Existing smoothed particle hydrodynamics (SPH) formulations for simulating continuous fluids have errors that may be divergent and it has been known for some time that the SPH equations do not satisfy low-order polynomial completeness conditions. Here SPH equations are derived that have convergent error terms and a correction method is presented for enforcing low-order polynomial completeness irrespective of how many completeness conditions are required. Discretization is achieved through division of the model domain, in its initial state, into sub-domains that have Lagrangian boundaries. It is shown that boundary integrals appearing in one derivation of the SPH equations may be treated as a convergent error. In simulations of basic fluid flows convergence and zeroth-order completeness are demonstrated, but significant instabilities and a failure to conserve energy are observed. Copyright © 2009 John Wiley & Sons, Ltd.