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Explicit Numerically Implementable Formulas for Poincaré–Steklov Operators

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Abstract

The paper presents explicit numerically implementable formulas for the Poincaré–Steklov operators, such as the Dirichlet–Neumann, Dirichlet–Robin, Robin1–Robin2, and Grinberg–Mayergoiz operators, related to the two-dimensional Laplace equation. These formulas are based on the lemma about a univalent isometric mapping of a closed analytic curve onto a circle. Numerical results for domains with very complex geometries were obtained for several test harmonic functions for the Dirichlet–Neumann and Dirichlet–Robin operators.

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Notes

  1. Boundary operators of the form \(u + b\frac{{\partial u}}{{\partial \nu }}\) first appeared in Isaac Newton’s (1642–1727) and Georgy Richman’s (1711–1753) works as an empirical dependence of the heat transfer process, but it was not without reason that the French Academy of Sciences twice awarded Victor Robin (1855–1897) the Francoeur Prize (1893 and 1897), as well as the Ponsel Prize (1895). Robin, considering the problem of the surface current density distribution in a conductor, reduced it to an integral equation (long before the advent of the theory of integral equations) and, in the case of a convex conductor, found a solution of this equation by the method of successive approximations.

  2. A short proof, preferable to the original [11] and [12], is given in footnote 4 of [13].

  3. The function \(U\) defined by series (12) depends on the parameter \(r\), as well as the coefficients ck and dk in formula (11). However, this dependence can only affect the rate of convergence of the series in formula (12), but in no way the results presented below and their proofs.

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This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. Lomonosov Moscow State University, 119991, Moscow, Russia

    A. S. Demidov

  2. Institute of Control Sciences, Russian Academy of Sciences, 117997, Moscow, Russia

    A. S. Samokhin

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  1. A. S. Demidov
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  2. A. S. Samokhin
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Correspondence to A. S. Demidov.

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Translated by E. Chernokozhin

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Demidov, A.S., Samokhin, A.S. Explicit Numerically Implementable Formulas for Poincaré–Steklov Operators. Comput. Math. and Math. Phys. 64, 237–247 (2024). https://doi.org/10.1134/S0965542524020040

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