Abstract
An algorithm is provided for the fast and accurate computation of the solution of the Bitsadze equation in the complex plane in the interior of the unit disk. The algorithm is based on the representation of the solution in terms of a double integral as it shown by Begehr [1,2], some recursive relations in Fourier space, and Fast Fourier Transforms. The numerical evaluation of integrals at points on a polar coordinate grid by straightforward summation for the double integral would require floating point operation per point. Evaluation of such integrals has been optimized in this paper giving an asymptotic operation count of per point on the average. In actual implementation, the algorithm has even better computational complexity, approximately of the order of per point. The algorithm has the added advantage of working in place, meaning that no additional memory storage is required beyond that of the initial data. This paper is a result of application of many of the original ideas described in Daripa [3].
Highlights
The solutions of many elliptic partial differential equations represent in terms of singular integrals in the complex plane in the interior of the unit disk, as the nonhomogeneous Cauchy-Riemann equations, the Beltrami equation, the Poisson equation, etc
We presented a fast algorithm to solve the Bitsadze equation in the unit disk under special boundary conditions in the complex plane, by constructed the fast algorithm to evaluate the singular integral transform (1.4)
The good performance of the algorithm is due to the use of scaling one-dimensional integral in the radial direction to produce the value of the singular integral over the entire domain
Summary
The solutions of many elliptic partial differential equations represent in terms of singular integrals in the complex plane in the interior of the unit disk, as the nonhomogeneous Cauchy-Riemann equations, the Beltrami equation, the Poisson equation, etc Solving these equations requires computing the values of the singular integrals. Daripa and Co-workers ([3,4,5,6,7,8,9]) presented fast algorithms to solve the singular integrals that arise in such solutions By these algorithms evaluation of singular integrals has been optimized, giving an asymptotic operation count of O N 2 ln ln N for N 2 points.
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