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

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