We present a family of approximation schemes, which we refer to as second-order maximum-entropy (max-ent) approximation schemes, that extends the first-order local max-ent approximation schemes to second-order consistency. This method retains the fundamental properties of first-order max-ent schemes, namely the shape functions are smooth, non-negative, and satisfy a weak Kronecker-delta property at the boundary. This last property makes the imposition of essential boundary conditions in the numerical solution of partial differential equations trivial. The evaluation of the shape functions is not explicit, but it is very efficient and robust. To our knowledge, the proposed method is the first higher-order scheme for function approximation from unstructured data in arbitrary dimensions with non-negative shape functions. As a consequence, the approximants exhibit variation diminishing properties, as well as an excellent behavior in structural vibrations problems as compared with the Lagrange finite elements, MLS-based meshfree methods and even B-Spline approximations, as shown through numerical experiments. When compared with usual MLS-based second-order meshfree methods, the shape functions presented here are much easier to integrate in a Galerkin approach, as illustrated by the standard benchmark problems
