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A NOTE TO FRIEDRICHS' INEQUALITY

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*Dana Ríhová-Skabrahová*

**Address.** Technical University Brno, Faculty of Technology - Department
of Mathematics, Mostn¡ 5139, 762 72 Zl¡n, CZECH REPUBLIC
**E-mail:** skabrahova@zlin.vutbr.cz

**Abstract.** The main aim of this paper is to derive continuous
and discrete forms of inequalities which are similar to Friedrichs' inequality
and to show that for $h$ sufficiently small the constant $C$ appearing
in discrete inequalities written for functions from finite element spaces
$X_h$ is independent of $h$. The discrete forms of Friedrichs' inequality
are restricted to two-dimensional domains in this paper. These inequalities
have applications in the theory of two-dimensional electromagnetic field
and in the analysis of the approximate solution of Maxwell's equations.

**AMSclassification.** 65N30

**Keywords.** Finite element method, two-dimensional electromagnetic
field, Friedrichs' inequality, discrete forms of Friedrichs' inequality