Averaging of directional derivatives in vertices of nonobtuse regular triangulations

Autor: Josef Dalík
Rok vydání: 2010
Předmět:
Zdroj: Numerische Mathematik. 116:619-644
ISSN: 0945-3245
0029-599X
DOI: 10.1007/s00211-010-0316-5
Popis: For a shape-regular triangulation $${\mathcal{T}_h}$$ without obtuse angles of a bounded polygonal domain $${\Omega\subset\Re^2}$$, let $${\mathcal L_h}$$ be the space of continuous functions linear on the triangles from $${\mathcal{T}_h}$$ and Πh the interpolation operator from $${C(\overline\Omega)}$$ to $${\mathcal L_h}$$. This paper is devoted to the following classical problem: Find a second-order approximation of the derivative $${\partial u/\partial z(a)}$$ in a direction z of a function $${u\in C^3(\overline\Omega)}$$ in a vertex a in the form of a linear combination of the constant directional derivatives $${\partial \Pi_h(u)/\partial z}$$ on the triangles surrounding a. An effective procedure for such an approximation is presented, its error is proved to be of the size O(h 2), an operator $${\mbox{W}_h: \mathcal L_h\longrightarrow\mathcal L_h\times\mathcal L_h}$$ relating a second-order approximation Wh [Πh (u)] of $${\nabla u}$$ to every $${u\in C^3(\overline\Omega)}$$ is constructed and shown to be a so-called recovery operator. The accuracy of the presented approximation is compared with the accuracies of the local approximations by other known techniques numerically.
Databáze: OpenAIRE
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