Averaging of directional derivatives in vertices of nonobtuse regular triangulations (Q711583)

Frequently, approximations of first derivatives of a function at the grid points of a given triangulation have to be provided, where the function itself can be evaluated at the grid points. This paper proposes an approximation of the first order derivative that is up to second order accurate if the function is smooth enough. For this purpose, the author assumes several properties of the triangulation in that it must be shape regular without obtuse angles. This allows the definition of rings in several variants. The author show that a so-called reduced ring yields a Lagrange six-tuple, which is also called poised set. Then these Lagrange six-tuples are used to derive a certain averaging of the available directional derivatives along the edges of the given triangulations. Subsequently the author shows that the resulting approximation of the first-order derivative is accurate up to second order. This property is verified with two small numerical tests. Finally it is proved that the proposed derivative approximation can be seen as a recovery operator as defined by \textit{M. Ainsworth} and \textit{A. Craig} [Numer. Math. 60, No. 4, 429--463 (1992; Zbl 0757.65109)].