The approximate solution of 2D Dirichlet problem in doubly connected domains
The authors describe a method for solving the Laplace equation subject to Dirichlet boundary conditions in a doubly connected domain \(\Omega \subset\mathbb{R}^2\) with \(0 \notin \Omega\). The starting point is to write the solution \(u\) in the form \(u(x,y)=\Re(B(z))+A\log(z)\), \(z=x+iy\), where \(B\) is analytic in \(\Omega\) and \(A\) is a real constant. \(B\) is represented using the Cauchy integral formula. The boundary of \(\Omega\) is given in parametric form and approximated by truncated Fourier polynomials. The determination of \(B\) on \(\partial\Omega\) results in solving two Fredholm integral equation of the second kind. Their solutions are also approximated by truncated Fourier series. At the end, a linear system for the unknown coefficients in the Fourier series is left to solve. Finally, \(A\) can be determined by a least square method. The numerical solution is improved for points near \(\partial\Omega\) in this way taking the singularity of the kernel in the Cauchy integral into account. Some numerical experiments are reported.
scientific article; zbMATH DE number 6991999