I am solving a linear stability problem using finite element discretization. Then, I have a generalised eigenvalue problem: $$ \lambda M x = J x.$$ I obtain complex eigenvalue and eigenvectors from Arpack using inverse shift method. I would like to compare the eigenvectors obtained with different meshes. To do this it is necessary to normalize norm and phase in a proper way. As concerns the norm of the eigenvector I normalize it imposing: $$ x^HMx = 1. $$ Now, I would need an integral norm for the phase too. I tried one but it doesn't seem to work. I cannot obtain similar eigenvectors with two different meshes.
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What do you mean by "compare eigenvectors"? If you just want to measure a concept of distance between them, I would use the angle between subspaces, which is independent of the normalization.
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$\begingroup$ I want to visualize the velocity field containted in the eigenvectors with Paraview. I would expect to see something similar or the same thing if the two eigenvectors are correctly normalized. Am I wrong? $\endgroup$ – Britomarti Jun 12 '15 at 18:02
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$\begingroup$ What kind of velocities do you have that have a complex phase? $\endgroup$ – Federico Poloni Jul 14 '15 at 17:57
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Please refer to some work of Peter Arbenz: http://people.inf.ethz.ch/arbenz/publications.html
He had worked out some generalized eigenvalue problems with constraint condition (I remembered that he used the Jacobi–Davidson Algorithm), his model problems came from the FEM for Maxwell equations.
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$\begingroup$ There are 109 references in your link. Would you mind adding a summary of what you want to highlight from there? $\endgroup$ – nicoguaro♦ Nov 22 '19 at 17:27
