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T. Wurzer, J. Melenk:
"Stability of the trace of the polynomial $L^2$-projection on triangles";
in: "ASC Report 36/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2010, ISBN: 978-3-902627-03-2.

This bachelor thesis deals with the stability and approximation of the trace of the
polynomial L2-projection on triangles. We consider the L2-projection 2D
N : L2(T) !
PN(T) onto PN(T), where T is the reference triangle f(x; y) : ��1 < x < 1;��1 < y <
��xg and show the following result
k 2D
N uk2
L2(��) CkukL2(T)kukH1(T); 8u 2 H1(T);
where we denote by �� one edge of @T.
At the end we will present a method to compute numerically the stability constant
C in the estimate above and show the computational results. We will also compute
two related stability constants, namely, the stability constant for the corresponding
one-dimensional statement
j( 1D
N u)( 1)j2 CkukL2(��1;1)kukH1(��1;1); 8u 2 H1(��1; 1);
where 1D
N : L2(��1; 1) ! PN(��1; 1) is the L2-projection onto the space of polynomials
of degree N, and the stability constant CN in the two-dimensional bound
k 2D
N uk2
L2(��) CNkuk2
H1(T); 8u 2 H1(T):
Here, CN is seen to be O(N).
1

http://www.asc.tuwien.ac.at/preprint/2010/asc36x2010.pdf