New approach for the inverse boundary value problem of Laplace's equation on a rectangle: technique renovation for the Grad-Shafranov (GS) reconstruction

Abstract:

In Li et al. [2013, New approach for solving the inverse boundary value problem of Laplace’s equation on a circle: Technique renovation of the Grad-Shafranov (GS) reconstruction, J. Geophys. Res. Space., 118, 2876-2881], a couple of Hilbert transform relations were applied to the study of the ill-posedness for the essential GS reconstructions. In this further study, a detailed derivation for these reciprocal relations are presented in case of the plane circular region, and then the reciprocal relations are extended to apply to the plane rectangular region after a conformal mapping procedure. While for the case of plane rectangular region, it is confronted by a traditional problem of the so-called corner singularities, which divided the extended reciprocal relations into four integrals with end-point singularities. With the help of the extended Euler-Maclaurin expansion, new quadrature schemes are developed for these singular integrals. Benchmark testing with the analytic solutions on a rectangle boundary has also show the efficiency and robustness of these extensions. The new solution approach is also developed with the introduced reciprocal relations, and an iterated Tikhonov regularization scheme is applied to deal with the ill-posed linear operators appearing in the discretization of the new approach. The special case on the rectangular boundary is benchmarked with the analytic solutions. Numerical experiments highlight the efficiency and robustness of the proposed method. A robust solution approach is expected to be developed based on these new results for the GS equation on any 2D region with partial-known boundary conditions.