On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations

Authors:
J. T. Chen;C. S. Wu;Y. T. Lee;K. H. Chen
Affiliations:
Department of Harbour and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan;Department of Harbour and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan;Department of Harbour and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan;Department of Information Management, Toko University, Chia-Yi 631, Taiwan
Venue:
Computers & Mathematics with Applications
Year:
2007

Citing 1
Cited 6

The method of fundamental solutions for the numerical solution of the biharmonic equation

Journal of Computational Physics

Numerical implementation of the EDEM for modified Helmholtz BVPs on annular domains

Journal of Computational and Applied Mathematics
Frequency response analyses in vibroacoustics using the method of fundamental solutions

Computational Mechanics
The Trefftz method using fundamental solutions for biharmonic equations

Journal of Computational and Applied Mathematics
Mathematical and numerical studies on meshless methods for exterior unbounded domain problems

Journal of Computational Physics
Adaptive multilayer method of fundamental solutions using a weighted greedy QR decomposition for the Laplace equation

Journal of Computational Physics
Efficient Trefftz collocation algorithms for elliptic problems in circular domains

Numerical Algorithms

Quantified Score

Hi-index	0.10

Visualization

Abstract

In this paper, it is proved that the two approaches, known in the literature as the method of fundamental solutions (MFS) and the Trefftz method, are mathematically equivalent in spite of their essentially minor and apparent differences in formulation. In deriving the equivalence of the Trefftz method and the MFS for the Laplace and biharmonic problems, it is interesting to find that the complete set in the Trefftz method for the Laplace and biharmonic problems are embedded in the degenerate kernels of the MFS. The degenerate scale appears using the MFS when the geometrical matrix is singular. The occurring mechanism of the degenerate scale in the MFS is also studied by using circulant. The comparison of accuracy and efficiency of the two methods was addressed.