# A refined finite element convergence theory for highly indefinite Helmholtz problems.

## Stefan Sauter

It is well known that standard *h*-version finite element
discretisations for
Helmholtz' equation suffer from the following stability condition:
"The mesh width *h* of the finite element mesh has to
satisfy *k ^{2}h \lesssim 1 *", where

*k*denotes the wave number. This condition rules out the reliable numerical solution of Helmholtz equation in three dimensions for large wave numbers

*k \gtrsim 50*.

In our talk, we will present a refined finite element theory for highly indefinite Helmholtz problems where the stability of the discretisation can be checked through an "almost invariance" condition.

As an application, we will consider a one-dimensional finite element
space for
the Helmholtz equation and apply our theory to prove stability under the
weakened condition *h k \lesssim 1* and optimal convergence estimates.