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Convergent meshfree approximation schemes of arbitrary order and smoothness

Bompadre, A. and Perotti, L. E. and Cyron, C. J. and Ortiz, M. (2012) Convergent meshfree approximation schemes of arbitrary order and smoothness. Computer Methods in Applied Mechanics and Engineering, 221-222 . pp. 83-103. ISSN 0045-7825. doi:10.1016/j.cma.2012.01.020. https://resolver.caltech.edu/CaltechAUTHORS:20120525-095216954

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Abstract

Local Maximum-Entropy (LME) approximation schemes are meshfree approximation schemes that satisfy consistency conditions of order one, i.e., they approximate affine functions exactly. In addition, LME approximation schemes converge in the Sobolev space W^(1,p), i.e., they are C^0-continuous in the conventional terminology of finite-element interpolation. Here we present a generalization of the Local Max-Ent approximation schemes that are consistent to arbitrary order, i.e., interpolate polynomials of arbitrary degree exactly, and which converge in W^(k,p), i.e., they are C^k-continuous to arbitrary order k. We refer to these approximation schemes as High Order Local Maximum-Entropy Approximation Schemes (HOLMES). We prove uniform error bounds for the HOLMES approximates and their derivatives up to order k. Moreover, we show that the HOLMES of order k is dense in the Sobolev space W^(k,p), for any 1⩽p<∞. The good performance of HOLMES relative to other meshfree schemes in selected test cases is also critically appraised.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.cma.2012.01.020DOIUNSPECIFIED
http://www.sciencedirect.com/science/article/pii/S004578251200031XPublisherUNSPECIFIED
ORCID:
AuthorORCID
Ortiz, M.0000-0001-5877-4824
Additional Information:© 2012 Elsevier B.V. Received 3 August 2011. Revised 25 November 2011. Accepted 31 January 2012. Available online 14 February 2012. The support of the Department of Energy National Nuclear Security Administration under Award Number DE-FC52-08NA28613 through Caltech’s ASC/PSAAP Center for the Predictive Modeling and Simulation of High Energy Density Dynamic Response of Materials is gratefully acknowledged. The third author (C.J.C.) gratefully acknowledges the support by the International Graduate School of Science and Engineering of the Technische Universitat Munchen.
Funders:
Funding AgencyGrant Number
Department of Energy (DOE) National Nuclear Security AdministrationDE-FC52-08NA28613
Technische Universitat Munchen International Graduate School of Science and EngineeringUNSPECIFIED
Subject Keywords:Meshfree interpolation; Convergence analysis; High-order interpolation; Smooth interpolation
DOI:10.1016/j.cma.2012.01.020
Record Number:CaltechAUTHORS:20120525-095216954
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20120525-095216954
Official Citation:A. Bompadre, L.E. Perotti, C.J. Cyron, M. Ortiz, Convergent meshfree approximation schemes of arbitrary order and smoothness, Computer Methods in Applied Mechanics and Engineering, Volumes 221–222, 1 May 2012, Pages 83-103, ISSN 0045-7825, 10.1016/j.cma.2012.01.020.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:31647
Collection:CaltechAUTHORS
Deposited By: Jason Perez
Deposited On:25 May 2012 20:28
Last Modified:09 Nov 2021 19:57

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