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Geometrically unfitted finite element methods for the Helmholtz equation

Swift, Luke James; (2018) Geometrically unfitted finite element methods for the Helmholtz equation. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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Abstract

It is well known that the standard Galerkin finite element method experiences difficulties when applied to the Helmholtz equation in the medium to high wave number regime unless a condition of the form hk^2<C is satisfied, where h is the mesh parameter and k is the wave number. This condition becomes even more difficult to enforce when coupling multiple domains which may have different wave numbers. Numerous stabilizations have been proposed in order to make computations under the engineering rule of thumb hk<C feasible. In this work I introduce a theoretical framework for analysing a class of stabilized finite element methods. I introduce two stabilized methods that both have an absolute stability property when considering Dirichlet or Neumann boundary conditions without condition on the mesh parameter or wave number and are shown to be stable for an appropriate choice of stabilization parameters when considering impedance boundary conditions. Numerically, I observ e a reduction of the pollution error for given problems provided the stabilization parameters are chosen appropriately. These stabilizations are then extended to a fictitious domain setting using cut elements and finally a multi-domain setting. The stabilizations proposed previously are used in the bulk of each domain and the coupling at the interface is handled using Nitsche's method. The coupling parameters are chosen to be complex with positive or negative imaginary part depending on the sign in the Robin boundary condition. The new fitted and unfitted domain decomposition methods have similar properties to the original stabilized methods and both enter a similar theoretical framework.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Geometrically unfitted finite element methods for the Helmholtz equation
Event: UCL (University College London)
Open access status: An open access version is available from UCL Discovery
Language: English
UCL classification: UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
URI: https://discovery.ucl.ac.uk/id/eprint/10042813
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