Geodesic
On uniform convergence of approximations of the double layer potential near the boundary of a two-dimensional domain
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 32 (2022) no. 1, pp. 26-43 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

On the basis of piecewise quadratic interpolation, semi-analytical approximations of the double layer potential near and on the boundary of a two-dimensional domain are obtained. To calculate the integrals formed after the interpolation of the density function, exact integration with respect to the variable $\rho=\left(r^2-d^2\right)^{1/2}$ is used, where $d$ and $r$ are the distances from the observed point to the boundary of the domain and to the boundary point of integration, respectively. The study proves the stable convergence of such approximations with the cubic velocity uniformly near the boundary of the class $C^5$, and also on the boundary itself. It is also proved that the use of standard quadrature formulas for calculating the integrals does not violate the uniform cubic convergence of approximations of the direct value of the potential on the boundary of the class $C^6$. With some simplifications, it is proved that the use of standard quadrature formulas for calculating the integrals entails the absence of uniform convergence of potential approximations inside the domain near any boundary point. The theoretical conclusions are confirmed by the results of the numerical solution of the Dirichlet problem for the Laplace equation in a circular domain.
Mots-clés : quadrature formula, uniform convergence.
Keywords: double layer potential, boundary element method, near singular integral, boundary layer effect 
@article{VUU_2022_32_1_a2,
     author = {Ivanov D.Yu.},
     title = {On uniform convergence of approximations of the double layer potential near the boundary of a two-dimensional domain},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {26--43},
     year = {2022},
     volume = {32},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2022_32_1_a2/}
}
TY  - JOUR
AU  - Ivanov D.Yu.
TI  - On uniform convergence of approximations of the double layer potential near the boundary of a two-dimensional domain
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2022
SP  - 26
EP  - 43
VL  - 32
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VUU_2022_32_1_a2/
LA  - ru
ID  - VUU_2022_32_1_a2
ER  - 
%0 Journal Article
%A Ivanov D.Yu.
%T On uniform convergence of approximations of the double layer potential near the boundary of a two-dimensional domain
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2022
%P 26-43
%V 32
%N 1
%U http://geodesic.mathdoc.fr/item/VUU_2022_32_1_a2/
%G ru
%F VUU_2022_32_1_a2
Ivanov D.Yu. On uniform convergence of approximations of the double layer potential near the boundary of a two-dimensional domain. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 32 (2022) no. 1, pp. 26-43. http://geodesic.mathdoc.fr/item/VUU_2022_32_1_a2/

[1] Brebbia C. A., Telles J. C. F., Wrobel L. C., Boundary element techniques. Theory and applications in engineering, Springer, Berlin, 1984 | DOI | MR | Zbl

[2] Lv J., Miao Y., Zhu H., “The distance $\sinh$ transformation for the numerical evaluation of nearly singular integrals over curved surface elements”, Computational Mechanics, 53:2 (2014), 359–367 | DOI | MR | Zbl

[3] Gao X.-W., Zhang J.-B., Zheng B.-J., Zhang C., “Element-subdivision method for evaluation of singular integrals over narrow strip boundary elements of super thin and slender structures”, Engineering Analysis with Boundary Elements, 66 (2016), 145–154 | DOI | MR | Zbl

[4] Gao X. W., Yang K., Wang J., “An adaptive element subdivision technique for evaluation of various 2D singular boundary integrals”, Engineering Analysis with Boundary Elements, 32:8 (2008), 692–696 | DOI | Zbl

[5] Krutitskii P. A., Fedotova A. D., Kolybasova V. V., “Quadrature formula for the simple layer potential”, Differential Equations, 55:9 (2019), 1226–1241 | DOI | DOI | MR | Zbl

[6] Xie G., Li K., Zhong Y., Li H., Hao B., Du W., Sun Ch., Wang H., Wen X., Wang L., “A systematic derived sinh based method for singular and nearly singular boundary integrals”, Engineering Analysis with Boundary Elements, 123 (2021), 147–153 | DOI | MR | Zbl

[7] Zhang J., Wang P., Lu Ch., Dong Y., “A spherical element subdivision method for the numerical evaluation of nearly singular integrals in 3D BEM”, Engineering Computations, 34:6 (2017), 2074–2087 | DOI | MR

[8] Krutitskii P. A., Kolybasova V. V., “Numerical method for the solution of integral equations in a problem with directional derivative for the Laplace equation outside open curves”, Differential Equations, 52:9 (2016), 1219–1233 | DOI | DOI | MR | Zbl

[9] Niu Zh., Cheng Ch., Zhou H., Hu Z., “Analytic formulations for calculating nearly singular integrals in two-dimensional BEM”, Engineering Analysis with Boundary Elements, 31:12 (2007), 949–964 | DOI | MR | Zbl

[10] Zhang Y.-M., Gu Y., Chen J.-T., “Stress analysis for multilayered coating systems using semi-analytical BEM with geometric non-linearities”, Computational Mechanics, 47:5 (2011), 493–504 | DOI | MR | Zbl

