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?

  • $\begingroup$ $x_i$ and $x_j$ refers to variables on which the function $h$ depends: $x_1$ $x_2$ when the the function depends on two spatial variables etc $\endgroup$ – AlphaXY Mar 21 '18 at 18:33
  • $\begingroup$ In order to understand Galerkin method you can take the case of Fourier serie truncation you have the decomposition $u(x,t)= \sum_{k\in \mathbb{Z}}c_ke^{ikx}$ in that case the Galerkin projector applied on $u$ truncate up to $n$, namely $$P_nu= \sum_{|k|\leq n}c_ke^{ikx}$$ $\endgroup$ – AlphaXY Mar 21 '18 at 18:40
  • $\begingroup$ Ok, so for 3D $u(x_i) = u(x_1, x_2, x_3)$. Right? But why the additional subscript $j$ if $i$ already denotes the three dimension? $\endgroup$ – user276833 Mar 21 '18 at 18:49

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


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