Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

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dc.contributor.advisorStynes, Martinen
dc.contributor.authorSun, Guangfu
dc.date.accessioned2014-08-27T15:41:28Z
dc.date.available2014-08-27T15:41:28Z
dc.date.issued1993
dc.date.submitted1993
dc.description.abstractThis thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.en
dc.description.statusNot peer revieweden
dc.description.versionAccepted Version
dc.format.mimetypeapplication/pdfen
dc.identifier.citationSun, G. 1993. Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations. PhD Thesis, University College Cork.en
dc.identifier.urihttps://hdl.handle.net/10468/1636
dc.language.isoenen
dc.publisherUniversity College Corken
dc.relation.urihttp://library.ucc.ie/record=b1212273
dc.rights© 1993, Guangfu Sunen
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/3.0/en
dc.subjectHigh order problem of reaction-diffusion typeen
dc.subjectHigh order problem of convection-diffusion typeen
dc.subjectSecond order interior turning point problemen
dc.subjectSemilinear reaction-diffusion problemen
dc.subject.lcshDifferential equations.en
dc.thesis.opt-outfalse
dc.titleUniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equationsen
dc.typeDoctoral thesisen
dc.type.qualificationlevelDoctoralen
dc.type.qualificationnamePhD (Science)en
ucc.workflow.supervisorcora@ucc.ie
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