Geodesic
The invertibility of the isoparametric mappings for triangular quadratic Lagrange finite elements
Applications of Mathematics, Tome 57 (2012) no. 5, pp. 445-462

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A reference triangular quadratic Lagrange finite element consists of a right triangle $\hat K$ with unit legs $S_1$, $S_2$, a local space $\hat {\mathcal L}$ of quadratic polynomials on $\hat K$ and of parameters relating the values in the vertices and midpoints of sides of $\hat K$ to every function from $\hat {\mathcal L}$. Any isoparametric triangular quadratic Lagrange finite element is determined by an invertible isoparametric mapping ${\mathcal F}_h=(F_1,F_2)\in \hat {\mathcal L}\times \hat {\mathcal L}$. We explicitly describe such invertible isoparametric mappings ${\mathcal F}_h$ for which the images ${\mathcal F}_h(S_1)$, ${\mathcal F}_h(S_2)$ of the segments $S_1$, $S_2$ are segments, too. In this way we extend the well-known result going back to W. B. Jordan, 1970, characterizing those invertible isoparametric mappings whose restrictions to the segments $S_1$ and $S_2$ are linear.
DOI : 10.1007/s10492-012-0026-7
Classification : 65M50, 65M60, 65N30, 65N50
Keywords: isoparametric triangular quadratic Lagrange finite element; invertible isoparametric mapping; initial or boundary value problems 
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Dalík, Josef. The invertibility of the isoparametric mappings for triangular quadratic Lagrange finite elements. Applications of Mathematics, Tome 57 (2012) no. 5, pp. 445-462. doi: 10.1007/s10492-012-0026-7

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