Fast spectral solver for the inversion of boundary data problem of Poisson equation in a doubly connected domain

In this paper, we study the inversion of the inner boundary data for Poisson equation in a doubly connected domain by fast Fourier ultraspherical spectral solver. The solver depends on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by an ultraspherical spectral method. Because this problem is seriously ill-posed, the Cauchy data with noise will lead to ill-conditioned linear system. Hence, we apply Tikhonov regularization to solve the obtained linear system and use generalized cross-validation (GCV) criterion to select regularization parameters. The accuracy and efficiency of the proposed method are illustrated by several numerical results of different regions.