Toby,
> My application case is similar to step-36 of the tutorial; except (i)
> My problem is a vector valued solution; which, I think, means that my
> lowest $N$ eigenvector solutions are all contained within a single
> vector-valued solution (which is the lowest eigenfunction of the
> generalized eigenspectrum problem).
No. The eigenvalue problem is of the form
L x = lambda M x
where x is the eigenvector and corresponds to *all* components of the solution
vector. For example, when you want to compute the Stokes eigenproblem, then L
is the complete Stokes operator, and x is a vector that contains both
velocity and pressure components. If you compute N eigenvectors, then each of
these N vectors has velocity and pressure components.
As for normalization: you know that the normalization is arbitrary. SLEPc
probably chooses the l2 norm of the vector x, whereas you may want to use a
norm that has something to do with the PDE. To do this, you get an x from
SLEPc with
|| x ||_{l2} = 1
and you need something with
|| U ||_V = 1
where U is the (vector-valued) function associated with the nodal values x.
For example, for the Stokes operator, U=(u,p) and the proper norm is probably
|| U ||_V^2 = || nabla u ||_{L2}^2 + || p ||_{L2}^2
and your rescaled eigenvector would be
x' = x / sqrt{x^T Y x}
where Y is a matrix that matches the norm above, i.e.
Y = [A 0; 0 M]
with A the Laplace matrix on the velocities and M the mass matrix on the
pressures. Note that you can build the matrix Y all at once, without ever
extracting individual components of the solution vector. If I understand your
case correctly -- you seem to imply that all components are equally
treated -- then Y will be block diagonal with mass matrices on the diagonal.
This matrix could be build using the MatrixCreator::mass_matrix function.
> cat DataComponentInterpretation::component_is_scalar be used for this?
This thing is only used for graphical output, nothing else.
Best
W.
-------------------------------------------------------------------------
Wolfgang Bangerth email: [email protected]
www: http://www.math.tamu.edu/~bangerth/
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