An error estimate for the finite difference scheme for one-phase obstacle problem

Arakelyan, Avetik; Barkhudaryan, Rafayel; Poghosyan, Michael

doi:10.3103/s1068362311030022

Cited by 4 publications

(10 citation statements)

References 9 publications

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“…This together with Proposition 1 implies the following asymptotic decay: ||u l − u l h || L ∞ (Ω h ) = O(h 2 ). Similar convergence rates have been obtained in [3,17,25] for the difference schemes of one-phase obstacle-like problems.…”

Section: Propositionsupporting

confidence: 79%

Convergence of the finite difference scheme for a general class of the spatial segregation of reaction–diffusion systems

Arakelyan

2018

Computers & Mathematics with Applications

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In this work we prove convergence of the finite difference scheme for equations of stationary states of a general class of the spatial segregation of reaction-diffusion systems with m ≥ 2 components. More precisely, we show that the numerical solution u l h , given by the difference scheme, converges to the l th component u l , when the mesh size h tends to zero, provided u l ∈ C 2 (Ω), for every l = 1, 2, . . . , m. In particular, our proof provides convergence of a difference scheme for the multi-phase obstacle problem.2010 Mathematics Subject Classification. 35R35, 65N06, 65N22, 92D25.

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Section: Propositionsupporting

confidence: 79%

Convergence of the finite difference scheme for a general class of the spatial segregation of reaction–diffusion systems

Arakelyan

2018

Computers & Mathematics with Applications

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“…Here ∆x and ∆t are space-and time-discretization steps. Recently in the works [12,13,14,15] finite difference scheme and the convergence results have been applied for the one-phase and two-phase elliptic obstacle problems.…”

Section: Finite Difference Scheme For the Iterative Algorithmmentioning

confidence: 99%

“…This is nothing else but the difference between the exact solution and the difference scheme for a parabolic obstacle problem (25). In recent years there has been given much attention to these type of estimates (see [11,13,18,19]). We will mainly follow the above mentioned work [11], which considers the problem for American option valuation.…”

Section: Finite Difference Scheme For the Iterative Algorithmmentioning

confidence: 99%

Numerical Treatment to a Non-local Parabolic Free Boundary Problem Arising in Financial Bubbles

Arakelyan¹,

Barkhudaryan

Shahgholian

et al. 2018

Bull. Iran. Math. Soc.

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In this paper we continue to study a non-local free boundary problem arising in financial bubbles. We focus on the parabolic counterpart of the bubble problem and suggest an iterative algorithm which consists of a sequence of parabolic obstacle problems at each step to be solved, that in turn gives the next obstacle function in the iteration. The convergence of the proposed algorithm is proved. Moreover, we consider the finite difference scheme for this algorithm and obtain its convergence. At the end of the paper we present and discuss computational results.Here the coefficients a i,j , b i , c are assumed to be continues and the matrix [a i,j (x)] is positive definite for all x ∈ Ω. Additionally we assume that the coefficients are 2010 Mathematics Subject Classification. 35R35; 35D40; 65M06; 91G80.

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“…The interested reader is also referred to the recent works [8,24]. For the numerical treatment of the same problems we refer to the works [2,3,4,7,27].…”

Section: Numerical Algorithm and Its Propertiesmentioning

confidence: 99%

“…In [3,4] the authors introduced the so-called Min-Max formulation for the usual twophase obstacle problem, which is very useful to define the notion of viscosity solution. Moreover, it turns out that the viscosity solution is equivalent to the weak solution to the two-phase obstacle problem.…”

Section: Numerical Algorithm and Its Propertiesmentioning

confidence: 99%

A numerical approach for a general class of the spatial segregation of reaction–diffusion systems arising in population dynamics

Arakelyan

Barkhudaryan

2016

Computers & Mathematics with Applications

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In the current work we consider the numerical solutions of equations of stationary states for a general class of the spatial segregation of reaction-diffusion systems with m ≥ 2 population densities. We introduce a discrete multi-phase minimization problem related to the segregation problem, which allows to prove the existence and uniqueness of the corresponding finite difference scheme. Based on that scheme, we suggest an iterative algorithm and show its consistency and stability. For the special case m = 2, we show that the problem gives rise to the generalized version of the so-called two-phase obstacle problem. In this particular case we introduce the notion of viscosity solutions and prove convergence of the difference scheme to the unique viscosity solution. At the end of the paper we present computational tests, for different internal dynamics, and discuss numerical results.2000 Mathematics Subject Classification. 35R35, 65N06, 65N22, 92D25.

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An error estimate for the finite difference scheme for one-phase obstacle problem

Cited by 4 publications

References 9 publications

Convergence of the finite difference scheme for a general class of the spatial segregation of reaction–diffusion systems

Convergence of the finite difference scheme for a general class of the spatial segregation of reaction–diffusion systems

Numerical Treatment to a Non-local Parabolic Free Boundary Problem Arising in Financial Bubbles

A numerical approach for a general class of the spatial segregation of reaction–diffusion systems arising in population dynamics

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