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 k2h \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.