# Article

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Keywords:
Cauchy-type integral; Dini continuous density; piecewise linear interpolation; uniform convergence; complex variable boundary element method
Summary:
The paper is a contribution to the complex variable boundary element method, shortly CVBEM. It is focused on Jordan regions having piecewise regular boundaries without cusps. Dini continuous densities whose modulus of continuity $\omega (\cdot )$ satisfies $$\limsup _{s\downarrow 0}\omega (s)\ln \frac {1}{s}=0$$ are considered on these boundaries. Functions satisfying the Hölder condition of order $\alpha$, $0<\alpha \leq 1$, belong to them. The statement that any Cauchy-type integral with such a density can be uniformly approximated by a Cauchy-type integral whose density is a piecewise linear interpolant of the original one is proved under the assumption that the mesh of the interpolation nodes is sufficiently fine and uniform. This result ensures the existence of approximate CVBEM solutions of some planar boundary value problems, especially of the Dirichlet ones.
References:
[1] Khubezhty, Sh. S.: Quadrature formulas for singular integrals with Cauchy kernel. Vladikavkaz. Mat. Zh. 10 (2008), 61-75 Russian \MR 2461693. MR 2461693
[2] II, T. V. Hromadka, Lai, C.: The Complex Variable Boundary Element Method in Engineering Analysis. Springler New York (1987).
[3] Whitley, R. J., II, T. V. Hromadka: Theoretical developments in the complex variable boundary element method. Eng. Anal. Bound. Elem. 30 (2006), 1020-1024. DOI 10.1016/j.enganabound.2006.08.002 | MR 1483319
[4] Whitley, R. J., II, T. V. Hromadka: The existence of approximate solutions for two-dimensional potential flow problems. Numer. Methods Partial Differ. Equations 12 (1996), 719-727. DOI 10.1002/(SICI)1098-2426(199611)12:6<719::AID-NUM5>3.0.CO;2-V | MR 1419772
[5] Lu, J. K.: Boundary Value Problems for Analytic Functions. World Scientific Publishing Company Singapore (1993). MR 1279172 | Zbl 0818.30027
[6] Privalov, I. I.: The boundary properties of analytical functions. CITTL Moscow (1950), Russian. MR 0047765
[7] Muskhelishvili, N. I.: Singular integral equations. Fizmatgiz Moscow (1962), Russian. MR 0355494 | Zbl 0103.07502

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