PROBLEM FOR THE HELMHOLTZ EQUATION

- Mathematics
- The ANZIAM Journal
- 2010

Abstract A preconditioned iterative method for the two-dimensional Helmholtz equation with Robbins boundary conditions is discussed. Using a finite-difference method to discretize the Helmholtz… Expand

Numerical Aspects of Iterative Solving of Linear Systems derived from Helmholtz's Problem

- Mathematics
- 2004

In this report, several numerical aspects and difficulties for solving a large linear system, derived from the Helmholtz equation are overviewed. The presentation starts with the derivation of the… Expand

A survey of finite element methods for time-harmonic acoustics

- Mathematics
- 2006

Many of the current issues and methodologies related to finite element methods for time-harmonic acoustics are reviewed. The effective treatment of unbounded domains is a major challenge. Most… Expand

The development of a finite volume method for modeling sound in coastal ocean environment

- Engineering
- OCEANS 2015 - MTS/IEEE Washington
- 2015

The rapid growth of renewable energy from offshore sources has raised concerns that underwater noise from construction and operation of offshore devices may interfere with communication of marine… Expand

Méthodes de décomposition de domaines pour des problèmes de propagation d'ondes hétérogènes

- Physics
- 2015

Dans cette these on considere des algorithmes de Schwarz appliques aux problemes harmoniques heterogenes (Maxwell et Helmholtz) en deux et trois dimensions. On fait l'etude pour une decomposition en… Expand

#### References

SHOWING 1-10 OF 22 REFERENCES

Preconditioned iterative solution of the 2D Helmholtz equation

- Mathematics
- 2002

Using a finite element method to solve the Helmholtz equation leads to a sparse system of equations which in three dimensions is too large to solve directly. It is also non-Hermitian and highly… Expand

Separation-of-variables as a preconditioner for an iterative Helmholtz solver

- Mathematics
- 2003

A preconditioned iterative method based on separation-of-variables for solving the Helmholtz equation in an inhomogeneous medium is tested. The preconditioner is constructed by approximating the… Expand

On a Class of Preconditioners for Solving the Helmholtz Equation

- Mathematics
- 2003

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is… Expand

An Iterative method for the Helmholtz equation

- Mathematics
- 1983

An iterative algorithm for the solution of the Helmholtz equation is developed. The algorithm is based on a preconditioned conjugate gradient iteration for the normal equations. The preconditioning… Expand

CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems

- Mathematics
- 1989

A Lanczos-type method is presented for nonsymmetric sparse linear systems as arising from discretisations of elliptic partial differential equations. The method is based on a polynomial variant of… Expand

A Petrov-Galerkin type method for solving Axk=b, where A is symmetric complex

- Physics
- 1990

Discretization of steady-state eddy-current equations may lead to linear system Ax=b in which the complex matrix A is not Hermitian, but may be chosen symmetric. In the positive definite Hermitian… Expand

Preconditioning indefinite discretization matrices

- Mathematics
- 1989

SummaryThe finite element discretization of many elliptic boundary value problems leads to linear systems with positive definite and symmetric coefficient matrices. Many efficient preconditioners are… Expand

Comparison of Krylov-type methods for complex linear systems applied to high-voltage problems

- Computer Science
- 1998

Comparisons between different Krylov-subspace methods, also in combination with residual smoothing techniques, and parameter studies are presented in this paper to find a sufficiently robust solution method for high-voltage power plants. Expand

GMRESR: a family of nested GMRES methods

- Computer Science, Mathematics
- Numer. Linear Algebra Appl.
- 1994

Recently Eirola and Nevanlinna have proposed an iterativ<: solution method for unsymmetric linear systems, in which the preconditioner is updated from step to step. Following their ideas we suggest… Expand

A Flexible Inner-Outer Preconditioned GMRES Algorithm

- Mathematics, Computer Science
- SIAM J. Sci. Comput.
- 1993

A variant of the GMRES algorithm is presented that allows changes in the preconditioning at every step, resulting in a result of the flexibility of the new variant that any iterative method can be used as a preconditionser. Expand