On derivative estimation and the solution of least squares problems (Q955074)

Two numerical algorithms are devised to approximate the gradient \(\nabla f\) of a function \(f(x,y)\) up to second-order accuracy where \(f\) determines a smooth surface and values \(f(x_i,y_i)\), \(i= 1,2,\dots,n\), are given on an irregular point set. Both methods establish an overdetermined set of \(p\) linear equations at each point \((x_i,y_i)\) whose unknown include \(\partial f/\partial x\), \(\partial f/\partial y\) at \((x_i,y_i)\). These equations are sovled by the least-squares method. The first method employs Taylor's expansion of second-order of \(f(x,y)\) about \((x_i,y_i)\) to \(p\) neighboring points \((x_{ij}, y_{ij})\), \(j= 1,2,\dots, p\), \(p> 5\). The second method was obtained by the authors when determining the gradient in connection with discrete approximation to the diffusion equation on a bounded region in \(\mathbb{R}^2\). Both methods are shown to have the same degree of error. Numerical results are given for two surface problems and a steady state diffusion problem on a square [cf. \textit{P. A. Jayantha} and \textit{I. W. Turner}, Int. J. Numer. Methods Heat Fluid Flow 13, No.~1, 31--56 (2003; Zbl 1123.76345)].