A meshless Galerkin method for Dirichlet problems using radial basis functions

In this paper, a numerical method is given for partial differential equations, which combines the use of Lagrange multipliers with radial basis functions. It is a new method to deal with difficulties that arise in the Galerkin radial basis function approximation applied to Dirichlet (also mixed) boundary value problems. Convergence analysis results are given. Several examples show the efficiency of the method using TPS or Sobolev splines.

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

[2]  Juhani Pitkäranta,et al.  The finite element method with Lagrange multipliers for domains with corners , 1981 .

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

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

[5]  Carsten Franke,et al.  Convergence order estimates of meshless collocation methods using radial basis functions , 1998, Adv. Comput. Math..

[6]  Holger Wendland,et al.  Error Estimates for Interpolation by Compactly Supported Radial Basis Functions of Minimal Degree , 1998 .

[7]  Rolf Stenberg,et al.  On some techniques for approximating boundary conditions in the finite element method , 1995 .

[8]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[9]  H. Wendland.  Sobolev-type error estimates for interpolation by radial basis functions , 1997 .

[10]  Robert Schaback,et al.  On unsymmetric collocation by radial basis functions , 2001, Appl. Math. Comput..

[11]  Holger Wendland,et al.  Meshless Galerkin methods using radial basis functions , 1999, Math. Comput..

[12]  Michael Griebel,et al.  A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic, and Hyperbolic PDEs , 2000, SIAM J. Sci. Comput..

[13]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[14]  Zongmin Wu,et al.  Additive Schwarz Domain Decomposition with Radial Basis Approximation , 2002 .

[15]  Robert Schaback,et al.  Improved error bounds for scattered data interpolation by radial basis functions , 1999, Math. Comput..

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

[17]  Helio J. C. Barbosa,et al.  Boundary Lagrange multipliers in finite element methods: Error analysis in natural norms , 1992 .

[18]  Juhani Pitkäranta,et al.  Local stability conditions for the Babuška method of Lagrange multipliers , 1980 .