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I'm trying to solve the problem below, part 1. is proved by a simple application of the Lax-Milgram Lemma (General Case, i.e, with inf-sup conditions), in part 3. I think I know how do it, but part 2. I don't know how to get it. Could someone please give me some hints?

Thanks in advance.


Let $(H, \langle\cdot,\cdot\rangle_H)$ a real Hilbert space and $A:H\times H\rightarrow \mathbb{R}$ a bounded bilineal form with subordinate operator ${\bf A}\in\mathcal{L}(H)$, i.e. $$A(\sigma,\tau)\ =\ \langle{\bf A}(\sigma), \tau\rangle_H.$$ Suppose there is operators ${\bf S}_1$, ${\bf S}_2\in\mathcal{L}(H)$ and constants $\alpha_1,\alpha_2 > 0$ such that $$\langle {\bf S}^*_1{\bf A}(\tau), \tau\rangle_H\ \geq\ \alpha_1\|\tau\|_H^2\quad \mbox{ and }\quad \langle{\bf A}{\bf S}_2(\tau), \tau\rangle_H\ \geq\ \alpha_2\|\tau\|_H^2\quad \forall\ \tau\in H.$$

  1. Show that for all $F\in H^{\prime}$ ($\;^\prime$ : dual), $\exists! \sigma\in H$ such that $$A(\sigma, \tau)\ =\ F(\tau)\qquad \forall\ \tau\in H,$$ and prove that there's $C > 0$, independient of $F$, such that $$\|\sigma\|_H\ \leq\ C\|F\|_{H^{\prime}}.$$
  2. Let $\{H_h\}_{h > 0}$ a countable family of finite-dimensional subspaces of $H$ such that $\lim\limits_{h\rightarrow 0}\text{dist}(\tau, H_h) = 0\quad \forall\ \tau\in H$, and, given $F\in H^{\prime}$, consider a Galerkin scheme: Find $\sigma_h\in H_h$ such that $$A(\sigma_h, \tau_h)\ =\ F(\tau_h)\qquad \forall\ \tau_h\in H_h.\quad\quad (*)$$ Suppose that for $i=1$ or for $i=2$ (but not for both), there's injective operators ${\bf S}_{i,h}\in\mathcal{L}(H_h)$ for all $h>0$, and constants $C_i$, $\delta > 0$, independients of $h$, such that $$\|{\bf S}_i(\tau_h) - {\bf S}_{i,h}(\tau_h)\|_H\ \leq\ C_ih^{\delta}\|{\bf S}_i(\tau_h)\|_H\quad\forall\ \tau_h\in H_h.$$ Show that there is $h_0 > 0$ such that for all $h \leq h_0$, the problem $(*)$ have a unique solution, is stable and verified the Cea's estimate.
  3. What do you say about the hypothesis for 1. and 2., if $A$ is symmetric?

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My professor gave me this hint: "Work with the supremum in the discrete inf-sup condition and find a lower bound with the discrete operator evaluated in the first variable and then add and subtract the corresponding continuous operator evaluated in the same variable." But still I could not solve it, can someone please help me. thanks –  FASCH Sep 19 '13 at 13:47

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