Abstract:
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 ∂tu=a2Δ2u−pu∂tu=a2Δ2u−pu
with constants a,p>0a,p>0 in a plane spatial domain ΩΩ (in a bounded one Ω+Ω+ or in its exterior
Ω−Ω−) on a finite time interval It≡[0,T]It≡[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 ττ, the BIEs are approximated by the piecewise-quadratic interpolation (PQI) of
the C0C0-semigroup of right shifts on time. Also, on the basis of the PQI, the approximation of the
multiplier e−pτe−pτ 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 ss; for the potentials, on both variables ss
and ττ. Then, the integration with respect to the variable ττ on the boundary elements (BEs) is
performed exactly. The integration with respect to the variable ss on the BE for the potentials is
performed approximately by using the Gaussian quadrature with γ⩾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 γ points.
The cubic convergence of approximate solutions of the IBVP at any point of the set Ω×It is
proved under conditions ∂Ω∈C5∩C2γ and w∈C31,3(∂Ω). It is also proved that such solutions are resistant to perturbations of the boundary function w in the norm of the space C01(∂Ω). Here,
Ckm,n(∂Ω)≡Ckm(∂Ω)∩C0m+n(∂Ω) and Ckm(∂Ω) is the Banach space of k times continuously differentiable on ∂Ω vector functions with values in Sobolev's space which is the domain of
definition of the operator Bm ((Bf)(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.
Keywords:
boundary integral equation, boundary element method, singular boundary elements, non-stationary heat conduction, collocation, operator, approximation, stability.
Received: 07.09.2017
Bibliographic databases:
Document Type:
Article
UDC:519.642.4
Language: Russian
Citation:
D. Yu. Ivanov, “On solving plane problems of non-stationary heat conduction by the collocation boundary element method”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2017, no. 50, 9–29
\Bibitem{Iva17}
\by D.~Yu.~Ivanov
\paper On solving plane problems of non-stationary heat conduction by the collocation boundary element method
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2017
\issue 50
\pages 9--29
\mathnet{http://mi.mathnet.ru/vtgu615}
\crossref{https://doi.org/10.17223/19988621/50/2}
\elib{https://elibrary.ru/item.asp?id=30778968}
Linking options:
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This publication is cited in the following 7 articles:
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
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
Ivanov D.Yu., “Utochnenie kollokatsionnogo metoda granichnykh elementov vblizi granitsy oblasti v sluchae dvumernykh zadach nestatsionarnoi teploprovodnosti s granichnymi usloviyami vtorogo i tretego roda”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2019, no. 57, 5–25