Galerkin approximations for parabolic PDE weak solution, getting a uniform bound

Question

(As usual $V \subset H$ are separable Hilbert spaces)

In a book I read this about existence of the solutions to parabolic PDEs:

the approximate solution $u_n(t)$ solves the equation $$(u_n', w_j) + (Au_n, w_j) = \langle f, w_j \rangle\tag{1}$$ for $j=1,...,N$. Here $w_j$ is the set of basis functions associated with the problem. We can write this as $$u_n' + Au_n = f$$ as an equality in $L^2(0,T;V').$

Surely this is not an equality in $L^2(0,T;V')$ but an equality in (I think) $L^2(0,T;V_n')$ where $V_n$ is the finite dimensional subset of $V$ spanned on $w_j$ for $j=1,...,N$? Because we only have that $(1)$ holds for everything in $V_n$ (by linearity).

The author then obtains a bound on $u_n'$ using this equality -- any elaboration would be useful.

I posted this in MSE (https://math.stackexchange.com/questions/418443/an-equality-in-l20-tv-weak-solution-to-pde-via-galerkin-approximations), please see for some discussion if you like.

Answer 1

First of all: I'm not sure what you mean by the expression $(Au,u)$. I'm only familiar with the notation $(A\nabla u, \nabla u)$ for elliptic operators. Or do you want to have also lower order terms and replace this expression by a bilinear form $B(u,u,t)$?

You actually find all the details in Evans' book. A bound for $u_n'$ in $L^2(V')$ can be obtained from the bound for $u_n$ in $L^2(V)$ by testing with an arbitrary function $v \in V$ and observing that $$ \begin{aligned} (u_n',v) = (u_n',P_mv) &= - B(u_n, P_mv) + (f,P_mv) \\ &\leq C (\|u_n\|_V + \|f\|_V) \|P_m v\|_V \\ &\leq C (\|u_n\|_V + \|f\|_V) \|v\|_V \\ \end{aligned} $$ This implies $\|u_n'\|_{V'} \leq C (\|u_n\|_V + \|f\|_V)$. Taking squares and integrating in time yields the assertion.

In any case, it would be nice to know the book you're refering to.