Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions

Abstract By extending Wendland’s meshless Galerkin methods using RBFs, we develop mixed methods for solving fourth-order elliptic and parabolic problems by using RBFs. Similar error estimates as classical mixed finite element methods are proved.

[1]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[2]  S. Atluri,et al.  The meshless local Petrov-Galerkin (MLPG) method , 2002 .

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

[4]  Y. Hon,et al.  Multiquadric method for the numerical solution of a biphasic mixture model , 1997 .

[5]  Kwok Fai Cheung,et al.  Multiquadric Solution for Shallow Water Equations , 1999 .

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

[7]  W. Madych,et al.  Multivariate interpolation and condi-tionally positive definite functions , 1988 .

[8]  M. Wheeler A Priori L_2 Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations , 1973 .

[9]  M. Golberg,et al.  Discrete projection methods for integral equations , 1996 .

[10]  Y. Hon,et al.  Overlapping domain decomposition method by radial basis functions , 2003 .

[11]  E. J. Kansa,et al.  Multizone decomposition for simulation of time-dependent problems using the multiquadric scheme , 1999 .

[12]  Harald Garcke,et al.  On A Fourth-Order Degenerate Parabolic Equation: Global Entropy Estimates, Existence, And Qualitativ , 1998 .

[13]  Zongmin Wu,et al.  Local error estimates for radial basis function interpolation of scattered data , 1993 .

[14]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[15]  E. Kansa Local, Point-Wise Rotational Transformations of the Conservation Equations into Stream-Wise Coordinates , 2002 .

[16]  Y. C. Hon,et al.  Numerical comparisons of two meshless methods using radial basis functions , 2002 .

[17]  Y. Chen,et al.  Mesh-free method for groundwater modeling , 2002 .

[18]  G. Allasia Approximating potential integrals by cardinal basis interpolants on multivariate scattered data , 2002 .

[19]  Jichun Li Full-Order Convergence of a Mixed Finite Element Method for Fourth-Order Elliptic Equations , 1999 .

[20]  F. J. Narcowich,et al.  Norms of inverses and condition numbers for matrices associated with scattered data , 1991 .

[21]  R. Schaback A unified theory of radial basis functions Native Hilbert spaces for radial basis functions II , 2000 .

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

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

[24]  Michael Griebel,et al.  Meshfree Methods for Partial Differential Equations , 2002 .

[25]  G. Fairweather Finite Element Galerkin Methods for Differential Equations , 1978 .

[26]  Robert Schaback,et al.  Operators on radial functions , 1996 .

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

[28]  John W. Barrett,et al.  Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility , 1999, Math. Comput..

[29]  Benny Y. C. Hon,et al.  An efficient numerical scheme for Burgers' equation , 1998, Appl. Math. Comput..

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

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