Singularly perturbed differential equations with discontinuous coefficients and concentrated factors

Angelova, Ivanka Tr.; Vulkov, Lubin G.

doi:10.1016/j.amc.2003.10.008

Cited by 10 publications

(6 citation statements)

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“…∞ is the global pointwise maximum norm and C is a constant independent of ε and N . In general, the gradients of the solution [2,3,13] become unbounded in the boundary/interface and corner layers as ε → 0; however, parameter-uniform numerical methods guarantee that the error in the numerical approximation is controlled solely by the size of N . Let us consider the boundary value problems for the reaction-diffusion model: where Ω = (−1, 1) × (0, 1), the diagonal matrix…”

Section: Introductionmentioning

confidence: 99%

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Uniform Convergence of Finite-Difference Schemes for Reaction-Diffusion Interface Problems

Angelova

Vulkov

2008

Large-Scale Scientific Computing

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Abstract. We consider a singularly perturbed reaction-diffusion equation in two dimensions (x, y) with concentrated source on a segment parallel to axis Oy. By means of an appropriate (including corner layer functions) decomposition, we describe the asymptotic behavior of the solution. Finite difference schemes for this problem of second and fourth order of local approximation on Shishkin mesh are constructed. We prove that the first scheme is almost second order uniformly convergent in the maximal norm. Numerical experiments illustrate the theoretical order of convergence of the first scheme and almost fourth order of convergence of the second scheme.2000 Mathematics Subject Classification: 65M06, 65M12.

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Section: Introductionmentioning

confidence: 99%

“…[1,2,5,6,7,8,9,10,12,13] and the references therein. Our interest lies in examine parameter-uniform numerical methods [13] of high order for singularly perturbed interface problems.…”

Section: Introductionmentioning

confidence: 99%

Uniform Convergence of Finite-Difference Schemes for Reaction-Diffusion Interface Problems

Angelova

Vulkov

2008

Large-Scale Scientific Computing

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“…In this paper, we consider the assumptions below to guarantee the interior layer of the solution to problem (1.1) and (1.2) at x = 0, ∀ t ∈ [0, T ],

\{\begin{matrix} left \\ a (0, t) = 0, & a_{x} (0, t) > 0, t \in [0, T], \\ |a_{x} (x, t)| \geq \frac{(||, a_{x} ((), 0, t))}{2}, & x \in [- 1, 1], t \in [0, T], \\ \frac{b (0, t)}{a_{x} (0, t)} > 0, & x \in t \in [0, T], \\ b (x, t) \geq b (0, t) > 0, & x \in [- 1, 1], t \in [0, T] . \end{matrix}

The interior layers may also originate from discontinuous data .…”

Section: Introductionmentioning

confidence: 99%

A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer

Munyakazi

Patidar

Sayi

2019

Numerical Methods Partial

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The objective of this paper is to construct and analyze a fitted operator finite difference method (FOFDM) for the family of time-dependent singularly perturbed parabolic convection-diffusion problems. The solution to the problems we consider exhibits an interior layer due to the presence of a turning point. We first establish sharp bounds on the solution and its derivatives. Then, we discretize the time variable using the classical Euler method. This results in a system of singularly perturbed interior layer two-point boundary value problems. We propose a FOFDM to solve the system above. Through a rigorous error analysis, we show that the scheme is uniformly convergent of order one with respect to both time and space variables. Moreover, we apply Richardson extrapolation to enhance the accuracy and the order of convergence of the proposed scheme. Numerical investigations are carried out to demonstrate the efficacy and robustness of the scheme. KEYWORDS error bounds, finite difference methods, interior layer, singularly perturbed problems, uniform convergence 1 Numer Methods Partial Differential Eq. 2019;35:2407-2422. wileyonlinelibrary.com/journal/num

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“…The interface problems are objects of intensive investigations and numerical methods construction during the past years, see [1,2,3,4,5,9,10,11,12,13,14,18] and references given there. In [1], the solution of a general interface problem is reduced to the solution of simpler interface problems of type (PC), (OCS).…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

“…In [1], the solution of a general interface problem is reduced to the solution of simpler interface problems of type (PC), (OCS). Conservative difference schemes are studied in [2,10], while the immersed interface method is developed in [12,14].…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

Finite Element Solution of Boundary Value Problems With Nonlocal Jump Conditions

Koleva¹

2008

Mathematical Modelling and Analysis

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Abstract. We consider stationary linear problems on non-connected layers with distinct material properties. Well posedness and the maximum principle (MP) for the differential problems are proved. A version of the finite element method (FEM) is used for discretization of the continuous problems. Also, the MP and convergence for the discrete solutions are established. An efficient algorithm for solution of the FEM algebraic equations is proposed. Numerical experiments for linear and nonlinear problems are discussed.

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Singularly perturbed differential equations with discontinuous coefficients and concentrated factors

Cited by 10 publications

References 5 publications

Uniform Convergence of Finite-Difference Schemes for Reaction-Diffusion Interface Problems

Uniform Convergence of Finite-Difference Schemes for Reaction-Diffusion Interface Problems

A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer

Finite Element Solution of Boundary Value Problems With Nonlocal Jump Conditions

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