Stabilised fine element methods for fictitious domain problems

  • Cheherazada Gonzalez Aguayo

Student thesis: Doctoral Thesis

Abstract

This thesis deals with the solution of the Laplace and heat equations on complicated domains. The approach follows the idea of the fictitious domain method, in which a larger (simpler) domain is introduced with the idea of avoiding the use of meshes that resolve the geometry. The first part of the thesis is dedicated to propose and analyse a new stabilised finite element method for the heat equation. The analysis, not available to date, is based on the introduction of a new projected initial condition that satisfies the boundary conditions of the original problem weakly. This allows us to prove inconditional stability and optimal convergence of the solution, thus avoiding the restriction linking the time discretisation and mesh width parameters present in previous references. In the second part of this thesis the methodology has been adapted and extended to cover the case in which the problem at hand is posed in a domain containing several inclusions of small size. For this case, the usual fictitious domain approach is no longer applicable, and then a new method that compensates for the lack of stability of the original one is proposed, analysed and tested numerically. The numerical analysis has been carried out for the steady state case, but its applicability to time dependent problems is sketched and shown by means of numerical experiments.
Date of Award24 Mar 2017
Original languageEnglish
Awarding Institution
  • University Of Strathclyde
SponsorsUniversity of Strathclyde & EPSRC (Engineering and Physical Sciences Research Council)
SupervisorGabriel Barrenechea (Supervisor) & John MacKenzie (Supervisor)

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