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 ∂_t u=cθ'∆₂u - p u with constants a, p > 0 in an open two-dimensional spatial domain Ω 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 Ω × [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 х ∈ Ω at which the potential is calculated to the integration point х' ∈ ∂Ω does not exceed approximately one-third of the radius of the Lyapunov circle R_λ, then we use exact integration with respect to a certain component ρ of the distance r: ρ ≡ (r² - d²)^½ (d is the distance from the point х ∈ Ω to the boundary ∂Ω). This exact integration is practically feasible for any analytically defined curve ∂Ω. In this integration, functions of the variable ρ are taken as the weighting functions and the rest of the integrand is approximated by quadratic interpolation on ρ. The functions of ρ are generated by the fundamental solution of the heat equation. The integrals with respect to s for r > R_λ3 are calculated using Gaussian quadrature with γ points. Under the condition ∂Ω ∈ C⁵∩C^2γ (γ ≥ 2), it is proved that the approximate solutions converge to an exact one with a cubic velocity uniformly in the domain Ω × [0, T]. It is also proved that the approximate solutions are stable to perturbations of the boundary function uniformly in the domain Ω × [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 ρ can substantially reduce the decrease in the accuracy of numerical solutions near the boundary ∂Ω, in comparison with the use of exclusively Gauss quadratures in calculating the potential. AMS Mathematical Subject Classification: 80М15, 65Е05

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