  The Meshless Local Petrov{Galerkin (MLPG) method is one of the popular
meshless methods that has been used very successfully to solve several types of
boundary value problems since the late nineties. In this paper, using a
generalized moving least squares (GMLS) approximation, a new direct MLPG
technique, called DMLPG, is presented. Following the principle of meshless
methods to express everything "entirely in terms of nodes", the generalized MLS
recovers test functionals directly from values at nodes, without any detour via
shape functions. This leads to a cheaper and even more accurate scheme. In
particular, the complete absence of shape functions allows numerical
integrations in the weak forms of the problem to be done over low{degree
polynomials instead of complicated shape functions. Hence, the standard MLS
shape function subroutines are not called at all. Numerical examples illustrate
the superiority of the new technique over the classical MLPG. On the
theoretical side, this paper discusses stability and convergence for the new
discretizations that replace those of the standard MLPG. However, it does not
treat stability, convergence, or error estimation for the MLPG as a whole. This
should be taken from the literature on MLPG.
