TL;DR: How does Cea Lemma work in approximating the error on finite element approximation?
I understand that there is no straight answer to my question. However, any explanation would be much appreciated.
Coming from engineering background, my math is not as rigour and would prefer if any explanation use simple math notation.
For example, let's say I have a linear approximation of a function u at element 1 having 2 nodes. This would look something like:
$$ u(x)\approx u_{h}(x)=u_{1}v_{h1}(x)+u_{2}v_{h2}(x) $$
$$ v_{h}\epsilon V_{h} $$
Note that in both Galerkin and Rayleigh–Ritz method, the $v_{h}(x)$ is used as the test function and called as the shape function in many textbooks.
The Cea lemma states that:
$$ \|u-u_{h}\|\le C\|u-v_{h}\| $$
I have read that this is actually variational crime. However, let's say is a good enough approximation that the Cea lemma still holds. Furthermore, at this point, unfortunately, other than wiki, I couldn't understand the complicated math concept and notation that many other resources on the internet present about Cea lemma.
My questions are :
To my knowledge the $\|u-u_{h}\|$ term is the approximation "local" error in $L_{2}$ norm. Is this correct? "Local" meaning only at that particular element.
To my knowledge, the $\|u-v_{h}\|$ term can be considered as the interpolation error. So does it represent the $L_{2}$ norm of the difference between the actual function u and the $v_{h}$? Meaning if implemented to the above approximation of u, it will look like:
$$ \text{At element 1}: \|u-v_{h}\| = \sqrt{(\int_{node1}^{node 2}(u-v_{h1})^{2} + (u-v_{h2})^{2} dx)} $$
How does this actually help with approximating the error? If we want the $\|u-u_{h}\|$ term, we need the C term and the $\|u-v_{h}\|$ term, both of which we don't know, do we?
Is the C term determine through refining the mesh and back calculate for it or what? I'm really confuse.