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Operators approximating partial derivatives at vertices of triangulations by averaging. (English) Zbl 1224.65057
Summary: Let \(\mathcal {T}_{h}\) be a triangulation of a bounded polygonal domain \(\Omega \subset \mathbb {R} ^2\), \(\mathcal {L}_{h}\) the space of the functions from \(C(\overline \Omega )\) linear on the triangles from \(\mathcal {T}_h\) and \(\Pi _h\) the interpolation operator from \(C(\overline \Omega )\) to \(\mathcal {L}_h\). For a unit vector \(z\) and an inner vertex \(a\) of \(\mathcal {T}_{h}\), we describe the set of vectors of coefficients such that the related linear combinations of the constant derivatives \(\partial \Pi _{h}(u)/\partial z\) on the triangles surrounding \(a\) are equal to \(\partial u/\partial z(a)\) 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 \(\nabla u(a)\) cannot be expressed as linear combinations of the constant gradients \(\nabla \Pi _{h}(u)\) on the triangles surrounding \(a\).
65D25 Numerical differentiation
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