FEM: How the weak form is related to an inner product

[Carla White]

Hi all, I am a final year maths student and am doing my dissertation in the finite element method. I have gotten a little stuck with some parts though.

I have the weak form as a(u,v)=l(v) where:
$$a(u,v)=\int_{\Omega}(\bigtriangledown u \cdot\bigtriangledown v)$$
and
$$l(v)=\int_\Omega fv-\int_{\partial\Omega_N} g_Nv$$.
I am trying to prove that a(u,v) is coercive and continuous and l(v) is continuous. I know to do this I need to relate these to an inner product with norm, I'm pretty sure this should be the L² norm:
$$||u||_2:=\left(\int_\Omega u^2\right)^{1/2}$$.
I'm struggling to understand how these relate though, and how I get from the linear and bilinear forms to the normed form. Could somebody either explain how it works or point me in the direction of somewhere that explains it.

Thanks
Carla

 

[RUber]

Hi Carla,
I am actually working on a similar problem right now.
Poincare's inequality is a good place to start.
I think that embedding theorems are useful for the second part as well. I haven't worked it all the way through yet, but I think if you can show that the gradient is in a closed and bounded space, e.g. $H^1_0(\Omega)$ and that space is also contained within $L^2(\Omega)$ you can conclude that the variational formulation is coercive.