[11] Gu Y., Chen W., Zhang B., Qu W., “Two general algorithms for nearly singular integrals in two dimensional anisotropic boundary element method”, Computational Mechanics, 53:6 (2014), 1223–1234 | DOI | MR | Zbl

[12] Niu Zh., Hu Z., Cheng Ch., Zhou H., “A novel semi-analytical algorithm of nearly singular integrals on higher order elements in two dimensional BEM”, Engineering Analysis with Boundary Elements, 61 (2015), 42–51 | DOI | MR | Zbl

[13] Cheng Ch., Pan D., Han Zh., Wu M., Niu Zh., “A state space boundary element method with analytical formulas for nearly singular integrals”, Acta Mechanica Solida Sinica, 31:4 (2018), 433–444 | DOI

[14] Krutitskii P. A., Reznichenko I. O., “Quadrature formula for the harmonic double layer potential”, Differential Equations, 57:7 (2021), 901–920 | DOI | DOI | MR | Zbl

[15] Zhang Y.-M., Gu Y., Chen J.-T., “Boundary element analysis of 2D thin walled structures with high-order geometry elements using transformation”, Engineering Analysis with Boundary Elements, 35:3 (2011), 581–586 | DOI | MR | Zbl

[16] Xie G., Zhang J., Qin X., Li G., “New variable transformations for evaluating nearly singular integrals in 2D boundary element method”, Engineering Analysis with Boundary Elements, 35:6 (2011), 811–817 | DOI | MR | Zbl

[17] Xie G., Zhang J., Dong Y., Huang Ch., Li G., “An improved exponential transformation for nearly singular boundary element integrals in elasticity problems”, International Journal of Solids and Structures, 51:6 (2014), 1322–1329 | DOI | MR

[18] Zhang Y., Li X., Sladek V., Sladek J., Gao X., “A new method for numerical evaluation of nearly singular integrals over high-order geometry elements in 3D BEM”, Journal of Computational and Applied Mathematics, 277 (2015), 57–72 | DOI | MR | Zbl

[19] Gong Y. P., Dong C. Y., Bai Y., “Evaluation of nearly singular integrals in isogeometric boundary element method”, Engineering Analysis with Boundary Elements, 75 (2017), 21–35 | DOI | MR | Zbl

[20] Gu Y., Chen W., Zhang Ch., “The $\sinh$ transformation for evaluating nearly singular boundary element integrals over high-order geometry elements”, Engineering Analysis with Boundary Elements, 37:2 (2013), 301–308 | DOI | MR | Zbl

[21] Gu Y., Zhang Ch., “Fracture analysis of ultra-thin coating/substrate structures with interface cracks”, International Journal of Solids and Structures, 225 (2021), 111074 | DOI

[22] Zhang Y., Gong Y., Gao X., “Calculation of 2D nearly singular integrals over high-order geometry elements using the sinh transformation”, Engineering Analysis with Boundary Elements, 60 (2015), 144–153 | DOI | MR | Zbl

[23] Gong Y., Dong Ch., Qin F., Hattori G., Trevelyan J., “Hybrid nearly singular integration for isogeometric boundary element analysis of coatings and other thin 2D structures”, Computer Methods in Applied Mechanics and Engineering, 367 (2020), 113099 | DOI | MR | Zbl

[24] 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”, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2019, no. 57, 5–25 (in Russian) | DOI

[25] Ivanov D. Yu., “On solving plane problems of non-stationary heat conduction by the collocation boundary element method”, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2017, no. 50, 9–29 (in Russian) | DOI

[26] Smirnov V. I., A course of higher mathematics, part 2, v. 4, Nauka, M., 1981 | MR

[27] Ivanov D. Yu., “A refinement of the boundary element collocation method near the boundary of a two-dimensional domain using semianalytic approximation of the double layer heat potential”, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2020, no. 65, 30–52 (in Russian) | DOI

[28] Dunford N., Schwartz J. T., Linear operators, v. 1, General theory, John Wiley and Sons, Hoboken, 1988 | MR | MR

[29] Ivanov D. Yu., “Stable solvability in spaces of differentiable functions of some two-dimensional integral equations of heat conduction with an operator-semigroup kernel”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2015, no. 38, 33–45 (in Russian) | DOI | Zbl

[30] Mikhlin S. G., Mathematical physics, an advanced course, North-Holland, Amsterdam, 1970 | MR | MR | Zbl

[31] Berezin I. S., Zhidkov N. P., Methods of computation, v. 1, Fizmatgiz, M., 1962 | MR

[32] Kolmogorov A. N., Fomin S. V., Elements of the theory of functions and functional analysis, Nauka, M., 1976 | MR

[33] Kato T., Perturbation theory for linear operators, Springer, Berlin–Heidelberg, 1995 | DOI | MR | Zbl

[34] Fikhtengol'ts G. M., Differential and integral calculus course, v. 2, Fizmatlit, M., 2003