A C1-P2 finite element without nodal basis
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 2, p. 175-192

A new finite element, which is continuously differentiable, but only piecewise quadratic polynomials on a type of uniform triangulations, is introduced. We construct a local basis which does not involve nodal values nor derivatives. Different from the traditional finite elements, we have to construct a special, averaging operator which is stable and preserves quadratic polynomials. We show the optimal order of approximation of the finite element in interpolation, and in solving the biharmonic equation. Numerical results are provided confirming the analysis.

DOI : https://doi.org/10.1051/m2an:2008002
Classification:  65N30,  73C35
Keywords: differentiable finite element, quadratic element, biharmonic equation, Strang's conjecture, criss-cross grid, averaging interpolation, non-derivative basis
@article{M2AN_2008__42_2_175_0,
     author = {Zhang, Shangyou},
     title = {A C1-P2 finite element without nodal basis},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {2},
     year = {2008},
     pages = {175-192},
     doi = {10.1051/m2an:2008002},
     zbl = {1145.65102},
     mrnumber = {2405144},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2008__42_2_175_0}
}
Zhang, Shangyou. A C1-P2 finite element without nodal basis. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 2, pp. 175-192. doi : 10.1051/m2an:2008002. http://www.numdam.org/item/M2AN_2008__42_2_175_0/

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