A Refined Finite Element Convergence Theory for Highly Indefinite Helmholtz Problems
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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≲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≳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≲1 and optimal convergence estimates.
AMS Subject Classifications65N12 65N15 65N30
KeywordsIndefinite problems Helmholtz equation finite element methods generalized FEM
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