This article is cited in 10 scientific papers (total in 10 papers)
MATHEMATICS
A refinement of the boundary element collocation method near the boundary of domain in the case of two-dimensional problems of non-stationary heat conduction with boundary conditions of the second and third kind
Abstract:
In this paper, we consider initial-boundary value problems (IBVPs) for the equation $\partial_tu=a^2\Delta_2u-pu$ with constants $a,p>0$ in an open two-dimensional spatial domain $\Omega$ with boundary conditions of the second and third kind at a zero initial condition. A fully justified collocation boundary element method is proposed, which makes it possible to obtain uniformly convergent in the space-time domain $\Omega\times[0,T]$ approximate solutions of the abovementioned IBVPs. The solutions are found in the form of the single-layer potential with unknown density functions determined from boundary integral equations of the second kind.
To ensure the uniform convergence, integration on arc-length $s$ when calculating the potential operator is carried out in two ways. If the distance $r$ from the point $x\in\Omega$ at which the potential is calculated to the integration point $x'\in\partial\Omega$ does not exceed approximately one-third of the radius of the Lyapunov circle $R_{\text{Л}}$, then we use exact integration with respect to a certain component $\rho$ of the distance $r:\,\rho\equiv(r^2-d^2)^{\frac12}$ ($d$ is the distance from the point $x\in\Omega$ to the boundary $\partial\Omega$). This exact integration is practically feasible for any analytically defined curve $\partial\Omega$. In this integration, functions of the variable $\rho$ are taken as the weighting functions and the rest of the integrand is approximated by quadratic interpolation on $\rho$. The functions of $\rho$ are generated by the fundamental solution of the heat equation. The integrals with respect to $s$ for $r>R_{\text{Л}}/3$ are calculated using Gaussian quadrature with $\gamma$ points.
Under the condition $\partial\Omega\in C^5\cap C^{2\gamma}$ ($\gamma\geqslant2$), it is proved that the approximate solutions converge to an exact one with a cubic velocity uniformly in the domain $\Omega\times [0, T]$. It is also proved that the approximate solutions are stable to perturbations of the boundary function uniformly in the domain $\Omega\times [0, T]$. The results of computational experiments on the solution of the IBVPs in a circular spatial domain are presented. These results show that the use of the exact integration with respect to $\rho$ can substantially reduce the decrease in the accuracy of numerical solutions near the boundary $\partial\Omega$, in comparison with the use of exclusively Gauss quadratures in calculating the potential.
Citation:
Ivanov D.Yu., “A refinement of the boundary element collocation method near the boundary of domain in the case of two-dimensional problems of non-stationary heat conduction with boundary conditions of the second and third kind”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2019, no. 57, 5–25
\Bibitem{Iva19}
\by Ivanov~D.Yu.
\paper A refinement of the boundary element collocation method near the boundary of domain in the case of two-dimensional problems of non-stationary heat conduction with boundary conditions of the second and third kind
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2019
\issue 57
\pages 5--25
\mathnet{http://mi.mathnet.ru/vtgu686}
\crossref{https://doi.org/10.17223/19988621/57/1}
\elib{https://elibrary.ru/item.asp?id=37113830}
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This publication is cited in the following 10 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
D. Yu. Ivanov, O. A. Platonova, MODERN APPROACHES IN ENGINEERING AND NATURAL SCIENCES: MAENS-2021, 2526, MODERN APPROACHES IN ENGINEERING AND NATURAL SCIENCES: MAENS-2021, 2023, 020007
A. I. Kanareykin, “COOLING OF AN INFINITE RECTANGULAR PLATE UNDER BOUNDARY CONDITIONS OF THE SECOND AND THIRD KIND”, CJ, 7:9(71) (2022), 36
Ivanov D.Yu., “O sovmestnom primenenii kollokatsionnogo metoda granichnykh elementov i metoda Fure dlya resheniya zadach teploprovodnosti v konechnykh tsilindrakh s gladkimi napravlyayuschimi”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2021, no. 72, 15–38
Ivanov D.Yu., “Utochnenie kollokatsionnogo metoda granichnykh elementov vblizi granitsy dvumernoi oblasti s pomoschyu poluanaliticheskoi approksimatsii teplovogo potentsiala dvoinogo sloya”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2020, no. 65, 30–52