OPERATORS APPROXIMATING PARTIAL DERIVATIVES AT VERTICES OF TRIANGULATIONS BY AVERAGING

Let T-h be a triangulation of a bounded polygonal domain Omega subset of R-2, L-h the space of the functions from C((Omega) over bar) linear on the triangles from T-h and Pi(h) the interpolation operator from C((Omega) over bar) to L-h. For a unit vector z and an inner vertex a of T-h, we describe the set of vectors of coefficients such that the related linear combinations of the constant derivatives delta Pi(h)(u)/delta z on the triangles surrounding a are equal to delta u/delta z(a) for all polynomials u of the total degree less than or equal to two. Then we prove that, generally, the values of the so-called recovery operators approximating the gradient del u(a) cannot be expressed as linear combinations of the constant gradients del Pi(h)(u) on the triangles surrounding a.