I am studying knowledge related to lumped mass finite elements. As is well known, lumped mass finite element methods higher than 2nd order on simplex mesh require the construction of new function spaces (here).
Specifically, on elements in the $k$-th order space, it is necessary to add bubble functions of order $k^{\prime}$ to ensure that the integration is accurate for $k+k^{\prime}-2$ -degree polynomials. Taking the 2nd-order finite element space as an example, we need to add a 3rd-order bubble function. Each element contains 7 basis functions, I would know that for the function $w_h$ in such a function space, How many times that function $w_h$ can be differentiated? Is it 2nd order or 3rd order?
More simply, for a function in such a space $\mathbb{P}_k \oplus [b]$, the derivatives beyond k derivatives should be equal to 0?
If there is any literature or knowledge about bubble functions, I would appreciate it if everyone could offer guidance. Thank you in advance.
