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.
How to use Galerkin method to prove uniqueness of solutions of hyperbolic equations?
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1Do you mean something like [this](http://www.springerlink.com/content/k671118863p44286/)? – 2011-06-22
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0Not exactly...I am searching a proof similar to the pages posted as the link above, but working on hyperbolic equations. – 2011-06-24
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0Can you give me a page number? In the pages opened up by the link I only see existence and regularity of weak solutions. – 2011-06-25
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0Am I misunderstanding something? I thought that this method (to prove existence of weak solutions by eigenfunction expansion) is called Galerkin method. – 2011-06-26
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0@ziyuang: you mentioned **uniqueness** in your question statement, but in the link you gave I only see **existence**. – 2011-06-26
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0Oh @Willie Wong, I am so sorry, it should be existence. Hope that not wasted you too much time. – 2011-06-26
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1Galerkin 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. – 2011-06-26
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1But 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. – 2011-06-26