Geodesic
Operators approximating partial derivatives at vertices of triangulations by averaging
Mathematica Bohemica, Tome 135 (2010) no. 4, pp. 363-372

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Let $\mathcal T_h$ be a triangulation of a bounded polygonal domain $\Omega \subset \Re ^2$, $\mathcal L_h$ 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 $a$.
Let $\mathcal T_h$ be a triangulation of a bounded polygonal domain $\Omega \subset \Re ^2$, $\mathcal L_h$ 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 $a$.
DOI : 10.21136/MB.2010.140827
Classification : 65D25
Keywords: partial derivative; high-order approximation; recovery operator 
Dalík, Josef. Operators approximating partial derivatives at vertices of triangulations by averaging. Mathematica Bohemica, Tome 135 (2010) no. 4, pp. 363-372. doi: 10.21136/MB.2010.140827
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