periodic boundary conditions and the FEM

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".

• Wrap your $[-5,5]$ segment into a circle. Intuitively, the endpoint should not be different from any other point, so $M_{10,10} = M_{i,i}$.The negative values you've obtained are due to wrapping and extrapolation outside of the finite element, I suppose. – uranix Jun 12 '15 at 13:32