Five methods for finding derivatives of finite element solutions are compared: (1) direct differentiation of the finite element approximating functions, with weighted averaging at element edges, (2) local smoothing by the Hinton-Campbell method, (3) an extension of the method of Zhu and Zienkiewicz, (4) global smoothing, and (5) a method based on Green's second identity, or equivalently on the Poisson integral. First order derivatives are examined in detail because most of the methods cannot determine higher derivatives. Where higher derivatives are available, the Poisson integral method is best in all cases. Finite element approximations to a known harmonic polynomial and a transcendental function were computed on second-order quadrilateral finite elements, and on a series of uniform triangular meshes up to fourth order. Fields in a deep electric machine slot were similarly analyzed. Direct differentiation gave low accuracy; the three smoothing methods yielded adequate results, and the Poisson integral method gave stable results of high accuracy. Computing times correlate inversely with the accuracy achieved.
Silvester et al. (Wed,) studied this question.