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The Finite Difference Method for Two-Phase Parabolic Obstacle-Like Problem: Like Problem
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
(English)Article in journal (Other academic) Submitted
##### Abstract [en]

In this paper for two-phase parabolic obstacle-like problem, $\Delta u -u_t=\lambda^+\cdot\chi_{\{u>0\}}-\lambda^-\cdot\chi_{\{u<0\}},\quad (t,x)\in (0,T)\times\Omega,$ where $T < \infty, \lambda^+ ,\lambda^- > 0$ are Lipschitz continuous functions, and $\Omega\subset\mathbb{R}^n$ is a bounded domain, we will introduce a certain variational form, which allows us to define a notion of viscosity solution. The uniqueness of viscosity solution is proved, and numerical nonlinear Gauss-Seidel method is constructed. Although the paper is devoted to the parabolic version of the two-phase obstacle-like problem, we prove convergence of discretized scheme to a unique viscosity solution for both two-phase parabolic obstacle-like and standard two-phase membrane problem. Numerical simulations are also presented.

Mathematics
##### Identifiers
OAI: oai:DiVA.org:kth-33575DiVA: diva2:416154
##### Note
QS 2011Available from: 2011-05-10 Created: 2011-05-10 Last updated: 2012-03-26Bibliographically approved
##### In thesis
1. The Finite Difference Methods for Multi-phase Free Boundary Problems
Open this publication in new window or tab >>The Finite Difference Methods for Multi-phase Free Boundary Problems
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis consist of an introduction and four research papers concerning numerical analysis for a certain class of free boundary problems.

Paper I is devoted to the numerical analysis of the so-called two-phase membrane problem. Projected Gauss-Seidel method is constructed. We prove general convergence of the algorithm as well as obtain the error estimate for the finite difference scheme.

In Paper II we have improved known results on the error estimates for a Classical Obstacle (One-Phase) Problem with a finite difference scheme.

Paper III deals with the parabolic version of the two-phase obstacle-like problem. We introduce a certain variational form, which allows us to definea notion of viscosity solution. The uniqueness of viscosity solution is proved, and numerical nonlinear Gauss-Seidel method is constructed.

In the last paper, we study a numerical approximation for a class of stationary states for reaction-diffusion system with m densities having disjoint support. The proof of convergence of the numerical method is given in some particular cases. We also apply our numerical simulations for the spatial segregation limit of diffusive Lotka-Volterra models in presence of high competition and inhomogeneous Dirichlet boundary conditions.

##### Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2011. vii, 23 p.
##### Series
Trita-MAT. MA, ISSN 1401-2278 ; 11:02
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-33543 (URN)
##### Public defence
2011-05-12, Sal F3., Lindstedtsvägen 26. KTH, Stocokholm, 13:00 (English)
##### Note
QC 20110510Available from: 2011-05-10 Created: 2011-05-10 Last updated: 2011-08-11Bibliographically approved

#### Open Access in DiVA

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http://arxiv.org/abs/1111.6287

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Arakelyan, Avetik
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