# Crouzeix-Raviart FE of degree 3
**URL:** https://community.freefem.org/t/crouzeix-raviart-fe-of-degree-3/1747
**Category:** General Discussion
**Created:** [May 16, 2022, 8:39am UTC](https://community.freefem.org/t/crouzeix-raviart-fe-of-degree-3/1747 "2022-05-16T08:39:08Z")
**Posts on this page:** 9
**Page:** 1
### Author:  [@Carina](https://community.freefem.org/u/Carina)
#### Post date: [May 16, 2022, 8:39am UTC](https://community.freefem.org/t/crouzeix-raviart-fe-of-degree-3/1747/1 "2022-05-16T08:39:08Z")
Hello there,
I am trying to implement a new finite element, Crouzeix-Raviart of degree 3 in dimension 2. But I actually don’t speak C++, so I kind of tried to combine P3 and P1nc (Crouzeix-Raviart of degree 1) and changed the basis of P1nc according to the definition.
[Element\_P3nc.zip](https://community.freefem.org/uploads/short-url/tAMSQsvnhost7QOtboRic0D0rB3.zip) (2.6 KB)
Obvouisly it doesn’t work yet. Could anyone help and take a look at what I did wrong and what I should change?
Thank you very much!
Definition of CR-element of degree 3:
Theorem 3.3 in [https://arxiv.org/pdf/2105.14981.pdf](https://arxiv.org/pdf/2105.14981.pdf)
So we have the normal Lagrange basis of degree 3, except on the vertices, where we use the edge bubble, 1/2 \* (5\*(1 - (lambda\_Ai)\*2)^3 - 3 \* (1 - (lambda\_Ai)\*2)) with lambda being the baricentric coordinate of the node Ai.
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### Author:  [@frederichecht](https://community.freefem.org/u/frederichecht)
#### Post date: [May 16, 2022, 11:10am UTC](https://community.freefem.org/t/crouzeix-raviart-fe-of-degree-3/1747/2 "2022-05-16T11:10:26Z")
I think the new finite element
- plugin Element\_P3pnc for new 2d finite element P3pnc (P3 + 2 bulles) nonconforming (continuite of P2 mod)
and add 2 examples with this new finite element
examples/plugin/cavityNewtowP3pnc.edp examples/plugin/testFE-P3pnc.edp
is this finite element
for more detail see tiger reach result of string “P3pnc” in the community.
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### Author:  [@Carina](https://community.freefem.org/u/Carina)
#### Post date: [May 16, 2022, 1:41pm UTC](https://community.freefem.org/t/crouzeix-raviart-fe-of-degree-3/1747/3 "2022-05-16T13:41:07Z")
Oh, I didn’t see the new update.
Yes, it seems like the same I tried to implement.
Thank you and have a nice day!
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### Author:  [@Carina](https://community.freefem.org/u/Carina)
#### Post date: [May 30, 2022, 11:36am UTC](https://community.freefem.org/t/crouzeix-raviart-fe-of-degree-3/1747/4 "2022-05-30T11:36:17Z")
@frederichecht unfortunately, the P3\_pnc does not really fit since in our case, we have 10 nodes instead of 12 (the same on the vertices/edged, but only one inside). Would it possible/easy to just erase the 2 additional nodes from the middle in the skript of P3\_pnc?
These are the nodes we need: {R2(0. , 0. ), R2( 1., 0.), R2(0. , 1.),
R2(2 / 3., 1 / 3.), R2(1 / 3., 2 / 3.), R2(0. , 2 / 3.),
R2(0. , 1 / 3.), R2(1 / 3., 0. ), R2(2 / 3., 0. ),
R2(1 / 3., 1 / 3.)}
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### Author:  [@frederichecht](https://community.freefem.org/u/frederichecht)
#### Post date: [June 7, 2022, 7:31am UTC](https://community.freefem.org/t/crouzeix-raviart-fe-of-degree-3/1747/5 "2022-06-07T07:31:52Z")
yes, but i have no time do this effectively I have add a bubble to P3
the only work is to recompile the corresponding array (10x10)
```
double C1[12][12] ={
{-1.2, 6, 1.2, -8.4, 10.8, 10.8, -46.8, -10.8, 20.4, 12, 0, -84 },
{-1.2, 1.2, 6, -73.2, 75.6, 73.2, -63.6, -73.2, 37.2, 12, -84, -84 },
{1.2, -15.6, -15.6, 298.8, -205.2, -306, 349.2, 198, -154.8, -180, 252, 504 },
{1.2, -1.2, 6, 73.2, -63.6, -73.2, 75.6, 37.2, -73.2, 12, 84, 84 },
{6, -1.2, 1.2, 10.8, -46.8, -8.4, 10.8, 20.4, -10.8, 12 , 84, 0 },
{-15.6, 1.2, -15.6, -306, 349.2, 298.8, -205.2, -154.8, 198, -180, -504, -252 },
{6, 1.2, -1.2, -46.8, 10.8, 20.4, -10.8, -8.4, 10.8, 12, -84, 0 },
{1.2, 6, -1.2, 20.4, -10.8, -46.8, 10.8, 10.8, -8.4, 12, 0, 84 },
{-15.6, -15.6, 1.2, 97.2, -54, 97.2, -54, 46.8, 46.8, -180, 252, -252 },
{-9.6, 4.8, 4.8, 189.6, 21.6, -192, 86.4, -24, -81.6, 60, 84, 168 },
{4.8, -9.6, 4.8, -192, 86.4, 189.6, 21.6, -81.6, -24, 60, -168, -84 },
{4.8, 4.8, -9.6, 2.4, -108, 2.4, -108, 105.6, 105.6, 60, 84, -84 },
```
}; \*/
which computed in exemple
[testFE-P3pnc.edp](https://community.freefem.org/uploads/short-url/s3BULSCiLHojnpwxNbmE5F5S7T5.edp) (5.7 KB)
after the modification is trivial!
