# A confusion about the bubble function in lumped mass FEM

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.

• I have lumped elements of order 15th with good results in the past. I have some examples here Nov 15, 2023 at 12:09
• @nicoguaro thx for u comment, but what I want is to use lumped mass FEM on simplex mesh instead of using Gauss-Lobatto on quadrilaterals and hexahedrons. Nov 15, 2023 at 15:44
• Independent of what type of elements you want to use, I am saying that your claim is not true. Nov 15, 2023 at 20:44

For the P2b element, that is, the quadratic triangle with a cubic bubble, all the basis functions contain the term $$\xi _{1} \xi _{2} \xi _{3}$$, where $$\xi_3=1-\xi_1-\xi_2$$. Thus, each basis function is an incomplete cubic polynomial. So, third-order derivates with respect to $$\xi_1$$ and $$\xi_2$$ are zeros, but third-order mixed derivatives, e.g., $$\frac{\partial^3 N_i}{\partial \xi_1^2\partial \xi_2}$$ are not zeros.