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Galerkin method is used heavily in finite element method, which can conveniently convert continuous problems to discrete ones. Particularly, Galerkin method can be used to prove uniqueness existence of solutions of some kinds of partial differential equations. But I can only find the detail of this for parabolic equations. I wonder how to have it applied to hyperbolic equations. Can anyone give some references? Thank you.

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Do you mean something like this? –  Willie Wong Jun 22 '11 at 9:00
    
Not exactly...I am searching a proof similar to the pages posted as the link above, but working on hyperbolic equations. –  ziyuang Jun 24 '11 at 17:46
    
Can you give me a page number? In the pages opened up by the link I only see existence and regularity of weak solutions. –  Willie Wong Jun 25 '11 at 13:44
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Galerkin method does not, strictly speaking, require use of eigenfunctions: when eigenfunctions are available you can use them to simplify the computation, which is very similar to why Fourier methods work. In any case, for hyperbolic equations one often constructs approximate solutions using Cauchy-Kovalevsky, which obviates the need for Galerkin type methods. In Courant-Hilbert's Methods of Mathematical Physics, the equivalent to Galerkin method (wave expansion) was dealt with briefly in only two pages. –  Willie Wong Jun 26 '11 at 16:41
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But I think Stig Larsson and Vidar Thomee's book Partial Differential Equations with Numerical Methods does address FEM for hyperbolic equations, so presumably something similar to what you are looking for is in there. –  Willie Wong Jun 26 '11 at 16:42
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