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Free Boundary Problems of Obstacle Type, a Numerical and Theoretical Study
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of five papers and it mainly addresses the theory and schemes to approximate the quadrature domains, QDs. The first deals with the uniqueness and some qualitative properties of the two QDs. The concept of two phase QDs, is more complicated than its one counterpart and consequently introduces significant and interesting open.

We present two numerical schemes to approach the one phase QDs in the paper. The first method is based on the properties of the free boundary the level set techniques. We use shape optimization analysis to construct second method. We illustrate the efficiency of the schemes on a variety of experiments.

In the third paper we design two finite difference methods for the approximation of the multi phase QDs. We prove that the second method enjoys monotonicity, consistency and stability and consequently it is a convergent scheme by Barles-Souganidis theorem. We also present various numerical simulations in the case of Dirac measures.

We introduce the QDs in a sub domain of and Rn study the existence and uniqueness along with a numerical scheme based on the level set method in the fourth paper.

In the last paper we study the tangential touch for a semi-linear problem. We prove that there is just one phase free boundary points on the flat part of the fixed boundary and it is also shown that the free boundary is a uniform C1-graph up to that part.

Abstract [sv]

Denna avhandling består av fem artiklar och behandlar främst teori och numeriska metoder för att approximera "quadrature domians", QDs. Den första artikeln behandlar entydighet och allmänna egenskaper hos tvåfas QDs. Begreppet tvåfas QDs, är mer komplicerat än enafasmotsvarigheten och introducerar därmed intressanta öppna problem.

Vi presenterar två numeriska metoder för att approximera enfas QDs i andra

artikeln. Den första metoden är baserad på egenskaperna hos den fria randen och nivå mängdmetoden. Vi använder forsoptimeringmanalys för att konstruera den andra metoden. Båda metoderna är testade i olika numeriska simuleringar.

I det tredje artikeln vi approximera flerafas QDs med konstruktionen tvåmetoder finita differens. Vi visar att den andra metoden har monotonicitat, konsistens och stabilitet och följaktligen är metoden konvergent tack vare Barles-Souganidis sats. Vi presenterar också olika numeriska simuleringar i fallet med Diracmåt.

Vi introducerar QDs i en delmängd av Rn och studerar existens och entydighet jämte en numerisk metod baserad på nivå mängdmetoden i fjärde pappret.

I det sista pappret studerar vi den tangentiella touchen för ett semilinjärt problem. Vi visar att det enbart är enafasrandpunkter på den platta delen av den fixerade randen. Vi visar också att den fria randen är en likformig C1-graf upp till den delen av den fixerade randen.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2012. , 67 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 79
Keyword [en]
Partial differential equations, Numerical analysis, Free boundary problems
National Category
Mathematical Analysis Computational Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-183393ISBN: 978-91-506-2316-1 (print)OAI: oai:DiVA.org:uu-183393DiVA: diva2:562616
Public defence
2012-12-14, Å80127, Ångström Laboratory, Lägerhyddsvägen 1, Uppsala, 13:00 (English)
Opponent
Supervisors
Available from: 2012-11-21 Created: 2012-10-25 Last updated: 2012-11-21Bibliographically approved
List of papers
1. Some properties of two-phase quadrature domains
Open this publication in new window or tab >>Some properties of two-phase quadrature domains
2011 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, no 10, 3386-3396 p.Article in journal (Refereed) Published
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-163099 (URN)10.1016/j.na.2011.02.014 (DOI)
Available from: 2012-02-07 Created: 2011-12-07 Last updated: 2017-12-08Bibliographically approved
2. Numerical Approximation of One Phase Quadrature Domains
Open this publication in new window or tab >>Numerical Approximation of One Phase Quadrature Domains
2013 (English)In: Numerical Methods for Partial Differential Equations, ISSN 0749-159X, E-ISSN 1098-2426, Vol. 29, no 5, 1709-1728 p.Article in journal (Other academic) Published
Abstract [en]

In this work, we present two numerical schemes for a free boundary problem called one phase quadrature domain. In the first method by applying the proprieties of given free boundary problem, we derive a method that leads to a fast iterative solver. The iteration procedure is adapted in order to work in the case when topology changes. The second method is based on shape reconstruction to establish an efficient Shape-Quasi-Newton-Method. Various numerical experiments confirm the efficiency of the derived numerical methods.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-170221 (URN)10.1002/num.21773 (DOI)000322203200013 ()
Available from: 2012-03-13 Created: 2012-03-09 Last updated: 2017-12-07Bibliographically approved
3. Numerical Schemes for Multi Phase Quadrature Domains
Open this publication in new window or tab >>Numerical Schemes for Multi Phase Quadrature Domains
2014 (English)In: International Journal of Numerical Analysis & Modeling, ISSN 1705-5105, Vol. 11, no 4, 726-744 p.Article in journal (Refereed) Published
Abstract [en]

In this work, numerical schemes to approximate the solution of one and multi phase quadrature domains are presented. We shall construct a monotone, stable and consistent finite difference method for both one and two phase cases, which converges to the viscosity solution of the partial differential equation arising from the corresponding quadrature domain theory. Moreover, we will discuss the numerical implementation of the resulting approach and present computational tests.

Keyword
Quadrature domain; Free boundary problem; Finite difference method; Degenerate elliptic equation
National Category
Mathematics
Research subject
Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-183391 (URN)000343624500004 ()
Available from: 2012-12-07 Created: 2012-10-25 Last updated: 2017-12-07Bibliographically approved
4. Quadrature domains in a subdomain of R^n, theory and a numerical approach
Open this publication in new window or tab >>Quadrature domains in a subdomain of R^n, theory and a numerical approach
(English)Manuscript (preprint) (Other academic)
Keyword
free boundary problems, quadrature domain, level set method
National Category
Mathematics
Research subject
Mathematics with specialization in Applied Mathematics; Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-183392 (URN)
Available from: 2012-12-07 Created: 2012-10-25 Last updated: 2012-12-07Bibliographically approved
5. Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions
Open this publication in new window or tab >>Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions
2014 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 52, no 1, 21-42 p.Article in journal (Refereed) Published
Abstract [en]

We study minimizers of the functional where B_{1}^{{\mathchoice {\raise .17ex\hbox {\scriptstyle +}} {\raise .17ex\hbox {\scriptstyle +}} {\raise .1ex\hbox {\scriptscriptstyle +}} {\scriptscriptstyle +}}}=\{x\in B_{1}: x_{1}>0\} , u=0 on {xB 1:x 1=0}, \lambda^{{\mathchoice {\raise .17ex\hbox {\scriptstyle \pm }} {\raise .17ex\hbox {\scriptstyle \pm }} {\raise .1ex\hbox {\scriptscriptstyle \pm }} {\scriptscriptstyle \pm }}} are two positive constants and 0<p<1. In two dimensions, we prove that the free boundary is a uniform C 1 graph up to the flat part of the fixed boundary and also that two-phase points cannot occur on this part of the fixed boundary. Here, the free boundary refers to the union of the boundaries of the sets {xu(x)>0}.

National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-170218 (URN)10.1007/s11512-012-0179-3 (DOI)000332797200003 ()
Available from: 2012-03-13 Created: 2012-03-09 Last updated: 2017-12-07Bibliographically approved

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