I'm following a few papers describing the Galerkin finite element method for a particular physical process. They all start with the same initial definition:
$$ h \approx \hat h(x,y,z,t) = \sum_{n=1}^N \phi_n(x,y,z) h_n(t) $$
where $h_n$ is the amplitude of $h$ at nodal point $n$ and $\phi_n$ is the base function at nodal point $n$. I think I follow this... it seems pretty straightforward. However, when they state the variational form and numerical form, new subscripts get added and I have no idea what they are referring to. This seems to be common enough that I believe it must be a standard notation in this field I'm not familiar with.
For example, here's a snippet of the variational form:
$$ \sum_e \int_\Omega \frac{\partial \hat h}{\partial x_j}\frac{\partial \phi_n}{\partial x_i}\ \mathrm{d}\Omega\ \dots $$
What might the subscripts $i$ and $j$ represent?
Then during the explanation of integration over the elements, the numerical form is presented. Here's a snippet of that:
$$ \sum_e \int_\Omega \phi_l \frac{\partial \phi_n}{\partial x_j} \frac{\partial \phi_m}{\partial x_i}\ \mathrm{d}\Omega\ \dots $$
What might the new subscripts $l$ and $m$ represent?
The multiplication of partial derivatives requires two index $i$ and $j$ take for example $$\frac{\partial h}{\partial x_1} \frac{\partial h}{\partial x_2}$$ You need an index $i$ to represent one and a different index to represent two or three