A multiple scale Trefftz method for the Laplace equation subjected to large noisy boundary data

In this paper, we develop an equal norm based multiple scale Trefftz method (MSTM) to solve the Laplace equation subjected to large noisy boundary data. When the complicated geometry with noisy perturbation and ill-conditioned system by using higher-order T-complete functions are encountered, numerical convergence is hard to reach. To tackle these complicated problems, we adopt the MSTM combined with the vector regularization method (VRM) to eliminate the higher-order numerical oscillation phenomena. Due to the inclusion of the characteristic length in the scheme, the ill-posed problem of the constructed Vandermonde matrix is reduced, and the number of terms in the T-complete functions can be increased to stabilize the numerical calculations. More importantly, the proposed approach can successfully overcome the ill-posedness of severely ill-conditioned matrices appearing in linear equations and thus, obtain the accurate numerical solution under a serious noise disturbance. The results reveal that the method presents a simple and stable way to deal with the highly ill-posed problem.