Stabilised fine element methods for fictitious domain problems

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This thesis deals with the solution of the Laplace and heat equations on complicateddomains. The approach follows the idea of the fictitious domain method, in which alarger (simpler) domain is introduced with the idea of avoiding the use of meshes thatresolve the geometry.The first part of the thesis is dedicated to propose and analyse a new stabilised finiteelement method for the heat equation. The analysis, not available to date, is basedon the introduction of a new projected initial condition that satisfies the boundaryconditions of the original problem weakly. This allows us to prove inconditional stabilityand optimal convergence of the solution, thus avoiding the restriction linking the timediscretisation and mesh width parameters present in previous references.In the second part of this thesis the methodology has been adapted and extended tocover the case in which the problem at hand is posed in a domain containing severalinclusions of small size. For this case, the usual fictitious domain approach is no longerapplicable, and then a new method that compensates for the lack of stability of theoriginal one is proposed, analysed and tested numerically. The numerical analysis hasbeen carried out for the steady state case, but its applicability to time dependentproblems is sketched and shown by means of numerical experiments.

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Miniaturansicht	Titel	Hochladedatum	Sichtbarkeit	Aktionen
	PDF of thesis T14694	2021-07-02	Öffentlich	Herunterladen