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I wonder if anyone can give me a reference to a paper/book that rigorously addresses how to use the Galerkin method to show existence/uniqueness of a PDE. The usual suspects (Evans, Renardy, ...) do not suffice for me.

I am getting confused with some sources saying we need weak-* convergence and others not so, and some sources do not address issues such as what the canonical way is to introduce the finite dimensional problem and how it becomes an ODE.

Thanks.

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I think there is more chance you get quick answers if you break down your question into a few more specific questions. –  timur Aug 20 '13 at 14:21

1 Answer 1

Would it be the Lax-Milgram theorem?

P. D. Lax, A. N. Milgram, Parabolic equations. Contributions to the theory of partial differential equations. Annals of Mathematics Studies, no. 33. Princeton University Press, 167-190, 1954.

These topics are usually covered in any rigorous book on the finite element methods, e.g.:

P. G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, 1978.

S. C. Brenner, L. R. Scott, The Mathematical Theory of Finite Element Methods. Springer, 2008.

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