I am trying to understand 1D $L^2$-projections using quadratic basis functions. Using 3 data points, and the Lagrange polynomial it is easy enough to see how to write out 3 basis functions. With the hat functions from the linear basis, it is easy to see how to expand a function. The hat functions are like delta-functions. With the quadratic basis I am having trouble writing out an explicit expression for the basis function because there are three functions in the same interval. (Fig 8.36 here : https://people.fh-landshut.de/~maurer/femeth/node265.html#SHP3).
The next step of what I am looking to do is construct the so called "mass matrix". Is there a way to visualize the quadratic basis like a hat-function? Once the basis functions are known, the mass matrix elements can be computed by looking at the overlap between the basis functions, L2-inner product.
Thanks ahead, any advice or comments appreciated.
Mass matrix structure for the linear-hat functions (x's denote a non-zero entry):
This information is based off of the accepted answer below, which gives good references for figuring this out.
Structure of the mass matrix with quadratic basis functions: