A numerical experiment on the exponential convergence of method of fundamental solutions and Trefftz method

• Main Authors: Lin,Poho, 林柏宏
• Other Authors: Tsai,Chiacheng
• Format: Others
• Language: zh-TW
• Published: 2012
• Online Access: http://ndltd.ncl.edu.tw/handle/79447396929752506866

碩士 === 國立高雄海洋科技大學 === 海洋環境工程研究所 === 100 === It is well known that both the method of fundamental solutions (MFS) and the Trefftz method are numerical methods of exponential convergence. In other words, the logarithmic error is proportional to the node number of spatial discretization or the number of the Trefftz basis. In this study, the exponential convergence of MFS is demonstrated by solving Laplace equation in domains of rectangles, ellipses, amoeba-like shapes, and rectangular cuboids. In the solution procedure, the sources of the MFS are located as far as possible and the instability resulted from the ill-conditioning of system matrix is avoided by using the multiple precision floating-point reliable (MPFR) library. The results converge faster for the cases of smoother boundary conditions and larger area/perimeter ratios. For the Trefftz method, it is found that numerical solutions obtained the LU decomposition is more stable. In addition, a formula is derived to predict the feasible range of the Trefftz characteristic length. The MPFR library is used to validate the range formula and to demonstrate the exponential convergence of the Trefftz method.