A Trefftz/MFS mixed-type method to solve the Cauchy problem of the Laplace equation. (January 2019)

• Record Type: Journal Article
• Authors: Liu, Chein-Shan; Wang, Fajie; Gu, Yan
• Is Part Of: Applied mathematics letters. Volume 87(2019)
• Page Start: 87
• Page End: 92
• Publication Date: 2019-01
• Subjects: Laplace equation -- Cauchy problem -- Trefftz method -- MFS test functions -- Mixed-type method; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2018.07.028
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: By using the multiple-scale Trefftz method (MSTM) to solve the Cauchy problem of the Laplace equation in an arbitrary bounded domain, we may lose the accuracy several orders when the noise being imposed on the specified Cauchy data is quite large. In addition to the linear equations obtained from the MSTM, the fundamental solutions play as the test functions being inserted into a derived boundary integral equation. Therefore, after merely supplementing a few linear equations in the mixed-type method (MTM), which is a well organized combination of the Trefftz method and the method of fundamental solutions (MFS), we can improve the ill-conditioned behavior of the linear equations system and hence increase the accuracy of the solution for the Cauchy problem significantly, as explored by two numerical examples.