I am trying to set up the mass matrix for a 1D system which I want to solve using finite elements. So the mass matrix is defined as
$$ M = \int{NN^T}dL, $$ where $N$ is the finite element linear basis functions. I use hat functions.
Say I have $10$ elements, corresponding to 11 nodes running from -5 to 5 so the spacing is 1. Node 1 is equal to node 11 since I want to employ periodic boundary conditions.
My issue is that I am not sure how to construct the mass matrix for the 10th node. As shown here, the elements for the 10th node will be (I use periodic boundary conditions, so $x_{N+1}=x_1$)
$$ M_{10,10} = \frac{x_{1}-x_{10}}{3} = -10/3\\ M_{10,1} = \frac{x_{1}-x_{10}}{6} = -10/6 $$ All other elements have positive values given by $1/3$ and $1/6$, respectively.
Are my values for $M_{10,10}$ and $M_{10,1}$ correct? I find it odd that their values are so much different than the values in the "bulk".