Convergence of corrected derivative methods for second-order linear partial differential equations

Approximations where the derivatives are corrected so as to satisfy linear completeness on the derivatives are investigated. These techniques are used in particle methods and other mesh-free methods. The basic approximation is a Shepard interpolant which possesses only constant completeness. The derivatives are then corrected to meet linear completeness conditions. It is shown that the resulting methods have order h convergence in the energy and L2 norms. The proof is subject to an inf{sup condition which is studied numerically. Numerical studies of the Poisson equation verify the convergence estimates. Copyright