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Features of the Nyström method for the Sherman-Lauricella equation on Piecewise Smooth Contours

Didenko, Victor and Helsing, Johan LU (2011) In East Asian Journal on Applied Mathematics 1(4). p.403-414
Abstract
The stability of the Nyström method for the Sherman-Lauricella equation on contours with corner points $c_j$, $j=0,1,...,m$ relies on the invertibility of certain operators $A_{c_j}$ belonging to an algebra of Toeplitz operators. The operators $A_{c_j}$ do not depend on the shape of the contour, but on the opening angle $\theta_j$ of the corresponding corner $c_j$ and on parameters of the approximation method mentioned. They have a complicated structure and there is no analytic tool to verify their invertibility. To study this problem, the original Nyström method is applied to the Sherman-Lauricella equation on a special model contour that has only one corner point with varying opening angle $\theta_j$. In the interval $(0.1\pi,1.9\pi)$,... (More)
The stability of the Nyström method for the Sherman-Lauricella equation on contours with corner points $c_j$, $j=0,1,...,m$ relies on the invertibility of certain operators $A_{c_j}$ belonging to an algebra of Toeplitz operators. The operators $A_{c_j}$ do not depend on the shape of the contour, but on the opening angle $\theta_j$ of the corresponding corner $c_j$ and on parameters of the approximation method mentioned. They have a complicated structure and there is no analytic tool to verify their invertibility. To study this problem, the original Nyström method is applied to the Sherman-Lauricella equation on a special model contour that has only one corner point with varying opening angle $\theta_j$. In the interval $(0.1\pi,1.9\pi)$, it is found that there are $8$ values of $\theta_j$ where the invertibility of the operator $A_{c_j}$ may fail, so the corresponding original Nyström method on any contour with corner points of such magnitude cannot be stable and requires modification. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Sherman-Lauricella equation, Nyström method, stability
in
East Asian Journal on Applied Mathematics
volume
1
issue
4
pages
403 - 414
publisher
Global Science Press
external identifiers
  • wos:000208793300004
  • scopus:84885899582
ISSN
2079-7370
DOI
10.4208/eajam.240611.070811a
language
English
LU publication?
yes
id
95f9318e-3d6d-4f67-9db3-a83379e575d8 (old id 2203901)
alternative location
http://www.maths.lth.se/na/staff/helsing/VJ2.pdf
date added to LUP
2011-12-28 16:51:47
date last changed
2017-03-15 09:39:35
@article{95f9318e-3d6d-4f67-9db3-a83379e575d8,
  abstract     = {The stability of the Nyström method for the Sherman-Lauricella equation on contours with corner points $c_j$, $j=0,1,...,m$ relies on the invertibility of certain operators $A_{c_j}$ belonging to an algebra of Toeplitz operators. The operators $A_{c_j}$ do not depend on the shape of the contour, but on the opening angle $\theta_j$ of the corresponding corner $c_j$ and on parameters of the approximation method mentioned. They have a complicated structure and there is no analytic tool to verify their invertibility. To study this problem, the original Nyström method is applied to the Sherman-Lauricella equation on a special model contour that has only one corner point with varying opening angle $\theta_j$. In the interval $(0.1\pi,1.9\pi)$, it is found that there are $8$ values of $\theta_j$ where the invertibility of the operator $A_{c_j}$ may fail, so the corresponding original Nyström method on any contour with corner points of such magnitude cannot be stable and requires modification.},
  author       = {Didenko, Victor and Helsing, Johan},
  issn         = {2079-7370},
  keyword      = {Sherman-Lauricella equation,Nyström method,stability},
  language     = {eng},
  number       = {4},
  pages        = {403--414},
  publisher    = {Global Science Press},
  series       = {East Asian Journal on Applied Mathematics},
  title        = {Features of the Nyström method for the Sherman-Lauricella equation on Piecewise Smooth Contours},
  url          = {http://dx.doi.org/10.4208/eajam.240611.070811a},
  volume       = {1},
  year         = {2011},
}