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Error Bounds for Least Squares Gradient Estimates

Least squares gradient estimates find application in many fields of computational science, particularly for the purposes of surface fitting and gradient reconstruction in computational fluid dynamics and data visualization. In this paper we derive error bounds for classical and weighted least squares gradient estimates. The bounds reflect how the number of points used in the least squares stencil and the smallest singular value of the least squares matrix impact the accuracy of these estimates. We show how an extrapolation method based on Householder transformations provides substantially tighter bounds. Numerical case studies are presented to elucidate our theory for a data set taken from Franke [Math. Comp., 38 (1982), pp. 181-200].