Lecture Notes in Computational Science and Engineering The third author (C.J.C.) gratefully acknowledges the support by the International Graduate School of Science and Engineering of the Technische Universität München. 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. © 2013 Springer-Verlag Berlin Heidelberg. The good performance of HOLMES relative to other meshfree schemes in selected test cases is also critically appraised. Moreover, we show that the HOLMES of order k is dense in the Sobolev Space W^(k,p), for any 1 ≤ p < ∞. We prove uniform error bounds for the HOLMES approximates and their derivatives up to order k.
We refer to these approximation schemes as High Order Local Maximum-Entropy Approximation Schemes (HOLMES). 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. 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. Local Maximum-Entropy (LME) approximation schemes are meshfree approximation schemes that satisfy consistency conditions of order 1, i.e., they approximate affine functions exactly.
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