Open Access

ARTICLE

A Multiple-Precision Study on the Modified Collocation Trefftz Method

Chia-Cheng Tsai¹, Po-Ho Lin²

1 Corresponding author, Department of Marine Environmental Engineering, National Kaohsiung Marine University, Kaohsiung 811, Taiwan, E-mail: tsaichiacheng@mail.nkmu.edu.tw
2 Department of Marine Environmental Engineering, National Kaohsiung Marine University, Kaohsiung 811, Taiwan

Computers, Materials & Continua 2012, 28(3), 231-260. https://doi.org/10.3970/cmc.2012.028.231

Abstract

Recently, Liu (CMES 21(2007), 53) developed the modified collocation Trefftz method (MCTM) by setting a characteristic length slightly larger than the maximum radius of the computational domain. In this study, we find that the range of admissible characteristic length can be significantly enlarged if the LU decomposition is applied for solving the resulted dense unsymmetric matrix. Furthermore, we discover a range formula for admissible characteristic length, in which the number of the T-complete functions, the shape of the computation domain, and the exponent bits of the involved floating-point arithmetic have been taken into consideration. In order to validate the prescribed formula for different exponent bits, the multiple precision floating-point reliable (MPFR) library is used. In addition, we find that the MCTM is a numerical method of exponential convergence. In other words, increasing the numbers of the T-complete functions can reduce the logarithmic error proportionally till the precision limit, which can be set up for the MPFR library. Numerical experiments are carried out to demonstrate that the proposed MCTM with the LU decomposition can solve the Laplace equation stably and accurately, even for a Cauchy problem. A multiple-precision comparison between the MCTM and the method of fundamental solution is also preformed.

---

Keywords

---