A Kansa-Radial Basis Function Method for Elliptic Boundary Value Problems in Annular Domains

We employ a Kansa-radial basis function (RBF) method for the numerical solution of elliptic boundary value problems in annular domains. This discretization leads, with an appropriate selection of collocation points and for any choice of RBF, to linear systems in which the matrices possess block circulant structures. These linear systems can be solved efficiently using matrix decomposition algorithms and fast Fourier transforms. A suitable value for the shape parameter in the various RBFs used is found using the leave-one-out cross validation algorithm. In particular, we consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy–Navier equations of elasticity. In addition to its simplicity, the proposed method can both achieve high accuracy and solve large-scale problems. The feasibility of the proposed techniques is illustrated by several numerical examples.

[1]  Andreas Karageorghis,et al.  A matrix decomposition MFS algorithm for certain linear elasticity problems , 2006, Numerical Algorithms.

[2]  Martin D. Buhmann,et al.  Radial Basis Functions: Theory and Implementations: Preface , 2003 .

[3]  G. Alexits.  Approximation theory , 1983 .

[4]  S. R. Simanca,et al.  On Circulant Matrices , 2012 .

[5]  E. Kansa,et al.  Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations , 2000 .

[6]  A. Karageorghis,et al.  Matrix decomposition RBF algorithm for solving 3D elliptic problems , 2009 .

[7]  Andreas Karageorghis.  Efficient Kansa-type MFS algorithm for elliptic problems , 2009, Numerical Algorithms.

[8]  Tobin A. Driscoll,et al.  Radial Basis Function Interpolation on Irregular Domain through Conformal Transplantation , 2010, J. Sci. Comput..

[9]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[10]  Elisabeth Larsson,et al.  Stable Computations with Gaussian Radial Basis Functions , 2011, SIAM J. Sci. Comput..

[11]  Robert Schaback,et al.  Error estimates and condition numbers for radial basis function interpolation , 1995, Adv. Comput. Math..

[12]  Andreas Karageorghis,et al.  Efficient MFS Algorithms for Problems in Thermoelasticity , 2013, J. Sci. Comput..

[13]  Cameron Thomas Mouat.  Fast algorithms and preconditioning techniques for fitting radial basis functions. , 2001 .

[14]  Dragan Poljak,et al.  Boundary Elements and Other Mesh Reduction Methods XXIX , 2007 .

[15]  L. Schumaker,et al.  Surface Fitting and Multiresolution Methods , 1997 .

[16]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[17]  Andreas Karageorghis.  Efficient MFS Algorithms for Inhomogeneous Polyharmonic Problems , 2011, J. Sci. Comput..

[18]  Andreas Karageorghis,et al.  Matrix decomposition MFS algorithms for elasticity and thermo-elasticity problems in axisymmetric domains , 2007 .

[19]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[20]  B. Fornberg,et al.  A numerical study of some radial basis function based solution methods for elliptic PDEs , 2003 .

[21]  Carlos Alberto Brebbia,et al.  Progress in Boundary Element Methods , 1981 .

[22]  A. U.S.,et al.  Stable Computation of Multiquadric Interpolants for All Values of the Shape Parameter , 2003 .

[23]  Gregory E. Fasshauer,et al.  On choosing “optimal” shape parameters for RBF approximation , 2007, Numerical Algorithms.

[24]  C. S. Chen,et al.  The Method of Particular Solutions for Solving Certain Partial Differential Equations , 2009 .

[25]  C. Fan,et al.  The method of approximate particular solutions for solving certain partial differential equations , 2012 .

[26]  C. S. Chen,et al.  A Kansa-RBF method for Poisson problems in annular domains , 2014 .

[27]  A. Cheng.  Multiquadric and its shape parameter—A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation , 2012 .

[28]  Nam Mai-Duy,et al.  Indirect RBFN method with thin plate splines for numerical solution of differential equations , 2003 .

[29]  Andreas Karageorghis.  The method of fundamental solutions for elliptic problems in circular domains with mixed boundary conditions , 2014, Numerical Algorithms.

[30]  Andreas Karageorghis,et al.  A matrix decomposition RBF algorithm: Approximation of functions and their derivatives , 2007 .

[31]  Graeme Fairweather,et al.  Matrix decomposition algorithms for elliptic boundary value problems: a survey , 2011, Numerical Algorithms.

[32]  Martin D. Buhmann,et al.  A new class of radial basis functions with compact support , 2001, Math. Comput..

[33]  S. C. Fan,et al.  Local multiquadric approximation for solving boundary value problems , 2003 .

[34]  Martin D. Buhmann,et al.  Radial Basis Functions , 2021, Encyclopedia of Mathematical Geosciences.

[35]  Wen Chen,et al.  The Localized RBFs Collocation Methods for Solving High Dimensional PDEs , 2013 .

[36]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.