Efficient numerical solution of the generalized Dirichlet-Neumann map for linear elliptic PDEs in regular polygon domains
The authors develop, for the two-dimensional Laplace equation, a collocation-type numerical method to study the Dirichlet-Neumann map which uses the derivative of the solution along a direction of an arbitrary angle to the boundary, to compute the derivative of the solution perpendicularly to this direction without solving the problem in the interior of the domain. They prove that in the case of regular polygon domains with the same type of boundary conditions on each side, the coefficient matrix is block circulant and independent of the choice of basis functions. These properties allow the use of the fast Fourier transform, for the efficient solution of the collocation linear system, with significant improvement of the performance for both direct and iterative methods considered. Numerical experiment are included.
scientific article; zbMATH DE number 6015762