A continuous finite element method with face penalty to approximate Friedrichs' systems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, p. 55-76

A continuous finite element method to approximate Friedrichs’ systems is proposed and analyzed. Stability is achieved by penalizing the jumps across mesh interfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence rates in the graph norm and suboptimal of order $\frac{1}{2}$ convergence rates in the ${L}^{2}$-norm. A variant of the method specialized to Friedrichs’ systems associated with elliptic PDE’s in mixed form and reducing the number of nonzero entries in the stiffness matrix is also proposed and analyzed. Finally, numerical results are presented to illustrate the theoretical analysis.

DOI : https://doi.org/10.1051/m2an:2007007
Classification:  65N30,  65N12,  74S05,  78M10,  76R99,  35F15
Keywords: finite elements, interior penalty, stabilization methods, Friedrichs' systems, first-order PDE's
@article{M2AN_2007__41_1_55_0,
author = {Burman, Erik and Ern, Alexandre},
title = {A continuous finite element method with face penalty to approximate Friedrichs' systems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {41},
number = {1},
year = {2007},
pages = {55-76},
doi = {10.1051/m2an:2007007},
zbl = {1129.65083},
language = {en},
url = {http://www.numdam.org/item/M2AN_2007__41_1_55_0}
}

Burman, Erik; Ern, Alexandre. A continuous finite element method with face penalty to approximate Friedrichs' systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, pp. 55-76. doi : 10.1051/m2an:2007007. http://www.numdam.org/item/M2AN_2007__41_1_55_0/

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