On solving plane problems of non-stationary heat conduction by the collocation boundary element method

In this paper, we propose a fully justified collocation boundary element method allowing one to obtain numerical solutions of internal and external initial-boundary value problems (IBVPs) with boundary conditions of the first, second, and third kind for the equation $\partial_tu=a^2\Delta_2u-pu$ with constants $a, p>0$ in a plane spatial domain $\Omega$ (in a bounded one $\Omega^+$ or in its exterior $\Omega^-$) on a finite time interval $I_t\equiv[0, T]$ at a zero initial condition. The solutions are found in the form of the double-layer potential for the Dirichlet IBVP and in the form of the simple layer potential for the Neumann–Robin IBVP with unknown density functions determined from the boundary integral equations (BIEs) of the second kind.
In this paper, instead of the usual piecewise-polynomial interpolation of the density function on time variable $\tau$, the BIEs are approximated by the piecewise-quadratic interpolation (PQI) of the $C_0$-semigroup of right shifts on time. Also, on the basis of the PQI, the approximation of the multiplier $e^{-p\tau}$ in kernels of the integral operators is carried out. In addition, the PQI of density functions is performed: for the BIE, only on arc-length $s$; for the potentials, on both variables $s$ and $\tau$. Then, the integration with respect to the variable $\tau$ on the boundary elements (BEs) is performed exactly. The integration with respect to the variable $s$ on the BE for the potentials is performed approximately by using the Gaussian quadrature with $\gamma\geqslant 2$ points. For the BIE, the integration with respect to the arc-length s is carried out in two ways. On singular BEs and on nearby singular BEs, adjacent to a singular BE in some fixed arc-length region, an exact integration with respect to the variable r is carried out ($r$ is the distance from the boundary point at which the integral is calculated as a function of parameter to the current boundary point of the integration). In this integration, functions of the variable $r$ are taken as the weighting functions. The functions of $r$ are generated by the fundamental solution of the heat equation and the rest of the integrand is approximated by quadratic interpolation on $r$. The integrals with respect to $s$ on the remaining BEs are calculated using the Gaussian quadrature with $\gamma$ points.
The cubic convergence of approximate solutions of the IBVP at any point of the set $\Omega\times I_t$ is proved under conditions $\partial\Omega\in C^5\cap C^{2\gamma}$ and $w\in C_{1,3}^3(\partial \Omega)$. It is also proved that such solutions are resistant to perturbations of the boundary function $w$ in the norm of the space $C_1^0(\partial\Omega)$. Here, $C_{m,n}^k(\partial\Omega)\equiv C_m^k(\partial\Omega)\cap C_{m+n}^0(\partial\Omega)$ and $C_m^k(\partial\Omega)$ is the Banach space of $k$ times continuously differentiable on $\partial\Omega$ vector functions with values in Sobolev's space which is the domain of definition of the operator $\mathbf{B^m}$ ($(\mathbf{B f})(t)=f'(t)$, $f(t=0)=0$).
In conclusion, results of the numerical experiments are presented. They confirm the cubic convergence of approximate solutions for all three IBVPs in a circular domain.