S. Esterhazy, J. Melenk:

"On Stability of Discretizations of the Helmholtz Equation";

in: "Numerical Analysis of Multiscale Problems", Springer Verlag, Berlin Heidelberg, 2012, ISBN: 978-3-642-22060-9.

We reviewthe stability properties of several discretizations of the Helmholtz

equation at large wavenumbers. For a model problem in a polygon, a complete kexplicit

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(logk), 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.

http://dx.doi.org/10.1007/978-3-642-22061-6_9

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