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# On Solving Three-dimensional Laplacian Problems in a Multiply Connected Domain Using the Multiple Scale Trefftz Method

Cheng-Yu Ku 1,2

Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan.
Corresponding author. Email: chkst26@mail.ntou.edu.tw

Computer Modeling in Engineering & Sciences 2014, 98(5), 509-541. https://doi.org/10.3970/cmes.2014.098.509

## Abstract

This paper proposes the numerical solution of three-dimensional Laplacian problems in a multiply connected domain using the collocation Trefftz method with multiple source points. A numerical solution for three-dimensional Laplacian problems was approximated by superpositioning T-complete functions formulated from 36 independent functions satisfying the governing equation in the cylindrical coordinate system. To deal with complicated problems for multiply connected domain, we adopted the generalized multiple source point boundary collocation Trefftz method which allows many source points in the Trefftz formulation without using the decomposition of the problem domain. In addition, to mitigate a severely ill-conditioned system of linear equations, this study adopted the newly developed multiple scale Trefftz method and the dynamical Jacobian-inverse free method. Numerical solutions were conducted for five three-dimensional groundwater flow problems in a simply connected domain, an infinite domain, a doubly connected domain, and a multiply connected domain. The results revealed that the proposed method can obtain accurate numerical solutions for three-dimensional Laplacian problems in a multiply connected domain, yielding a superior convergence in numerical stability to that of the conventional Trefftz method.

## Keywords

APA Style
, C.K. (2014). On solving three-dimensional laplacian problems in a multiply connected domain using the multiple scale trefftz method. Computer Modeling in Engineering & Sciences, 98(5), 509-541. https://doi.org/10.3970/cmes.2014.098.509
Vancouver Style
CK. On solving three-dimensional laplacian problems in a multiply connected domain using the multiple scale trefftz method. Comput Model Eng Sci. 2014;98(5):509-541 https://doi.org/10.3970/cmes.2014.098.509
IEEE Style
C.K. , "On Solving Three-dimensional Laplacian Problems in a Multiply Connected Domain Using the Multiple Scale Trefftz Method," Comput. Model. Eng. Sci., vol. 98, no. 5, pp. 509-541. 2014. https://doi.org/10.3970/cmes.2014.098.509

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