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Erschienen in:
2012 | OriginalPaper | Buchkapitel
On Stability of Discretizations of the Helmholtz Equation
verfasst von : S. Esterhazy, J. M. Melenk
Erschienen in: Numerical Analysis of Multiscale Problems
Verlag: Springer Berlin Heidelberg
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We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete
k
-explicit stability (including
k
-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size
h
and the approximation order
p
are selected such that
kh
∕
p
is sufficiently small and
p
=
O
(log
k
), and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation.