After reading the FreeFEM documentation, I could not find any function or element corresponding to the KMV (Kong-Mulder-Veldhuizen) element, as described on DefElement: https://defelement.org/elements/kong-mulder-veldhuizen.html. This element is particularly useful for implementing mass-lumped triangular or tetrahedral finite elements. Is there currently a method to use it in FreeFEM, or is this planned for a future update?
In the link you mention, the description corresponds to (at least if continuity is required, there is no mention on this)
degree 1 : P1
degree 2 : P2b (P2+bubble)
The only “difference” is that the normalisation (factors 6 or 40…) is unusual.
This could be managed with a diagonal matrix multiplication.
Thank you for your kind reply. I agree the KMV element and quadrature are trivial for degree=1. For degree=2, I will review the information you shared.
I should clarify a key point I omitted earlier: implementing mass-lumped triangular elements requires both special basis functions and a matching quadrature rule, as detailed in this article: Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation | Journal of Engineering Mathematics . Thus, given that a special quadrature rule is also required for degree >= 2, what is the recommended approach to implement this custom quadrature rule? Thank you again.
Different quadrature rules are defined in FreeFem for triangles.
They are described in
p.226
On this page the keywords are slightly erased. Their full names can be seen at
I think it is not possible for the user to define new quadrature rules.
The P2b element is described in the pdf above, p.197
Thank you for your reply. Based on our discussion, it seems that implementing mass-lumped triangular or tetrahedral elements in FreeFEM is currently non-trivial. I will continue learning this software and look forward to future developments. It was a very helpful discussion. Thank you again.