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We can use a Galerkin method to show that there is a solution to a PDE. So suppose $w_j$ is the basis functions.

I am interested in regularity of solutions. In the book by Evans, he differentiates a weak form involving the finite dimensional approximations (say) $u_m = \sum_{j=1}^mc_j(t)w_j(x)$. So the equation $$(u_m', w_j) + B(u_m, w_j) = f(w_j)$$ he differentiates to get $$(u_m'', w_j) + B(u_m', w_j) = f'(w_j)$$

Then he messes around with this equation and ends up showing that $u_m''$ lies in $L^2(0,T; H^{-1})$.

Questions: 1) How can we differentiate $u_m$ twice? How do we know that $c_j$ is differentiable twice? Even once? 2) Is it enough to show that $u_m''$ lies in the above mentioned space to know that $u_m''$ exists? How do we know that the limit of $u_m''$ as $m \to \infty$ is $u''$??? How to make sense of $u''$??

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$u_m''$ actually is for $x$, not for $t$.So we do not need $c_j$ to be differentiable here. See $u_m = \sum_{j=1}^m c_j(t)w_j(x)$. And $w_j$ should be differentiable. –  Yimin Mar 5 '13 at 23:06
    
@Yimin i don't think so. pretty sure it's differentiation wrt. $t$. –  george.s Mar 5 '13 at 23:16
    
ok, then which chapter is this part in the book? I will look it up later. –  Yimin Mar 6 '13 at 0:00
    
@Yimin try 7.1.3. –  george.s Mar 7 '13 at 21:07

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