Abstract:
On the basis of piecewise quadratic interpolation, semi-analytical approximations of the double layer potential near and on the boundary of a two-dimensional domain are obtained. To calculate the integrals formed after the interpolation of the density function, exact integration with respect to the variable ρ=(r2−d2)1/2 is used, where d and r are the distances from the observed point to the boundary of the domain and to the boundary point of integration, respectively. The study proves the stable convergence of such approximations with the cubic velocity uniformly near the boundary of the class C5, and also on the boundary itself. It is also proved that the use of standard quadrature formulas for calculating the integrals does not violate the uniform cubic convergence of approximations of the direct value of the potential on the boundary of the class C6. With some simplifications, it is proved that the use of standard quadrature formulas for calculating the integrals entails the absence of uniform convergence of potential approximations inside the domain near any boundary point. The theoretical conclusions are confirmed by the results of the numerical solution of the Dirichlet problem for the Laplace equation in a circular domain.
Keywords:
quadrature formula, double layer potential, boundary element method, near singular integral, boundary layer effect, uniform convergence.
Citation:
Ivanov D.Yu., “On uniform convergence of approximations of the double layer potential near the boundary of a two-dimensional domain”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 32:1 (2022), 26–43
\Bibitem{Iva22}
\by Ivanov~D.Yu.
\paper On uniform convergence of approximations of the double layer potential near the boundary of a two-dimensional domain
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2022
\vol 32
\issue 1
\pages 26--43
\mathnet{http://mi.mathnet.ru/vuu797}
\crossref{https://doi.org/10.35634/vm220103}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4415768}
Linking options:
https://www.mathnet.ru/eng/vuu797
https://www.mathnet.ru/eng/vuu/v32/i1/p26
This publication is cited in the following 2 articles:
D. Yu. Ivanov, “On the Uniform Convergence of Approximations to the Tangential and Normal Derivatives of the Single-Layer Potential Near the Boundary of a Two-Dimensional Domain”, Comput. Math. and Math. Phys., 64:7 (2024), 1504
D. Yu. Ivanov, “On uniform convergence of semi-analytic solution of Dirichlet problem for dissipative Helmholtz equation in vicinity of boundary of two-dimensional domain”, Ufa Math. J., 15:4 (2023), 76–99