Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method

In this paper, the inverse Cauchy problem for Laplace equation defined in an arbitrary plane domain is investigated by using the collocation Trefftz method (CTM) with a better postconditioner. We first introduce a multiple-scale Rk in the T-complete functions as a set of bases to expand the trial solution. Then, the better values of Rk are sought by using the concept of an equilibrated matrix, such that the resulting coefficient matrix of a linear system to solve the expansion coefficients is best-conditioned from a view of postconditioner. As a result, the multiple-scale Rk can be determined exactly in a closed-form in terms of the collocated points used in the collocation to satisfy the boundary conditions. We test the present method for both the direct Dirichlet problem and the inverse Cauchy problem. A significant reduction of the condition number and the effective condition number can be achieved when the present CTM is used, which has a good efficiency and stability against the disturbance from large random noise, and the computational cost is much saving. Some serious cases of the inverse Cauchy problems further reveal that the unknown data can be recovered very well, although the overspecified data are provided only at a 20% of the overall boundary.