Radial basis functions for solving near singular Poisson problems

In this paper, we investigate the use of radial basis functions for solving Poisson problems with a near-singular inhomogeneous source term. The solution of the Poisson problem is first split into two parts: near-singular solution and smooth solution. A method for evaluating the near-singular particular solution is examined. The smooth solution is further split into a particular solution and a homogeneous solution. The MPS-DRM approach is adopted to evaluate the smooth solution. Copyright © 2003 John Wiley & Sons, Ltd.

[1]  I. Sloan,et al.  Collocation methods for the second kind integral equations with non-compact operators , 1989 .

[2]  Gennady Mishuris.  Interface crack and nonideal interface concept (Mode III) , 2001 .

[3]  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 .

[4]  Zongmin Wu,et al.  Compactly supported positive definite radial functions , 1995 .

[5]  Mahadevan Ganesh,et al.  Particular solutions of 3D Helmholtz-type equations using compactly supported radial basis functions , 2000 .

[6]  C. S. Chen,et al.  Particular solutions of Helmholtz-type operators using higher order polyhrmonic splines , 1999 .

[7]  Gregory E. Fasshauer,et al.  Solving differential equations with radial basis functions: multilevel methods and smoothing , 1999, Adv. Comput. Math..

[8]  Gennady Mishuris,et al.  Asymptotic Behaviour of the Elastic Solution near the Tip of a Crack Situated at a Nonideal Interface , 2001 .

[9]  C. S. Chen,et al.  Some comments on the use of radial basis functions in the dual reciprocity method , 1998 .

[10]  M. Floater,et al.  Multistep scattered data interpolation using compactly supported radial basis functions , 1996 .

[11]  Andreas Karageorghis.  Modified methods of fundamental solutions for harmonic and biharmonic problems with boundary singularities , 1992 .

[12]  C. Micchelli.  Interpolation of scattered data: Distance matrices and conditionally positive definite functions , 1986 .

[13]  Graeme Fairweather,et al.  The method of fundamental solutions for elliptic boundary value problems , 1998, Adv. Comput. Math..

[14]  Gennady Mishuris.  Stress singularity at a crack tip for various intermediate zones in bimaterial structures (mode III) , 1999 .

[15]  Andreas Poullikkas,et al.  Methods of fundamental solutions for harmonic and biharmonic boundary value problems , 1998 .

[16]  Gennady Mishuris,et al.  On boundary value problems in fracture of elastic Composites , 1995, European Journal of Applied Mathematics.

[17]  Michael A. Golberg,et al.  Some recent results and proposals for the use of radial basis functions in the BEM , 1999 .

[18]  Holger Wendland,et al.  Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree , 1995, Adv. Comput. Math..

[19]  Carlos Alberto Brebbia,et al.  Dual reciprocity method using compactly supported radial basis functions , 1999 .

[20]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .