TY - JOUR
T1 - Features of the Nyström Method for the Sherman-Lauricella Equation on Piecewise Smooth Contours
AU - Victor D. Didenko & Johan Helsing
JO - East Asian Journal on Applied Mathematics
VL - 4
SP - 403
EP - 414
PY - 2018
DA - 2018/02
SN - 1
DO - http://doi.org/10.4208/eajam.240611.070811a
UR - https://global-sci.org/intro/article_detail/eajam/10912.html
KW - Sherman-Lauricella equation, Nystrom method, stability.
AB - <p style="text-align: justify;">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 <span style="text-align: justify;">$A_{c_j}$</span> do not depend
on the shape of the contour, but on the opening angle $θ_j$ 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 $θ_j$ . In the
interval ($0.1π$, $1.9π$), it is found that there are 8 values of $θ_j$ where the invertibility of
the operator <span style="text-align: justify;">$A_{c_j}$</span> may fail, so the corresponding original Nyström method on any contour
with corner points of such magnitude cannot be stable and requires modification.</p>