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(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

M = \int{NN^T}dL,

[/tex]

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 [itex]x_{N+1}=x_1[/itex])

[tex]

M_{10,10} = \frac{x_{1}-x_{10}}{3} = -10/3\\

M_{10,1} = \frac{x_{1}-x_{10}}{6} = -10/6

[/tex]

All other elements have positive values given by 1/3 and 1/6, respectively.

Are my values for [itex]M_{10,10}[/itex] and [itex]M_{10,1}[/itex] correct? I find it odd that their values are so much different than the values in the "bulk".

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# FEM: periodic boundary conditions (1D)

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