The Laplace equation in three dimensions by the method of fundamental solutions and the method of particular solutions

Abstract For Laplace's equation in a bounded simply-connected domain Ω in 3D, the method of fundamental solutions (MFS) is studied in this paper. Although some numerical computations can be found in Chen et al. [10] , the theoretical analysis is much behind (Li [23] only for unit sphere Ω). Our efforts are devoted to exploring a strict error analysis of the MFS. The error bounds are derived, and the optimal polynomial convergence rates can be achieved. Numerical experiments are carried out to support the analysis made, and several useful locations of source nodes are investigated numerically. The analysis in this paper may lay a theoretical basis of the MFS for 3D problems, as Bogomolny [8] and [24] for 2D problems. Besides, the method of particular solutions (MPS) in [26] is also studied by using the spherical harmonic functions (SHF). The optimal polynomial convergence rates and the exponential growth of condition number (Cond) are obtained. The source nodes are located based on the abscissas of quadrature rules on surfaces; they are “grid-like”. Since most of 3D problems, in reality, can not be simplified to 2D problems, and since the MFS has more advantages for 3D problems in algorithm simplicity and wide application, the study in this paper is essential and important to the MFS.

[1]  Yimin Wei,et al.  The method of fundamental solutions for the Helmholtz equation , 2019, Applied Numerical Mathematics.

[2]  Hung-Tsai Huang,et al.  Neumann problems of 2D Laplace's equation by method of fundamental solutions , 2017 .

[3]  Jun Lu,et al.  A novel meshless method for fully nonlinear advection-diffusion-reaction problems to model transfer in anisotropic media , 2018, Appl. Math. Comput..

[4]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[5]  Bengt Fornberg,et al.  A Stable Algorithm for Flat Radial Basis Functions on a Sphere , 2007, SIAM J. Sci. Comput..

[6]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1966 .

[7]  Y. T. Lee,et al.  On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations , 2007, Comput. Math. Appl..

[8]  C. Schwab,et al.  Boundary Element Methods , 2010 .

[9]  Holger Wendland,et al.  Scattered Data Approximation: Conditionally positive definite functions , 2004 .

[10]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[11]  Wenzhen Qu,et al.  Singular Boundary Method: Three RegularizationApproaches and ExteriorWave Applications , 2014 .

[12]  V. I. Lebedev,et al.  Quadratures on a sphere , 1976 .

[13]  Robert Schaback,et al.  Solving the 3D Laplace equation by meshless collocation via harmonic kernels , 2013, Adv. Comput. Math..

[14]  Roland Glowinski,et al.  An introduction to the mathematical theory of finite elements , 1976 .

[15]  Xin Li.  Convergence of the method of fundamental solutions for Poisson’s equation on the unit sphere , 2008, Adv. Comput. Math..

[16]  G. Lohöfer.  Inequalities for the Associated Legendre Functions , 1998 .

[17]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[18]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[19]  Zi-Cai Li,et al.  The method of fundamental solutions for annular shaped domains , 2009 .

[20]  Ji Lin,et al.  Simulation of Seismic Wave Scattering by Embedded Cavities in an Elastic Half-Plane Using the Novel Singular Boundary Method , 2018, Advances in Applied Mathematics and Mechanics.

[21]  Pedro R. S. Antunes.  Reducing the ill conditioning in the method of fundamental solutions , 2018, Adv. Comput. Math..

[22]  Pedro R. S. Antunes.  Numerical calculation of eigensolutions of 3D shapes using the method of fundamental solutions , 2011 .

[23]  Jun Lu,et al.  Fast simulation of multi-dimensional wave problems by the sparse scheme of the method of fundamental solutions , 2016, Comput. Math. Appl..

[24]  K. Atkinson,et al.  Spherical Harmonics and Approximations on the Unit Sphere: An Introduction , 2012 .

[25]  A. Bogomolny.  Fundamental Solutions Method for Elliptic Boundary Value Problems , 1985 .

[26]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[27]  Yan Li,et al.  On choosing the location of the sources in the MFS , 2015, Numerical Algorithms.

[28]  Wen Chen,et al.  Recent Advances in Radial Basis Function Collocation Methods , 2013 .

[29]  A. A. Samarskii,et al.  Equations of Mathematical Physics , 1964 .

[30]  Kendall Atkinson,et al.  Numerical integration on the sphere , 1982, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[31]  I. Saavedra,et al.  Multipole fast algorithm for the least‐squares approach of the method of fundamental solutions for three‐dimensional harmonic problems , 2003 .

[32]  Yi Xu,et al.  Simulation of linear and nonlinear advection-diffusion-reaction problems by a novel localized scheme , 2020, Appl. Math. Lett..

[33]  Liping Zhang,et al.  The modified method of fundamental solutions for exterior problems of the Helmholtz equation; spurious eigenvalues and their removals , 2019, Applied Numerical Mathematics.

[34]  A. Cheng,et al.  Trefftz and Collocation Methods , 2008 .

[35]  Zi-Cai Li,et al.  Effective condition number for numerical partial differential equations , 2008, Numer. Linear Algebra Appl..