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.
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References
Andreev, V.: On the accuracy of grid approximation to nonsmooth solutions of a singularly perturbed reaction-diffusion equation in the square. Differential Equations 42, 954–966 (2006) (in Russian)
Angelova, I., Vulkov, L.: Singularly perturbed differential equations with discontinuous coefficients and concentrated factors. Appl. Math. Comp. 158, 683–701 (2004)
Angelova, I., Vulkov, L.: High-Order Difference Schemes for Elliptic Problems with Intersecting Interfaces. Appl. Math. Comp. 158(3), 683–701 (2007)
Angelova, I., Vulkov, L.: Marchuk identity-type second order difference schemes of 2-D and 3-D elliptic problems with intersected interfaces. In: Krag. J. Math (2007)
Braianov, I., Vulkov, L.: Numerical solution of a reaction-diffusion elliptic interface problem with strong anisotropy. Computing 71(2), 153–173 (2003)
Clavero, C., Gracia, J.: A compact finite difference scheme for 2D reaction-diffusion singularly perturbed problems. JCAM 192, 152–167 (2006)
Clavero, C., Gracia, J., O’Riordan, E.: A parameter robust numerical method for a two dimensionsl reaction-diffusion problem. Math. Comp. 74, 1743–1758 (2005)
Dunne, R., O’Riordan, E.: Interior layers arising in linear singularly perturbed differential equations with discontinuous coefficients, MS-06-09, 1–23 (2006)
Grisvard, P.: Boundary value Problems in Non-smooth Domains, Pitman, London (1985)
Han, H., Kellog, R.: Differentiability properties of solutions of the equation \(-\varepsilon ^{2} \triangle u +ru = f(x,y)\) in a square. SIAM J. Math. Anal. 21, 394–408 (1990)
Jovanović, B., Vulkov, L.: Regularity and a priori estimates for solutions of an elliptic problem with a singular source. In: J. of Diff. Eqns. (submitted)
Li, J., Wheller, M.: Uniform superconvergence of mixed finite element methods on anisotropically refined grids. SINUM 38(3), 770–798 (2000)
Miller, J., O’Riordan, E., Shishkin, G.: Fitted numerical methods for singular perturbation problems. World-Scientific, Singapore (1996)
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Angelova, I.T., Vulkov, L.G. (2008). Uniform Convergence of Finite-Difference Schemes for Reaction-Diffusion Interface Problems. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2007. Lecture Notes in Computer Science, vol 4818. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78827-0_75
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DOI: https://doi.org/10.1007/978-3-540-78827-0_75
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78825-6
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