The impact of smooth Wgrids in the numerical solution of singular perturbation twopoint boundary value problems
(2012) In Applied Mathematics and Computation 218(10). p.60456055 Abstract
 This paper develops a semianalytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed twopoint boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the Wgrid, 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 Wgrid compared to the wellknown piecewise... (More)
 This paper develops a semianalytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed twopoint boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the Wgrid, 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 Wgrid compared to the wellknown 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 Wgrid, 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:
http://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, Wgrid, 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
 00963003
 DOI
 10.1016/j.amc.2011.11.086
 language
 English
 LU publication?
 yes
 id
 9c898ed291ba417cb3e13a16e3f294d9 (old id 2358567)
 date added to LUP
 20120224 07:53:24
 date last changed
 20190730 01:06:28
@article{9c898ed291ba417cb3e13a16e3f294d9, abstract = {This paper develops a semianalytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed twopoint boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the Wgrid, 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 Wgrid compared to the wellknown 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 Wgrid, 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 = {00963003}, keyword = {Singular perturbation,Boundary value problems,Finite difference,method,Galerkin method,Adaptive grid,Wgrid,Grid density,Shishkin,mesh}, language = {eng}, number = {10}, pages = {60456055}, publisher = {Elsevier}, series = {Applied Mathematics and Computation}, title = {The impact of smooth Wgrids in the numerical solution of singular perturbation twopoint boundary value problems}, url = {http://dx.doi.org/10.1016/j.amc.2011.11.086}, volume = {218}, year = {2012}, }