Error bounds for GMLS derivatives approximations of Sobolev functions

Abstract

This paper provides the error estimates for generalized moving least squares (GMLS) derivatives approximations of a Sobolev function in L p norms and extends them for local weak forms of DMLPG methods. Sometimes they are called diffuse or uncertain derivatives, but precisely they are direct approximants of exact derivatives which possess the optimal rates of convergence. GMLS derivatives approximations are different from the standard derivatives of MLS approximation. While they are much easier to evaluate at considerably lower cost, in this article the same orders of convergence with comparison to the standard derivatives are obtained for them.

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“…It is worthy to note that, by this modification we do not lose the order of convergence. This has been analytically proven in [15,16] for different definitions of functionals, specially for the local weak forms of DMLPG.…”

“…They are different from the standard derivatives (2.4), and in meshless literature they are called diffuse or uncertain derivatives. But [15] and [16] prove the optimal rate of convergence for them toward the exact derivatives, and thus there is nothing diffuse or uncertain about them. As suggested in [15], they can be called GMLS derivative approximations.…”