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

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.