Geodesic
On uniform convergence of semi-analytic solution of Dirichlet problem for dissipative Helmholtz equation in vicinity of boundary of two-dimensional domain
Ufa mathematical journal, Tome 15 (2023) no. 4, pp. 76-99

Voir la notice de l'article provenant de la source Math-Net.Ru

In the framework of the collocation boundary element method, we propose a semi-analytic approximation of the double-layer potential, which ensures a uniform cubic convergence of the approximate solution to the Dirichlet problem for the Helmholtz equation in a two-dimensional bounded domain or its exterior with a boundary of class $C^5$. In order to calculate integrals on boundary elements, an exact integration over the variable $\rho:=(r^2-d^2)^{1/2}$ is used, where $r$ and $d$ are the distances from the observed point to integration point and to the boundary of the domain, respectively. Under some simplifications we prove that the use of a number of traditional quadrature formulas leads to a violation of the uniform convergence of potential approximations in the vicinity of the boundary of the domain. The theoretical conclusions are confirmed by a numerical solving of the problem in a circular domain.
Keywords: double layer potential, Dirichlet problem, Helmholtz equation, boundary integral equation, almost singular integral, boundary layer phenomenon
Mots-clés : quadrature formula, uniform convergence. 
@article{UFA_2023_15_4_a5,
     author = {D. Yu. Ivanov},
     title = {On uniform convergence of semi-analytic solution of {Dirichlet} problem for dissipative {Helmholtz} equation in vicinity of boundary of two-dimensional domain},
     journal = {Ufa mathematical journal},
     pages = {76--99},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2023_15_4_a5/}
}
TY  - JOUR
AU  - D. Yu. Ivanov
TI  - On uniform convergence of semi-analytic solution of Dirichlet problem for dissipative Helmholtz equation in vicinity of boundary of two-dimensional domain
JO  - Ufa mathematical journal
PY  - 2023
SP  - 76
EP  - 99
VL  - 15
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/UFA_2023_15_4_a5/
LA  - en
ID  - UFA_2023_15_4_a5
ER  - 
%0 Journal Article
%A D. Yu. Ivanov
%T On uniform convergence of semi-analytic solution of Dirichlet problem for dissipative Helmholtz equation in vicinity of boundary of two-dimensional domain
%J Ufa mathematical journal
%D 2023
%P 76-99
%V 15
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/UFA_2023_15_4_a5/
%G en
%F UFA_2023_15_4_a5
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 mathematical journal, Tome 15 (2023) no. 4, pp. 76-99. http://geodesic.mathdoc.fr/item/UFA_2023_15_4_a5/