On derivative estimation and the solution of least squares problems

Surface interpolation finds application in many aspects of science and technology. Two specific areas of interest are surface reconstruction techniques for plant architecture and approximating cell face fluxes in the finite volume discretisation strategy for solving partial differential equations numerically. An important requirement of both applications is accurate local gradient estimation. In surface reconstruction this gradient information is used to increase the accuracy of the local interpolant, while in the finite volume framework accurate gradient information is essential to ensure second order spatial accuracy of the discretisation.In this work two different least squares strategies for approximating these local gradients are investigated and the errors associated with each analysed. It is shown that although the two strategies appear different, they produce the same least squares error. Some carefully chosen case studies are used to elucidate this finding.