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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 $$a$$.
##### MSC:
 65D25 Numerical differentiation
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