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A Singularly Perturbed Reaction-Diffusion Problem with Incompatible Boundary-Initial Data

  • J. L. Gracia
  • E. O’Riordan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)

Abstract

A singularly perturbed reaction-diffusion parabolic problem with an incompatibility between the initial and boundary conditions is examined. A finite difference scheme is considered which utilizes a special finite difference operator and a piecewise uniform Shishkin mesh. Numerical results are presented for both nodal and global pointwise convergence, using bilinear interpolation and, also, an interpolation method based on the error function. These results show that the method is not globally convergent when bilinear interpolation is used but they indicate that, for the test problem considered, it is globally convergent using the second type of interpolation.

Keywords

Test Problem Global Convergence Parabolic Problem Bilinear Interpolation Singularly Perturb 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Chen, Q., Qin, Z., Temam, R.: Treatment of incompatible initial and boundary data for parabolic equations in higher dimensions. Math. Comp. 80(276), 2071–2096 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust computational techniques for boundary layers. CRC Press (2000)Google Scholar
  3. 3.
    Gracia, J.L., O’Riordan, E.: A singularly perturbed parabolic problem with a layer in the initial condition, Appl. Math. Comp. 219, 498–510 (2012)MathSciNetGoogle Scholar
  4. 4.
    Hemker, P.W., Shishkin, G.I.: Approximation of parabolic PDEs with a discontinuous initial condition. East-West J. Numer. Math. 1, 287–302 (1993)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hemker, P.W., Shishkin, G.I.: Discrete approximation of singularly perturbed parabolic PDEs with a discontinuous initial condition. Comp. Fluid Dynamics J. 2, 375–392 (1994)Google Scholar
  6. 6.
    Stynes, M., O’Riordan, E.: A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem. J. Math. Anal. Appl. 214, 36–54 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zadorin, A.I., Zadorin, N.A.: Quadrature formulas for functions with a boundary layer component. Zh. Vychisl. Mat. Mat. Fiz. 51, 1952–1962 (2011) (Russian) ; Translation in Comput. Math. Math. Phys. 51, 1837–1846 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • J. L. Gracia
    • 1
    • 2
  • E. O’Riordan
    • 1
    • 2
  1. 1.IUMA. Department of Applied MathematicsUniversity of ZaragozaSpain
  2. 2.School of Mathematical SciencesDublin City UniversityIreland

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