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Volume 16, Issue 1
An Accurate Numerical Solution of a Two Dimensional Heat Transfer Problem with a Parabolic Boundary Layer

C. Clavero, J.J.H. Miller, E. O'Riordan & G.I. Shishkin

J. Comp. Math., 16 (1998), pp. 27-39.

Published online: 1998-02

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  • Abstract

A singularly perturbed linear convection-diffusion problem for heat transfer in two dimensions with a parabolic boundary layer is solved numerically. The numerical method consists of a special piecewise uniform mesh condensing in a neighbourhood of the parabolic layer and a standard finite difference operator satisfying a discrete maximum principle. The numerical computations demonstrate numerically that the method is $ε$-uniform in the sense that the rate of convergence and error constant of the method are independent of the singular perturbation parameter $ε$. This means that no matter how small the singular perturbation parameter $ε$ is, the numerical method produces solutions with guaranteed accuracy depending solely on the number of mesh points used. 

  • Keywords

Linear convection-diffusion, parabolic layer, piecewise uniform mesh, finite difference.

  • AMS Subject Headings

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

jm@incaireland.org (J.J.H. Miller)

eugene.oriordan@dcu.ie (E. O'Riordan)

  • BibTex
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  • TXT
@Article{JCM-16-27, author = {Clavero , C.Miller , J.J.H.O'Riordan , E. and Shishkin , G.I.}, title = {An Accurate Numerical Solution of a Two Dimensional Heat Transfer Problem with a Parabolic Boundary Layer}, journal = {Journal of Computational Mathematics}, year = {1998}, volume = {16}, number = {1}, pages = {27--39}, abstract = {

A singularly perturbed linear convection-diffusion problem for heat transfer in two dimensions with a parabolic boundary layer is solved numerically. The numerical method consists of a special piecewise uniform mesh condensing in a neighbourhood of the parabolic layer and a standard finite difference operator satisfying a discrete maximum principle. The numerical computations demonstrate numerically that the method is $ε$-uniform in the sense that the rate of convergence and error constant of the method are independent of the singular perturbation parameter $ε$. This means that no matter how small the singular perturbation parameter $ε$ is, the numerical method produces solutions with guaranteed accuracy depending solely on the number of mesh points used. 

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9139.html} }
TY - JOUR T1 - An Accurate Numerical Solution of a Two Dimensional Heat Transfer Problem with a Parabolic Boundary Layer AU - Clavero , C. AU - Miller , J.J.H. AU - O'Riordan , E. AU - Shishkin , G.I. JO - Journal of Computational Mathematics VL - 1 SP - 27 EP - 39 PY - 1998 DA - 1998/02 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9139.html KW - Linear convection-diffusion, parabolic layer, piecewise uniform mesh, finite difference. AB -

A singularly perturbed linear convection-diffusion problem for heat transfer in two dimensions with a parabolic boundary layer is solved numerically. The numerical method consists of a special piecewise uniform mesh condensing in a neighbourhood of the parabolic layer and a standard finite difference operator satisfying a discrete maximum principle. The numerical computations demonstrate numerically that the method is $ε$-uniform in the sense that the rate of convergence and error constant of the method are independent of the singular perturbation parameter $ε$. This means that no matter how small the singular perturbation parameter $ε$ is, the numerical method produces solutions with guaranteed accuracy depending solely on the number of mesh points used. 

Clavero , C.Miller , J.J.H.O'Riordan , E. and Shishkin , G.I.. (1998). An Accurate Numerical Solution of a Two Dimensional Heat Transfer Problem with a Parabolic Boundary Layer. Journal of Computational Mathematics. 16 (1). 27-39. doi:
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