the array is
```auto
-6 6 6 54 54 -54 18 -54 -42 12
-6 6 6 54 54 -54 -42 -54 18 12
30 -30 -30 -270 -270 270 90 270 90 -180
6 -6 6 -54 -42 54 54 18 -54 12
6 -6 6 -54 18 54 54 -42 -54 12
-30 30 -30 270 90 -270 -270 90 270 -180
6 6 -6 18 -54 -42 -54 54 54 12
6 6 -6 -42 -54 18 -54 54 54 12
-30 -30 30 90 270 90 270 -270 -270 -180
0 0 0 0 0 0 0 0 0 60
```
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### Author:  [@frederichecht](https://community.freefem.org/u/frederichecht)
#### Post date: [June 7, 2022, 11:50am UTC](https://community.freefem.org/t/crouzeix-raviart-fe-of-degree-3/1747/6 "2022-06-07T11:50:27Z")
I have code this finite element in [GitHub - FreeFem/FreeFem-sources at develop](https://github.com/FreeFem/FreeFem-sources/tree/develop)
see plugin `Element_P3nc.cpp` add the small test `examples/plugin/testFE-P3nc.edp`
This is a very first version
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### Author:  [@Carina](https://community.freefem.org/u/Carina)
#### Post date: [June 22, 2022, 10:16am UTC](https://community.freefem.org/t/crouzeix-raviart-fe-of-degree-3/1747/7 "2022-06-22T10:16:34Z")
@frederichecht Perfect!
But I noticed, that some of the basis are not what I have in my case. So instead of l0^3 it would be 1/2 \* (5\* l0^3 -3\*l0) and the same for l1^3 and l2^3. I actually just replaced this to calculate the Matrix C1 and then also in Element\_P3nc for l12 (line 147). Does it actually work with just replacing the basis? Do I also need to chance something for D12 (line 229) because of the change of the basis or anything else?
The new Matrix C1 would be:
[calculate\_C-Matrix.edp](https://community.freefem.org/uploads/short-url/1tF63tlj16SQwondpM78IQPQIcJ.edp) (2.4 KB)
And my changes to the Element\_P3nc:
[Element\_P3nc.cpp.zip](https://community.freefem.org/uploads/short-url/zITAGh8lNP9gYFQSu2aXmptwh2D.zip) (3.3 KB)
Does this all make sense programming wise?
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### Author:  [@Carina](https://community.freefem.org/u/Carina)
#### Post date: [June 24, 2022, 7:28am UTC](https://community.freefem.org/t/crouzeix-raviart-fe-of-degree-3/1747/8 "2022-06-24T07:28:26Z")
There was an error in calculating the Matrix C1, so here is the new version:
[calculate\_C-Matrix.edp](https://community.freefem.org/uploads/short-url/1tF63tlj16SQwondpM78IQPQIcJ.edp) (2.4 KB)
And with this the new Element\_P3nc:
[Element\_P3nc.cpp.zip](https://community.freefem.org/uploads/short-url/ufpkHHhmdjoiMPxBv303sHouRBp.zip) (3.2 KB)
But while testing this with [testFE-P3nc.edp](https://community.freefem.org/uploads/short-url/t8B4dsEZGB2drXej8YoUvI1AJOG.edp) (3.8 KB), I get an error for the last part:
err=136418
current line = 150
Exec error : exec assert
– number :1
Exec error : exec assert
– number :1
err code 8 , mpirank 0
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### Author:  [@frederichecht](https://community.freefem.org/u/frederichecht)
#### Post date: [June 25, 2022, 6:22pm UTC](https://community.freefem.org/t/crouzeix-raviart-fe-of-degree-3/1747/9 "2022-06-25T18:22:51Z")
I sorry I Have no computer so I will correct in week,