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

  1. Jun 20, 2009 #1

    TNT

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    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 875637baf5e31ac9738ab7dd31003652.png 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
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
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