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Computational Science, Engineering & Technology Series
ISSN 1759-3158
Edited by: F. Magoulès
Chapter 5

Boundary Conditions and Iterative Schemes for the Helmholtz Equation in Unbounded Regions

E. Turkel

School of Mathematical Sciences, Tel Aviv University, Israel

Full Bibliographic Reference for this chapter
E. Turkel, "Boundary Conditions and Iterative Schemes for the Helmholtz Equation in Unbounded Regions", in F. Magoulès, (Editor), "Computational Methods for Acoustics Problems", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 5, pp 127-158, 2008. doi:10.4203/csets.18.5
Keywords: Helmholtz equation, Krylov methods, preconditioning, absorbing boundary conditions.

We consider the numerical solution of the Helmholtz equation in unbounded regions. Given an interior discretisation this requires the construction of an artificial surface to bound the domain of interest and the specification of a boundary condition that limits the reflections of waves back into the interior region. We discuss various options for this absorbing boundary condition. The resultant complex valued system of non-Hermitian equations needs to be solved. For high frequencies a direct solver is no longer feasible. The iterative method is usually a Krylov space technique. To speed the convergence a preconditioner is introduced. Various types of preconditioners are described.

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