# Weak, Strong convergence of the discretized solutions of a variational problem

1. Jun 20, 2009

### TNT

1. The problem statement, all variables and given/known data

Consider the Galerkin discretization of an abstract variational problem where the Hilbert space V is separable.
http://en.wikipedia.org/wiki/Galerkin_method#Introduction_with_an_abstract_problem"

Each of the subspaces Vn is generated by the first n terms of a sequence of elements of the separable Hilbert space V. This sequence is such that each of these subspaces will be dense in V. The problem is proving that there is a subsequence of the bounded sequence of discretized solutions [PLAIN]http://upload.wikimedia.org/math/9/e/2/9e220730654354c2c9daac9e379babaf.png, [Broken] that converges weakly to the solution of the variational abstract problem [PLAIN]http://upload.wikimedia.org/math/d/a/4/da4760886b448bf1f9ecd7e0fd994bce.png,[/URL] [Broken] then proving that the same subsequence converges strongly and finally proving that the sequence of discretized solutions http://upload.wikimedia.org/math/9/e/2/9e220730654354c2c9daac9e379babaf.png converges to the solution of the variational abstract problem [PLAIN]http://upload.wikimedia.org/math/d/a/4/da4760886b448bf1f9ecd7e0fd994bce.png.[/URL] [Broken]
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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