Applications of the modified Trefftz method for the Laplace equation

In this article, a solution to nonlinear moving boundary problems in heterogeneous geological media using the meshless method is proposed. The free surface flow is a moving boundary problem governed by Laplace equation but has nonlinear boundary conditions. We adopt the collocation Trefftz method (CTM) to approximate the solution using Trefftz base functions, satisfying the Laplace equation. An iterative scheme in conjunction with the CTM for finding the phreatic line with over–specified nonlinear moving boundary conditions is developed. To deal with flow in the layered heterogeneous soil, the domain decomposition method is used so that the hydraulic conductivity in each subdomain can be different. The method proposed in this study is verified by several numerical examples. The results indicate the advantages of the collocation meshless method such as high accuracy and that only the surface of the problem domain needs to be discretized. Moreover, it is advantageous for solving nonlinear moving boundary problems with heterogeneity with extreme contrasts in the permeability coefficient.

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The MFS versus the Trefftz method for the Laplace equation in 3D

Engineering Analysis with Boundary Elements ◽

10.1016/j.enganabound.2017.06.006 ◽

2017 ◽

Vol 83 ◽

pp. 133-140 ◽

Cited By ~ 7

Author(s):

Hui Lv ◽

Fang Hao ◽

Yong Wang ◽

C.S. Chen

Keyword(s):

Laplace Equation ◽

Trefftz Method

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A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation

Engineering Analysis with Boundary Elements ◽

10.1016/j.enganabound.2007.12.002 ◽

2008 ◽

Vol 32 (9) ◽

pp. 778-785 ◽

Cited By ~ 53

Author(s):

Chein-Shan Liu

Keyword(s):

Cauchy Problem ◽

Laplace Equation ◽

Trefftz Method

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Singular Boundary Method for Various Exterior Wave Applications

International Journal of Computational Methods ◽

10.1142/s0219876215500115 ◽

2015 ◽

Vol 12 (02) ◽

pp. 1550011 ◽

Cited By ~ 14

Author(s):

Zhuo-Jia Fu ◽

Wen Chen ◽

Ji Lin ◽

Alexander H.-D. Cheng

Keyword(s):

Fundamental Solution ◽

Helmholtz Equation ◽

Laplace Equation ◽

International Workshop ◽

Fundamental Solutions ◽

Singular Boundary ◽

Intensity Factors ◽

Trefftz Method ◽

Boundary Method ◽

Singular Boundary Method

This paper presents an improved singular boundary method (ISBM) to various exterior wave applications. The SBM is mathematically simple, easy-to-program, meshless and introduces the concept of source intensity factors to eliminate the singularity of the fundamental solutions. In this study, we first derive the source intensity factors of the exterior Helmholtz equation by means of the source intensity factors of the exterior Laplace equation due to the same order of the singularities on their fundamental solutions. The source intensity factors of the exterior Laplace equation can be determined using the reference technique [Chen, W. and Gu, Y. [2011] "Recent advances on singular boundary method," Joint international workshop on Trefftz method VI and method of fundamental solution II, Taiwan]. Numerical illustrations demonstrate the efficiency and accuracy of the proposed scheme on four benchmark exterior wave examples.

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