Approximations by the Cauchy-type integrals with piecewise linear densities. (English) Zbl 1274.30146

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 \[ \lim \sup _{s\downarrow 0}\omega (s)\ln \frac {1}{s}=0 \] 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.


30E10 Approximation in the complex plane
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
Full Text: DOI Link


[1] Sh. S. Khubezhty: Quadrature formulas for singular integrals with Cauchy kernel. Vladikavkaz. Mat. Zh. 10 (2008), 61–75. (In Russian.) · Zbl 1324.30090
[2] T.V. Hromadka II, C. Lai: The Complex Variable Boundary Element Method in Engineering Analysis. Springler, New York, 1987. · Zbl 0609.65078
[3] R. J. Whitley, T.V. Hromadka II: Theoretical developments in the complex variable boundary element method. Eng. Anal. Bound. Elem. 30 (2006), 1020–1024. · Zbl 1195.76299
[4] R. J. Whitley, T.V. Hromadka II: The existence of approximate solutions for twodimensional potential flow problems. Numer. Methods Partial Differ. Equations 12 (1996), 719–727. · Zbl 0862.76044
[5] J.K. Lu: Boundary Value Problems for Analytic Functions. World Scientific Publishing Company, Singapore, 1993. · Zbl 0818.30027
[6] I. I. Privalov: The boundary properties of analytical functions. CITTL, Moscow, 1950. (In Russian.)
[7] N. I. Muskhelishvili: Singular integral equations. Fizmatgiz, Moscow, 1962. (In Russian.)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.