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HOLMES: Convergent Meshfree Approximation Schemes of Arbitrary Order and Smoothness

Bompadre, Agustín and Perotti, Luigi E. and Cyron, Christian J. and Ortiz, Michael (2012) HOLMES: Convergent Meshfree Approximation Schemes of Arbitrary Order and Smoothness. In: Meshfree Methods for Partial Differential Equations VI. Lecture Notes in Computational Science and Engineering. No.89. Springer , Berlin, pp. 111-126. ISBN 978-3-642-32978-4. https://resolver.caltech.edu/CaltechAUTHORS:20200519-145536180

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Abstract

Local Maximum-Entropy (LME) approximation schemes are meshfree approximation schemes that satisfy consistency conditions of order 1, i.e., they approximate affine functions exactly. In addition, LME approximation schemes converge in the Sobolev space W^(1,p), i.e., they are C⁰-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:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/978-3-642-32979-1_7DOIArticle
https://rdcu.be/b4g9EPublisherFree ReadCube access
ORCID:
AuthorORCID
Ortiz, Michael0000-0001-5877-4824
Additional Information:© 2013 Springer-Verlag Berlin Heidelberg. First Online: 27 September 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 Universität München.
Group:GALCIT
Funders:
Funding AgencyGrant Number
Department of Energy (DOE)DE-FC52-08NA28613
Technische Universität MünchenUNSPECIFIED
Subject Keywords:High-order meshfree interpolation
Series Name:Lecture Notes in Computational Science and Engineering
Issue or Number:89
Record Number:CaltechAUTHORS:20200519-145536180
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200519-145536180
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:103328
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:19 May 2020 22:06
Last Modified:19 May 2020 22:06

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