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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S.N. Atluri and S. Shen, The Meshless Local Petrov–Galerkin (MLPG) Method (Tech Science Press, Encino, CA, 2002).
J.W. Barrett and J.F. Blowey, Finite element approximation of the Cahn–Hilliard equation with concentration dependent mobility, Math. Comp. 68 (1999) 487–517.
T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods: An overview and recent developments, Comput. Methods Appl. Mech. Engrg. 139 (1996) 3–47.
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, Berlin, 1994).
M.D. Buhmann, Radial basis functions, Acta Numerica 9 (2000) 1–38.
G. Fairweather, Finite Element Galerkin Methods for Differential Equations (Marcel Dekker, New York, 1978).
G.E. Fassauer, Solving partial differential equation by collocation with radial basis functions, in: Proc. of Chamonix, eds. A. LeMehaute, C. Rabut and L. Schumaker (Vanderbilt Univ. Press, Nashville, TN, 1997) pp. 131–138.
C. Franke and R. Schaback, Convergence order estimates of meshless collocation methods using radial basis functions, Adv. Comput. Math. 8 (1998) 381–399.
M.A. Golberg and C.S. Chen, Discrete Projection Methods for Integral Equations (Computational Mechanics Publications, Southampton, 1997).
M. Griebel and M.A. Schweitzer, eds., Meshfree Methods for Partial Differential Equations (Springer, Berlin, 2003).
Y.C. Hon, K.F. Cheung, X.Z. Mao and E.J. Kansa, Multiquadric solution for shallow water equations, ASCE J. Hydraulic Engrg. 125 (1999) 524–533.
Y.C. Hon, M.W. Lu, W.M. Xue and Y.M. Zhu, Multiquadric method for the numerical solution of a biphasic mixture model, Appl. Math. Comput. 88 (1997) 153–175.
Y.C. Hon and X.Z. Mao An efficient numerical scheme for Burgers’ equation, Appl. Math. Comput. 95 (1998) 37–50.
Y.C. Hon and X.Z. Mao A radial basis function method for solving options pricing model, Financial Engrg. 8 (1999) 31–49.
Y.C. Hon and Z. Wu Additive Schwarz domain decomposition with radial basis approximation, Internat. J. Appl. Math. 4 (2000) 81–98.
E. Kansa and Y.C. Hon, eds., Radial basis functions and partial differential equations, Comput. Math. Appl. 43 (2002) 275–619.
J. Li, Full-order convergence of a mixed finite element method for fourth-order elliptic equations, J. Math. Anal. Appl. 230 (1999) 329–349.
J. Li, C.S. Chen, D. Pepper and Y. Chen, Mesh-free method for groundwater modeling, in: Proc. of 24th World Conf. on Boundary Element Methods Incorporating Meshless Solutions Seminar, eds. C.A. Brebbia, A. Tadeu and V. Popov (WIT Press, Southampton, 2002) pp. 145–154.
J. Li, Y.C. Hon and C.S. Chen, Numerical comparisons of two meshless methods using radial basis functions, Engrg. Anal. Boundary Elements 26 (2002) 205–225.
W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl. 4 (1988) 77–89.
C.A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx. 2 (1986) 11–22.
F.J. Narcowich and J.D. Ward, Norms of inverses and condition numbers for matrices associated with scattered data, J. Approx. Theory 64 (1991) 69–94.
R.D. Passo, H. Garcke and G. Grun, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal. 29 (1998) 321–342.
M.J.D. Powell, The theory of radial basis function approximation in 1990, in: Advances in Numerical Analysis, ed. W. Light (Oxford Science, Oxford, 1992) pp. 105–210.
R. Schaback, A unified theory of radial basis functions: Native Hilbert spaces for radial basis functions II, J. Comput. Appl. Math. 121 (2000) 165–177.
R. Schaback and Z. Wu, Operators on radial functions, J. Comput. Appl. Math. 73 (1996) 257–270.
H. Wendland, Numerical solution of variational problems by radial basis functions, in: Approximation Theory, Vol. 9, eds. C.K. Chui and L.L. Schumaker (Vanderbilt Univ. Press, Nashville, TN, 1998) pp. 361–368.
H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998) 258–272.
H. Wendland, Meshless Galerkin methods using radial basis functions, Math. Comp. 68 (1999) 1521–1531.
M.F. Wheeler, A priori L 2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973) 723–759.
S.M. Wong, Y.C. Hon, T.S. Li, S.L. Chung and E.J. Kansa, Multizone decomposition of time-dependent problems using the multiquadric scheme, Comput. Math. Appl. 37 (1999) 23–43.
Z. Wu, Multivariate compactly supported positive definite radial functions, Adv. Comput. Math. 4 (1995) 283–292.
Z. Wu, Solving PDE with radial basis functions, in: Advances in Computational Mathematics, eds. Z. Chen, Y. Li, C.A. Micchelli and Y. Xu (Marcel Dekker, New York, 1999) pp. 537–543.
Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993) 13–27.
X. Zhou, Y.C. Hon and J. Li, Overlapping domain decomposition method by radial basis functions, Appl. Numer. Math. 44 (2003) 243–257.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Z. Wu and B.Y.C. Hon
AMS subject classification
35G15, 65N12
Rights and permissions
About this article
Cite this article
Li, J. Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions. Adv Comput Math 23, 21–30 (2005). https://doi.org/10.1007/s10444-004-1807-7
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10444-004-1807-7