Analytical study and numerical experiments for Laplace equation with overspecified boundary conditions

Abstract

In this paper, the window function, e−αk2, is applied to regularize the divergent problem which occurs in the Laplace equation with overspecified boundary conditions in an infinite strip region. To deal with this ill-posed problem, the corner of the L-curve is chosen as the compromise point to determine the optimal α of the Gaussian window, e−αk2, so that the high wave-number (k) content can be suppressed instead of engineering judgement using the concept of a cutoff wave-number. From the examples shown, it is found that a reasonable solution of the unknown boundary potential can be reconstructed, and that both high wave-number content and divergent results can be avoided by using the proposed regularization technique.