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General DG-Methods for Highly Indefinite Helmholtz Problems
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Journal Article
General DG-Methods for Highly Indefinite Helmholtz Problems
2013
Overview
We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in
R
d
,
d
∈
{
1
,
2
,
3
}
. The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the
hp
-version of the finite element method explicitly in terms of the mesh width
h
, polynomial degree
p
and wavenumber
k
. It is shown that the optimal convergence order estimate is obtained under the conditions that
kh
/
p
is sufficiently small and the polynomial degree
p
is at least
O
(
log
k
)
. On regular meshes, the first condition is improved to the requirement that
kh
/
p
be sufficiently small.
Publisher
Springer US,Springer Nature B.V
