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Localized method of fundamental solutions for three-dimensional inhomogeneous elliptic problems: theory and MATLAB code

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Abstract

In this paper we investigate the application of the localized method of fundamental solutions (LMFS) for solving three-dimensional inhomogeneous elliptic boundary value problems. A direct Chebyshev collocation scheme (CCS) is employed for the approximation of the particular solutions of the given inhomogeneous problem. The Gauss–Lobatto collocation points are used in the CCS to ensure the pseudo-spectral convergence of the method. The resulting homogeneous equations are then calculated by using the LMFS. In the framework of the LMFS, the computational domain is divided into a set of overlapping local subdomains where the traditional MFS formulation and the moving least square method are applied. The proposed CCS-LMFS produces sparse and banded stiffness matrix which makes the method possible to perform large-scale simulations on a desktop computer. Numerical examples involving Poisson, Helmholtz as well as modified-Helmholtz equations (with up to 1,000,000 unknowns) are presented to illustrate the efficiency and accuracy of the proposed method.

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Abbreviations

\( L \) :

Linear differential operator

\( \Omega \) :

Original computational domain

\( u_{p} \) :

Particular solution

\( u_{h} \) :

Homogeneous solution

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Omega } \) :

A cubic which contains \( \Omega \)

\( T_{n} \) :

nth Chebyshev polynomial

N 1, N 2, N 3 :

Degrees of Chebyshev’s polynomials

M :

Number of roots of Chebyshev polynomial

\( \Omega_{s} \) :

Local subdomain

\( N_{s} \) :

Number of collocation points inside \( \Omega_{s} \)

\( M_{s} \) :

Number of sources associated with \( \Omega_{s} \)

\( R_{s} \) :

Radius of the local artificial sphere

\( G \) :

Fundamental solution

\( B(u) \) :

Residual function

\( \xi \) :

Root of Chebyshev polynomial

\( \phi \) :

Unknown coefficients in Chebyshev method

\( \varphi \) :

Unknown coefficients in MFS formulation

\( \omega \) :

Weighting function

m, n, l :

Indexes in a sum in Chebyshev method

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Acknowledgements

The work described in this paper was supported by the National Natural Science Foundation of China (Nos. 11872220, 71571108), Projects of International (Regional) Cooperation and Exchanges of NSFC (No. 71611530712), and the Shandong Provincial Natural Science Foundation (Nos. ZR2017JL004, ZR2017BA003, ZR2017MF055).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, People’s Republic of China

    Yan Gu, Wenzhen Qu & Fajie Wang

  2. Institute of Mechanics for Multifunctional Materials and Structures, Qingdao University, Qingdao, 266071, People’s Republic of China

    Yan Gu, Wenzhen Qu, Fajie Wang & Chuanzeng Zhang

  3. Department of Harbor and River Engineering and Computation and Simulation Center, National Taiwan Ocean University, Keelung, 20224, Taiwan

    Chia-Ming Fan

  4. Department of Civil Engineering, University of Siegen, Paul-Bonatz-Str. 9-11, 57076, Siegen, Germany

    Chuanzeng Zhang

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  1. Yan Gu
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Correspondence to Yan Gu.

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Appendix A: Source code written in MATLAB_R2018a

Appendix A: Source code written in MATLAB_R2018a

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Gu, Y., Fan, CM., Qu, W. et al. Localized method of fundamental solutions for three-dimensional inhomogeneous elliptic problems: theory and MATLAB code. Comput Mech 64, 1567–1588 (2019). https://doi.org/10.1007/s00466-019-01735-x

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