# Terminology: Galerkin approximation

Dear all, I'm preparing a paper in which I'm trying to prove that my numerical approximation (Galerkin) of some mechanical problem indeed provides an approximation of the solution. In order to do so, I'm trying to invoke as many known theorems as possible (I'm no mathematician --only a mechanician who's trying to do things the proper way...). I'm basing my work on the book by Ern and Guermond (Theory and practice of finite elements) which provides me with exactly what I need. However, I find difficult to separate their personal terminology from "generally accepted" terminology. Could you help me there ?

Many thanks, Sebastien

The variational problem at hand is

$\text{find}\quad u\in\mathbb V\quad\text{such that}\quad a(u, v)=f(v)\quad\text{for all}\quad v\in\mathbb V,$

and the discretized version

$\text{find}\quad u^h\in\mathbb V^h\quad\text{such that}\quad a^h(u^h, v^h)=f^h(v^h)\quad\text{for all}\quad v^h\in\mathbb V^h.$

I would like to make sure that the following terminology: approximability, consistency and asymptotic consistency is widely accepted. If not, what is the generally accepted name for these concepts?

### Approximability

$\displaystyle \forall u\in\mathbb V, \lim_{h\rightarrow 0}\inf_{u^h\in\mathbb V^h}||u-u^h|| = 0$

### Consistency

$\displaystyle \forall v^h\in\mathbb V^h, a^h(u, v^h)=f^h(v^h)$

### Asymptotic consistency

When $a^h$ is uniformly bounded, the approximation method is said to be asymptotically consistent if there is an operator $\Pi^h$ such that for all $v\in\mathbb V$, $||\Pi^hv-v||\leq c\inf_{v^h\in\mathbb V^h}||v-v^h||$ where $c$ is independent of $v$, and

$$\lim_{h\rightarrow 0}\sup_{v^h\in\mathbb V^h}\frac{|f^h(v^h)-a^h(\Pi^hu, v^h)|}{||v^h||}=0.$$

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What is your question? – user17762 May 18 '11 at 16:59
@Sivaram: edited the question... – Sebastien May 19 '11 at 1:53