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Efficient computation and applications of the Calderón projector

Scroggs, Matthew William; (2020) Efficient computation and applications of the Calderón projector. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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Abstract

The boundary element method (BEM) is a numerical method for the solution of partial differential equations through the discretisation of associated boundary integral equations.BEM formulations are commonly derived from properties of the Calderón projector, a blocked operator containing four commonly used boundary integral operators. In this thesis, we look in detail at the Calderón projector, derive and analyse a novel use of it to impose a range of boundary conditions, and look at how it can be efficiently computed. Throughout, we present computations made using the open-source software library Bempp, many features of which have been developed as part of this PhD. We derive a method for weakly imposing boundary conditions on BEM, inspired by Nitsche’s method for finite element methods. Formulations for Laplace problems with Dirichlet, Neumann, Robin, and mixed boudary conditions are derived and analysed. For Robin and mixed boundary conditions, the resulting formulations are simpler than standard BEM formulations, and convergence at a similar rate to standard methods is observed. As a more advanced application of this method, we derive a BEM formulation for Laplace’s equation with Signorini contact conditions. Using the weak imposition framework allows us to naturally impose this more complex boundary condition; the ability to do this is a significant advantage of this work. These formulations are derived and analysed, and numerical results are presented. Using properties of the Calderón projector, methods of operator preconditioning for BEM can be derived. These formulations involve the product of boundary operators. We present the details of a discrete operator algebra that allows the easy calculation of these products on the discrete level. This operator algebra allows for the easy implementation of various formulations of Helmholtz and Maxwell problems, including regularised combined field formulations that are immune to ill-conditioning near eigenvalues that are an issue for other formulations. We conclude this thesis by looking at weakly imposing Dirichlet and mixed Dirichlet–Neumann boundary condition on the Helmholtz equation. The theory for Laplace problems is extended to apply to Helmholtz problems, and an application to wave scattering from multiple scatterers is presented.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Efficient computation and applications of the Calderón projector
Event: UCL
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: © The Author 2020. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request.
UCL classification: UCL
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/10090630
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