First eigenvalue not matching, but all others are

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member 428835
Hi PF!

I am applying a spectral technique on a system of fluid dynamics problems. Specifically, I am looking for the characteristic frequencies, which turn out to be the eigenvalues of a matrix system ##M = \lambda K## for ##n\times n## matrices ##M,K##, which comes from a variational Rayleigh-Ritz procedure, reducing the differential eigenvalue problem to an algebraic one. The Ritz technique I apply can be compared to an analytic solution. For ##\lambda_{2-5}## I show less than 1% difference, but ##\lambda_1## can be off by 25%. Why could this be happening?
 
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mathman said:
Sounds like a physics question, not math.
I'm not sure how. The analytic solution is valid for a parameter value ##\alpha = \pi/2##. The input I select is ##\alpha = 89 \pi/2##. When doing so, all eigenvalues agree except the first. My suspicion lies in the math, though I've no clue why.
 
mathman said:
As an abstract math problem there is nothing wrong with the result. I have no knowledge of physics problem. However on a circle ##89\pi/2=\pi/2##.
Sorry, I meant ##89 \pi/180## compared to ##\pi/2##
 
mathman said:
89/180 is close to 1/2. Since I don't know what is going on, I can't add anything more.
Have you ever seen something like this before? Where the first eigenvalue is off by 25% from a Ritz method where the higher eigenvalues are accurate within 1%?
 
I'm taking shots mostly in the dark, but I might be inclined to expect you don't have a good set of basis functions to represent the solution corresponding to your first eigenvalue. Have you tried expanding the space of your approximation, i.e. getting 6 or more eigenvalues, to see if the discrepancy begins to decrease?
 
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Haborix said:
I'm taking shots mostly in the dark, but I might be inclined to expect you don't have a good set of basis functions to represent the solution corresponding to your first eigenvalue. Have you tried expanding the space of your approximation, i.e. getting 6 or more eigenvalues, to see if the discrepancy begins to decrease?
This is good advice! Unfortunately, the first basis function I believe is correct, as it's the first Bessel function that satisfies a set of Neumann boundary conditions. And you know how that process goes, you get one correct and you get them all, especially when using symbolic programming. And I've used up to 10 terms, where iterative convergence is smaller than 1% error.