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Achieng, Pauline
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Achieng, Pauline
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Mathematics and Applied MathematicsFaculty of Science & Engineering
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Computational Mathematics
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Analysis of the Robin-Dirichlet iterative procedure for solving the Cauchy problem for elliptic equations with extension to unbounded domainsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2020 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Linköping: Linköping University Electronic Press, 2020. , p. 10
##### Series

Linköping Studies in Science and Technology. Licentiate Thesis, ISSN 0280-7971 ; 1891
##### National Category

Computational Mathematics
##### Identifiers

URN: urn:nbn:se:liu:diva-170835DOI: 10.3384/lic.diva-170835ISBN: 9789179297565 (print)OAI: oai:DiVA.org:liu-170835DiVA, id: diva2:1479114
##### Presentation

2020-12-03, Planck, F-building, entrance 57, Campus Valla, Linköping, 10:00 (English)
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##### Note

##### List of papers

In this thesis we study the Cauchy problem for elliptic equations. It arises in many areas of application in science and engineering as a problem of reconstruction of solutions to elliptic equations in a domain from boundary measurements taken on a part of the boundary of this domain. The Cauchy problem for elliptic equations is known to be ill-posed.

We use an iterative regularization method based on alternatively solving a sequence of well-posed mixed boundary value problems for the same elliptic equation. This method, based on iterations between Dirichlet-Neumann and Neumann-Dirichlet mixed boundary value problems was first proposed by Kozlov and Maz’ya [13] for Laplace equation and Lame’ system but not Helmholtz-type equations. As a result different modifications of this original regularization method have been proposed in literature. We consider the Robin-Dirichlet iterative method proposed by Mpinganzima et.al [3] for the Cauchy problem for the Helmholtz equation in bounded domains.

We demonstrate that the Robin-Dirichlet iterative procedure is convergent for second order elliptic equations with variable coefficients provided the parameter in the Robin condition is appropriately chosen. We further investigate the convergence of the Robin-Dirichlet iterative procedure for the Cauchy problem for the Helmholtz equation in a an unbounded domain. We derive and analyse the necessary conditions needed for the convergence of the procedure.

In the numerical experiments, the precise behaviour of the procedure for different values of k^{2} in the Helmholtz equation is investigated and the results show that the speed of convergence depends on the choice of the Robin parameters, μ_{0} and μ_{1}. In the unbounded domain case, the numerical experiments demonstrate that the procedure is convergent provided that the domain is truncated appropriately and the Robin parameters, μ_{0} and μ_{1} are also chosen appropriately.

Funding Agencies: International Science Programme (ISP) and the Eastern Africa Universities Mathematics Programme (EAUMP).

Available from: 2020-10-26 Created: 2020-10-26 Last updated: 2020-10-28Bibliographically approved1. Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations$(function(){PrimeFaces.cw("OverlayPanel","overlay1479111",{id:"formSmash:j_idt566:0:j_idt570",widgetVar:"overlay1479111",target:"formSmash:j_idt566:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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