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How to calculate the convergence rate of Gauss-Seidel line iteration?

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Old   November 2, 2009, 23:08
Default How to calculate the convergence rate of Gauss-Seidel line iteration?
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For the two dimensional Poissonís equation,
second-order centered difference with uniform grid is used to discretize. Then, we can obtain
a[v(n,i+1,j)-2v(n+1,i,j)+ v(n+1,i-1,j)]+c[v(n+1,i,j+1)-2v(n+1,i,j)+ v(n+1,i,j-1)]=0
where a<<c (e.g. a=0.01, c=1), hence y-line iteration should be used.
By using the Fourier expansion of the above equation, we can obtain the convergence rate mu(alpha,beta) as
mu(alpha,beta)= | a/[ 2(a+c)-c*cos(beta)-exp(I*alpha) ] | .
According to some book, the maximal value of the above equation is
max(mu)=max{1/sqrt(5), a/(a+2c)}.

I canít understand how to calculate max(mu). Anybody know? I have tried to analyze the eigenvalues of my iteration matrix, the one with maximal norm is about 0.005, however, the convergence rate in my actual numerical experiment is closed to 1/sqrt(5). How to calculate the convergence rate 1/sqrt(5) theoretically?

Thank you so much.
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