Spline representation of incompressible flow

A region occupied by an incompressible fluid is considered. The present study is concerned with the development of a foolproof and efficient procedure for recovering the velocity components at any point of the region, taking into account the assumption that the integral of the normal component of fluid velocity over each face of a rectilinear subdivision of that region is given. The considered task arises, for instance, in the context of a computation in which these integrals appear as the result of solving a finite-element or finite-difference discretization of the equations of fluid flow. Attention is given to a simple solution, lemmas concerning natural splines, the outline of a procedure in two dimensions, an adjustment for solid walls on the boundary, and an extension to three dimensions.