The impact of smooth W-grids in the numerical solution of singular perturbation two-point boundary value problems
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/2358567
- author
- Söderlind, Gustaf LU and Singh Yadaw, Arjun
- organization
- publishing date
- 2012
- 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
- additional info
- The information about affiliations in this record was updated in December 2015. The record was previously connected to the following departments: Numerical Analysis (011015004)
- id
- 9c898ed2-91ba-417c-b3e1-3a16e3f294d9 (old id 2358567)
- date added to LUP
- 2016-04-01 10:01:19
- date last changed
- 2022-01-25 19:00:33
@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 << 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.}},
author = {{Söderlind, Gustaf and Singh Yadaw, Arjun}},
issn = {{0096-3003}},
keywords = {{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}},
doi = {{10.1016/j.amc.2011.11.086}},
volume = {{218}},
year = {{2012}},
}