## Higher order quadrature rules on simplex geometry [extension]

I have seen that the simplex quadrature rules in 3D are restricted to the polynomial order range 0-5. This is fine for lower order lagrange finite elements, for example. In the case of an order 4 lagrange basis and assembling of a mass-matrix it might not be sufficient any more. This is not an exotic case and might be of interest to some users.

I propose to extend the quadrature rules up to order 12 similar to the 2D case, and reduce the nr. of quadrature points for some of the implemented rules. Following the publication

A SET OF SYMMETRIC QUADRATURE RULES ON TRIANGLES AND TETRAHEDRA, L. Zhang and T. Cui and H. Liu, (2009)

symmetric quadrature rules with positive weights can be constructed up to high order. The authors have provided a solver library, to solve the corresponding non-linear least-squares problem and have provided high precision rules in their finite element library *PHG*.

Calculating symmetric rules by myself I found the following combinations:

**Current dune implementation (3D):**

order | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|

num points | 15 | 15 | 60 | 60 | 96 | 114 | 175 |

(rules for order > 5 are constructed by tensor-product approaches)

**Proposed rules:**

order | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|

num points | 14 | 24 | 35 | 48 | 61 | 95 |

But you could also construct even better rules (for the higher orders) with less points. See, e.g., the file `src/quad.c`

in the PHG-Library (LGPL-License)

For the 2D case, the rules are all fine. One could extend the rules to higher order, easily, if necessary. The only exception is the rule for order 3. There, currently also negative quadrature weights are present in the dune implementation. If this is a problem for someone, one could replace this with a positive weighted rule, but with a more quadrature points.

In the mentioned publication above, the rules are formulated in terms of so called *permutation stars*. This is quite nice and would also allow to group the quadrature rule, since points in a permutation star have all the same weight and thus maybe a reduction of the the number of arithmetic operations could be achieved.