I had seen that github issue/pull-request, but didn't realise that
"saddle-point problems = (indefinite + symmetric)". These three words do
not appear in close-proximity to each other anywhere in the slides,
tutorials or video lectures. While I acknowledge the importance of
mathematical terminology, perhaps it would be quite useful to
de-jargonise this for the newcomers?
Or just link to wikipedia:
https://en.wikipedia.org/wiki/Ladyzhenskaya%E2%80%93Babu%C5%A1ka%E2%80%93Brezzi_condition
That's what I'm doing here:
https://github.com/dealii/dealii/pull/9651/files
Now, my question becomes "how to test for definiteness and symmetric
nature of my DAEs?", i.e. simply by looking at the weak form of the
equation which consists of only symbolic notations, can one somehow
infer the definiteness/symmetry of the matrix?
Yes. If your bilinear form satisfies
a(u,v)=a(v,u)
then your matrix is symmetric. If in addition
a(u,u) >= c \|u\|
with some norm of u on the right hand side, then the matrix is also
positive definite. Both is the case for Laplace equation, for example.
If the second condition only applies to the top left block, but the rest
of the problem has the structure shown in the wikipedia article above,
then you have a saddle point problem.
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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