HOLMES: Convergent Meshfree Approximation Schemes of Arbitrary Order and Smoothness

  • Agustín Bompadre
  • Luigi E. Perotti
  • Christian J. Cyron
  • Michael Ortiz
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 89)


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 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 \(1 \leq p < \infty \). The good performance of HOLMES relative to other meshfree schemes in selected test cases is also critically appraised.


High-order meshfree interpolation 



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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Agustín Bompadre
    • 1
  • Luigi E. Perotti
    • 1
  • Christian J. Cyron
    • 2
  • Michael Ortiz
    • 1
  1. 1.Graduate Aerospace LaboratoriesCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Institute for Computational MechanicsTechnische Universität MünchenGarchingGermany

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