Conformal capacity and polycircular domains (Q2087505)

A multiply connected planar domain with boundaries consisting of unions of finitely many circular arcs is called a polycircular domain. In this paper, the authors mainly study the numerical conformal mapping of polycircular domains. They apply two numerical methods, the hp-finite element method (FEM) and the Boundary Integral Equation (BIE) method with the generalized Neumann Kernel, to present numerical results on capacities of multiply connected polycircular domains. The hp-FEM method allows one to incorporate a priori knowledge of the singularities into highly non-quasiuniform meshes, within which the polynomial order can vary from element to element. Here one would expect exponential convergence in the natural norm if the discretization is refined properly. The BIE method converges exponentially for analytic boundaries and algebraically for piecewise smooth boundaries. One of the objectives of this paper is to present a comparison where the BIE is compared against the hp-FEM. The auhors obtain surprisingly good agreements between the two methods. The obtained results show that the BIE indeed gives accurate results for domains with corners even if the angles at the corners are small. Experimental error estimates are provided for the computed capacity and, when possible, the rate of convergence under refinement of discretization is analyzed. The results obtained with these two methods agree with high accuracy.