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Given that some of the mass lumping techniques, for example, row-sum lumping does not produce practically viable lumped mass matrices for all the element shapes, what are the techniques used for mass lumping for higher-order elements? I am particularly interested in applications in CFD.

I know that deal.ii has a tutorial on the projection method where the diffusion term is treated implicity. So, mass lumping is not necessary here. FreeFem uses explicit step for the diffusion term but demonstrates the implementation with P1 elements only.

Are there any libraries that solve the diffusion term explicitly using lumped-mass matrices for the velocity field when LBB-stable combinations using higher-order elements are employed?

I would appreciate any useful references on the topic. Thanks in advance.

Chenna K
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  • I have used nodal points as quadrature points for elements up to 8×8 nodes. See: https://scicomp.stackexchange.com/a/19714/9667 – nicoguaro Mar 12 '21 at 21:38
  • Thanks, @nicoguaro! Lobatto points are okay for quad & hex elements but are expensive for the triangular element (6/3=2 times) and the tetrahedral element (10/4=2.5 times) when compared with Gauss points. – Chenna K Mar 12 '21 at 23:15
  • You don't need to use Lobatto points, you could use your regular nodes as quadrature points. That would reduce the accuracy of your integral, though. But that's the whole point of mass lumping anyways. – nicoguaro Mar 12 '21 at 23:59
  • Lobatto points coincide with the nodes anyway, at least for lower-order elements. The idea of mass lumping is to obtain a diagonal matrix to ease the task of inversion in explicit schemes; the use of Lobatto points (or nodes) is one such technique to obtain lumped mass matrices for the Lagrange family of elements. Gauss quadrature is sufficient when using Bezier elements; we can use row-sum lumping.

    I am wondering if you know of any FEM library that employs lumped-mass matrices for CFD, for example, in projection schemes.

     – Chenna K Mar 13 '21 at 00:33
  • @ChennaK "The use of Lobatto points (or nodes) is one such technique to obtain lumped mass matrices for the Lagrange family of elements." This is not correct. Nicoguaro's link, especially point 3, should be sufficient for your task. Note that for all nodal points there exist a unique quadrature rule (Legendre quadrature) which result in a diagonal mass matrix, not only for (Legendre) Lobatto nodes. The difference is only the integration accuracy, which is far superior if you use (Legendre) Gauss and (Legendre) Lobatto nodes. – ConvexHull Mar 13 '21 at 15:14
  • Thank you, @ConvexHull! As shown there, the method in point 3 yields zeros for corner nodes of the 6-noded triangular element. How can I invert zeros? So, I don't understand how you can say that it is sufficient for my task. Moreover, quadrature using nodal points does not yield lumped masses for Bezier elements since they are not interpolatory at control points. – Chenna K Mar 13 '21 at 16:41
  • @ConvexHull, more than the techniques for mass-lumping, I am interested in learning about libraries or useful papers that use mass lumping for problems in CFD. – Chenna K Mar 13 '21 at 16:43

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