A refined finite element convergence theory for highly indefinite Helmholtz problems
It is well known that standard h-version finite element discretisations using lowest order elements for Helmholtz' equation suffer from the following stability condition: "The mesh width h of the finite element mesh has to satisfy k(2)h less than or similar to 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 greater than or similar to 50. In our paper. 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 hk less than or similar to 1 and optimal convergence estimates.
helmholtz equation indefinite problems finite element methods generalized fem