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):Edit: 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:
L2-Projection using quadratic basis functions
interpolation
With linear elements, you put the shape functions together into little hat-shaped basis functions and overlap them like this:With quadratic shape functions, you do something similar and overlap them in a similar way, but you gain an interior point inside the element that doesn't overlap with its neighboring element:*Images from http://hplgit.github.io/INF5620/doc/pub/sphinx-fem/._main_fem003.html and http://nptel.ac.in/courses/112104116/lecture-8/8_3.htm, respectively.