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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.
##### MSC:
 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
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