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The impact of smooth W-grids in the numerical solution of singular perturbation two-point boundary value problems

Söderlind, Gustaf LU and Singh Yadaw, Arjun (2012) In Applied Mathematics and Computation 218(10). p.6045-6055
Abstract
This paper develops a semi-analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two-point boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the W-grid, which depends on the perturbation parameter epsilon << 1. For problems on [0, 1] with a boundary layer at one end point, the local mesh width h(i) = x(i+1) - x(i), with 0 = x(0) < x(1) < ... < x(N) = 1, is condensed at either 0 or 1. Two simple 2nd order finite element and finite difference methods are combined with the new mesh, and computational experiments demonstrate the advantages of the smooth W-grid compared to the well-known piecewise... (More)
This paper develops a semi-analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two-point boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the W-grid, which depends on the perturbation parameter epsilon << 1. For problems on [0, 1] with a boundary layer at one end point, the local mesh width h(i) = x(i+1) - x(i), with 0 = x(0) < x(1) < ... < x(N) = 1, is condensed at either 0 or 1. Two simple 2nd order finite element and finite difference methods are combined with the new mesh, and computational experiments demonstrate the advantages of the smooth W-grid compared to the well-known piecewise uniform Shishkin mesh. For small epsilon, neither the finite difference method nor the finite element method produces satisfactory results on the Shishkin mesh. By contrast, accuracy is vastly improved on the W-grid, which typically produces the nominal 2nd order behavior in L(2), for large as well as small values of N, and over a wide range of values of epsilon. We conclude that the smoothness of the mesh is of crucial importance to accuracy, efficiency and robustness. Published by Elsevier Inc. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Singular perturbation, Boundary value problems, Finite difference, method, Galerkin method, Adaptive grid, W-grid, Grid density, Shishkin, mesh
in
Applied Mathematics and Computation
volume
218
issue
10
pages
6045 - 6055
publisher
Elsevier
external identifiers
  • wos:000298968300021
  • scopus:84655169792
ISSN
0096-3003
DOI
10.1016/j.amc.2011.11.086
language
English
LU publication?
yes
id
9c898ed2-91ba-417c-b3e1-3a16e3f294d9 (old id 2358567)
date added to LUP
2012-02-24 07:53:24
date last changed
2017-02-13 13:10:00
@article{9c898ed2-91ba-417c-b3e1-3a16e3f294d9,
  abstract     = {This paper develops a semi-analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two-point boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the W-grid, which depends on the perturbation parameter epsilon &lt;&lt; 1. For problems on [0, 1] with a boundary layer at one end point, the local mesh width h(i) = x(i+1) - x(i), with 0 = x(0) &lt; x(1) &lt; ... &lt; x(N) = 1, is condensed at either 0 or 1. Two simple 2nd order finite element and finite difference methods are combined with the new mesh, and computational experiments demonstrate the advantages of the smooth W-grid compared to the well-known piecewise uniform Shishkin mesh. For small epsilon, neither the finite difference method nor the finite element method produces satisfactory results on the Shishkin mesh. By contrast, accuracy is vastly improved on the W-grid, which typically produces the nominal 2nd order behavior in L(2), for large as well as small values of N, and over a wide range of values of epsilon. We conclude that the smoothness of the mesh is of crucial importance to accuracy, efficiency and robustness. Published by Elsevier Inc.},
  author       = {Söderlind, Gustaf and Singh Yadaw, Arjun},
  issn         = {0096-3003},
  keyword      = {Singular perturbation,Boundary value problems,Finite difference,method,Galerkin method,Adaptive grid,W-grid,Grid density,Shishkin,mesh},
  language     = {eng},
  number       = {10},
  pages        = {6045--6055},
  publisher    = {Elsevier},
  series       = {Applied Mathematics and Computation},
  title        = {The impact of smooth W-grids in the numerical solution of singular perturbation two-point boundary value problems},
  url          = {http://dx.doi.org/10.1016/j.amc.2011.11.086},
  volume       = {218},
  year         = {2012},
}