On generalized moving least squares and diffuse derivatives

The moving least squares (MLS) method provides an approximation û of a function u based solely on values u(xj) of u on scattered ‘meshless’ nodes xj. Derivatives of u are usually approximated by derivatives of û. In contrast to this, we directly estimate derivatives of u from the data, without any detour via derivatives of û. This is a generalized MLS technique, and we prove that it produces diffuse derivatives as introduced by Nyroles et al. (1992, Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech., 10, 307–318). Consequently, these turn out to be efficient direct estimates of the true derivatives, without anything ‘diffuse’ about them, and we prove optimal rates of convergence towards the true derivatives. Numerical examples confirm this, and we finally show how the use of shifted and scaled polynomials as basis functions in the generalized and standard MLS approximation stabilizes the algorithm.