Abstract
In this paper, we investigate the consistency and the approximation properties of h-p clouds methods. For this purpose, a special partition of unity function space in which inverse inequalities can be established is constructed. The optimal error estimate for the h-p clouds Galerkin methods is then established. The convergence rates are measured by the radius of influence domains of weight functions instead of the mesh size as usually used in the finite element analysis.
Subsidized by the Special Fund for Major State Basic Research Projects and State Educational Ministry.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Babuška. I and J. M. Melenk, The partition of unity finite element method, Int. J. Num. Meth. Eng. 40(1997), pp. 727–758.
Belytschko, T., Y. Y. Lu and L. Gu. Element-free Galerkin methods, Int. J. Numer . Meth. Engr 37 (1994), pp. 229–256.
T. Belytsho, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods: An overview and recent developments.
S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods . Springer-Verlag, New York, 1994.
Chen J. S. and Wang H. P, New boundary condition treatments in meshfree computation problems, Comp. Meth. Appl. Nech. Engr, 187 2000.
Chen, J. S., Wu, C. T., Yoon, S., and You, Y, “A Stabilized Conforming Nodal Integration for Galerkin Meshfree Methods,” International Journal for Numerical Methods in Engineering, 50 (2001), pp. 435–466.
Duarte, C. A. and J. T. Oden. H-p clouds-an h-p meshless method. Numer . Meth. Part. Diff. Equat, (1996), pp. 1–34.
J. Gosz and W. K. Liu, Admissible approximations for essential boundary conditions in the reproducing kernel method, Comp. Mech 19 (1996), pp. 120–135.
M. Griebel and M. A. Schweitzer. A particle-partition of unity method for the solution of elliptic, parabolic and hyperbolic pdes.
W. Han and X. Meng. Error analysis of the Reproducing Kernel Particle Method, Computer Methods in Applied Mechanics and Engineering, 190 (2001), pp. 6157–6181.
W. Han, G. J. Wagner and W. K. Liu, Convergence analysis of a hierarchical enrichment of Dirichlet boundary conditions in a meshfree method, to appear.
Hu Jun. Analysis for a kind of h-p clouds Galerkin method and lower approximation of eigenvalues. MSc. Dissertationion, Xiangtan University, 2001.
Hu Jun and Huang Yun Qing. Analysis for a kind of meshless Galerkin Method, The Fourth International Conference on Engineering and Computation, 2001, Beijing, China.
W. K. Liu, S. Jun, and Y. F. Zhang. Reproducing kernel particle methods. Int. J. Numer . Meth. Engr, 20 (1995), pp. 1081–1106.
W. K. Liu, S. Li, and T. Betytscho. Moving least-square reproducing kernel methods. part I: methodology and convergence, Computer Methods in Applied Mechanics and Engineering 143(1997), pp. 113–154.
Krongauz Y., Belytschko T., Enforcement of Essential Boundary Conditions in Meshless Approximations Using Finite Elements, Computer Methods in Applied Mechanics and Engng, 131 (1996), pp. 133–145.
P. Krysl and T.Belytschko. Element-free Galerkin method: Convergence of the continuous and discontinuous shape functions. Comp. Meth. Appl. Mech. Engr. 148 (1996), pp. 257– 277
B. Nayroles, G. Touzot, and P. Villon. Generalizing the finite element method: diffuse approximation and diffuse elements. Comp. Mech. 10 (1992), pp. 307–318.
G. J. Wagner and W. K. Liu. Hierarchical enrichment for bridging scales and mesh-free boundary conditions. Int. J. Numer. Meth. Engr 50 (2001), pp. 507–524.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media New York
About this paper
Cite this paper
Hu, J., Huang, Y., Xue, W. (2002). An Optimal Error Estimate for an H-P Clouds Galerkin Method. In: Chan, T.F., Huang, Y., Tang, T., Xu, J., Ying, LA. (eds) Recent Progress in Computational and Applied PDES. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0113-8_16
Download citation
DOI: https://doi.org/10.1007/978-1-4615-0113-8_16
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-4929-7
Online ISBN: 978-1-4615-0113-8
eBook Packages: Springer Book Archive