On the equivalence of method of fundamental solutions and Trefftz method for Laplace equation

Abstract

In this paper, it is proved that the two approaches for Laplace problems, known in the literature as the method of fundamental solution (MFS) and the Trefftz method, are mathematically equivalent in spite of their essentially minor and apparent differences in the formulation. It is interesting to find that the T-complete set in the Trefftz method for the interior and exterior problems are imbedded in the degenerate kernels of MFS. By designing a circular-domain problem, the unknown coefficients of each method correlate by a mapping matrix after considering the degenerate kernels for the fundamental solutions in the MFS and the T-complete function in the Trefftz method. The mapping matrix is composed of a rotation matrix and a geometric matrix which depends on the source location. The degenerate scale for the Laplace equation appears using the MFS when the geometric matrix is singular. The ill-posed problem in MFS also stems from the geometric matrix when the fictitious source is distributed far away from the real boundary. Finally, the efficiency of MFS is compared with the Trefftz method under the same number of degrees of freedom. 本文主要以 Trefftz 法與基本解法來探討兩者在數學上之等效性， 並由文中可得知 Trefftz的完整解集合不論是在內域問題或外域問題皆可由基本解法中的退化核函數中求得。文中設計一個圓形範例做說明，利用退化核函數展開基本解所得到之係數矩陣與 Trefftz 法中所得到之係數矩陣相互比較後，可產生一映射矩陣。此映射矩陣根據源點的位置分佈所構成並可分解為一旋轉矩陣與幾何矩陣。在拉普拉斯問題中，我們可藉由幾何矩陣之奇異性說明在基本解法中退化尺度問題所產生的機制，並且當虛擬源點與實際邊界相差甚遠時所發生矩陣病態行為，亦可由此看出。最後，本文在相同自由度數目的情況下，對於兩者的收斂速率的優劣亦作探討。