Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T16:30:36.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Error estimates for Modified Local Shepard's Formulasin Sobolev spaces

Published online by Cambridge University Press:  15 November 2003

Carlos Zuppa
Carlos Zuppa*
Affiliation:
Departamento de Matemáticas, Universidad Nacional de San Luis, Chacabuco y Pedernera, 5700 San Luis, Argentina. [email protected].
Get access

Abstract

Interest in meshfree methods in solving boundary-value problems has grownrapidly in recent years. A meshless method that has attracted considerableinterest in the community of computational mechanics is built around theidea of modified local Shepard's partition of unity. For these kinds ofapplications it is fundamental to analyze the order of the approximation inthe context of Sobolev spaces. In this paper, we study two differenttechniques for building modified local Shepard's formulas, and we provide atheoretical analysis for error estimates of the approximation in Sobolevnorms. We derive Jackson-type inequalities for h-p cloud functionsusing the first construction. These estimates are important in the analysisof Galerkin approximations based on local Shepard's formulas or h-pcloud functions.

Keywords

Error estimatesShepard's formulasJacksoninequalitiesSobolev spaces.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 6 , November 2003 , pp. 973 - 989
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams, Sobolev Spaces. Academic Press, Inc., Orlando (1975).
S.C. Brener and L.R. Scott, The Mathematical Theory of Finite Elements Methods. Springer-Verlag, New York (1994).
P.G. Ciarlet, The Finite Elements Method for Elliptic Problems. North-Holland, Amsterdam (1978).
C.A. Duarte and J.T. Oden, Hp clouds-a meshless method to solve boundary-value problems. Technical Report 95-05, TICAM, The University of Texas at Austin (1995).
Duarte, C.A. and Oden, J.T., H-p clouds-an h-p meshless method. Numer. Methods Partial Differential Equations 1 (1996) 1–34.
Duarte, C.A.M., Liszka, T.J. and Tworzydlo, W.W., hp-meshless cloud method. Comput. Methods Appl. Mech. Engrg. 139 (1996) 263–288. CrossRef
Durán, R.G., On polynomial approximation in Sobolev spaces. SIAM J. Numer. Anal. 20 (1983) 985–988. CrossRef
Han, W. and Meng, X., Error analysis of the reproducing kernel particle method. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6157–6181. CrossRef
Lu, Y.Y., Belyschko, T. and Element-free Galerkin, L. Gu methods. Internat. J. Numer. Methods Engrg. 37 (1994) 229–256.
E. Oñate, R. Taylor, O.C. Zienkiewicz and S. Idelshon, Moving least square approximations for the solutions of differential equations. Technical Report, CIMNE, Santa Fé, Argentina (1995).
Renka, R.J., Multivariate interpolation of large sets of scattered data. ACM Trans. Math. Software 14 (1988) 139–148. CrossRef
L.L. Schumaker, Fitting surfaces to scattered data, in Approximation Theory II, Academic Press, Inc., New York (1970).
D.D. Shepard, A Two Dimensional Interpolation Function for Irregularly Spaced Data. Proc. 23rd Nat. Conf. ACM (1968).
Verfúrth, R., A note on polynomial approximation in Sobolev spaces. ESAIM: M2AN 33 (1999) 715–719. CrossRef
C. Zuppa, Error estimates for modified local Shepard's formulaes. Appl. Numer. Math. (to appear).
Zuppa, C., Good quality point sets and error estimates for moving least square approximations. Appl. Numer. Math. 47 (2003) 575–585. CrossRef