Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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AN SDFEM FOR A CONVECTION-DIFFUSION PROBLEM WITH NEUMANN BOUNDARY CONDITION AND DISCONTINUOUS SOURCE TERM

  • Babu, A. Ramesh (Department of Mathematics, School of Mathematical Sciences, Bharathidasn University, Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA University) ;
  • Ramanujam, N. (Department of Mathematics, School of Mathematical Sciences, Bharathidasn University)
  • Published : 2010.01.30
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Abstract

In this article, we consider singularly perturbed Boundary Value Problems(BVPs) for second order Ordinary Differential Equations (ODEs) with Neumann boundary condition and discontinuous source term. A parameter-uniform error bound for the solution is established using the Streamline-Diffusion Finite Element Method (SDFEM) on a piecewise uniform meshes. We prove that the method is almost second order of convergence in the maximum norm, independently of the perturbation parameter. Further we derive superconvergence results for scaled derivatives of solution of the same problem. Numerical results are provided to substantiate the theoretical results.

Keywords

References

  1. T. J. R. Hughes and A. Brooks, A multidimensional upwind scheme with no crosswind diffusion. In T.J. R. Hughes, editor, Finite Element Methods for convection dominated Flows, pages 19-35. AMD, Vo1.34,ASME, New York, 1979.
  2. P. A. Farrell, A. F. Hegarty, E. O'Riordan, G. I. Shishkin, Robust computational techniques for boundary layers, Chapman Hall/ CRC, Boca Raton, 2000.
  3. H-G. Roos, T. LinB, Sufficient conditions for uniform convergence on layer-adapted grids,Computing 63 (1999) 27-45. https://doi.org/10.1007/s006070050049
  4. H-G. Roos, M. Stynes, L. Tobiska, Numerical methods for singularly perturbed differential equations, Volume 24 of Springer series in Computational Mathematics (2nd Edition), Springer-Verlag, Berlin, 2008.
  5. H-G. Roos, Helena Zarin, A second-order scheme for singularly perturbed differential equations with discontinuous source term, J.Numer. Math. ,Vol. 10, No. 4 (2002) 275-289. https://doi.org/10.1515/JNMA.2002.275
  6. P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan, G. I. Shishkin Global maximum norm parameter-uniform numerical method for a singularly perturbed convection- diffusion problem with discontinuous convection- coeffient, Mathematical and Computer Modelling., Vol. 40 (2005) 1375-1390.
  7. Zhongdi Cen, A hybrid scheme for a singularly perturbed convection-diffusion problem with discontinuous convection coeffient, Applied Mathematics and Computation, 169, (2005), 689-699. https://doi.org/10.1016/j.amc.2004.08.051
  8. P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O'Riordan, G.I. Shishkin, Singularly perturbed convection diffusion problems with boundary and weak interior layers, Journal of Computational and Applied Mathematics, v. 166, n. 1, 2004, 133-151. https://doi.org/10.1016/j.cam.2003.09.033
  9. H.-G. Roos, Torsten LinB, Gradient Recoverν for Singularly Perturbed Boundary value Problems I: One-Dimensional Convection-Diffuaion , Computing 66 (2001) , 163 - 178. https://doi.org/10.1007/s006070170033
  10. E.P. Doolan, J.J.H.Miller, W. H. A. Schilders, Uniform numerical methods for problemswith initial and boundary layers, Boole, Dublin, 1980.
  11. H-G. Roos, Helena Zarin, The streamline-diffusion method for a convection-diffusion problem with a point source, J. Comp. Appl. Math. ,Vol.10, No.4(2002) 275-289.
  12. V. B. Andreev, The Green function and A priori estimates of solutions of monotone three-point singularly perturbed finite-difference schemes, Numerical Methods, Differential Equations, Vol. 37, No. 7, (2001), 923-933. https://doi.org/10.1023/A:1011949419389
  13. A. H. Nayfeh, Introduction to Perturbation Methods, Wiley, New York, 1981.
  14. H. Han and R. B. Kellog, Differentiability properties of solution of the equation $-\varepsilon^{2}{\Delta}u+ru = f(x,y)$, SIAM Journal of Mathematical Anaysis, Vol. 21, No. 2, (1990), 394-408. https://doi.org/10.1137/0521022