Convergence issues for cubic FE_SimplexP on triangles

[Arthur Bawin]

Good morning deal.ii developers,

I was doing convergence tests with a manufactured solution with simplicial meshes and did not obtain the expected convergence with cubic elements on triangles (FE_SimplexP<2>(3)). I wrote a very simple interpolation convergence test to check: on tetrahedra,  all degrees converge with the expected rates in L2, Linf and H1 (semi)norms; on triangles the results are good for P1 and P2 on triangles, but not for P3, where I get first order convergence for L2, Linf and no convergence in H1:

P1 on triangles
Number of dofs: 25
Number of dofs: 81
Number of dofs: 289
Number of dofs: 1089
nElm          L2_u                   Li_u                   H1_u          
  32 5.938414e-03    -    - 2.425255e-02    -    - 1.549249e-01    -    -
 128 1.498810e-03 3.96 1.99 6.769010e-03 3.58 1.84 7.782930e-02 1.99 0.99
 512 3.755853e-04 3.99 2.00 1.780487e-03 3.80 1.93 3.896037e-02 2.00 1.00
2048 9.395142e-05 4.00 2.00 4.561509e-04 3.90 1.96 1.948590e-02 2.00 1.00

P2 on triangles
Number of dofs: 81
Number of dofs: 289
Number of dofs: 1089
Number of dofs: 4225
nElm          L2_u                   Li_u                   H1_u          
  32 3.112575e-04    -    - 6.997055e-04    -    - 1.041667e-02    -    -
 128 3.890718e-05 8.00 3.00 8.746319e-05 8.00 3.00 2.604167e-03 4.00 2.00
 512 4.863398e-06 8.00 3.00 1.093290e-05 8.00 3.00 6.510417e-04 4.00 2.00
2048 6.079247e-07 8.00 3.00 1.366612e-06 8.00 3.00 1.627604e-04 4.00 2.00

P3 on triangles
Number of dofs: 169
Number of dofs: 625
Number of dofs: 2401
Number of dofs: 9409
nElm          L2_u                   Li_u                   H1_u          
  32 1.541667e-02    -    - 7.120958e-02    -    - 5.117887e-01    -     -
 128 8.815536e-03 1.75 0.81 4.355310e-02 1.64 0.71 5.712863e-01 0.90 -0.16
 512 4.690321e-03 1.88 0.91 2.389578e-02 1.82 0.87 6.001976e-01 0.95 -0.07
2048 2.416438e-03 1.94 0.96 1.249422e-02 1.91 0.94 6.144240e-01 0.98 -0.03

P1 on tets
Number of dofs: 125
Number of dofs: 729
Number of dofs: 4913
Number of dofs: 35937
 nElm           L2_u                   Li_u                   H1_u          
   384 8.095068e-03    -    - 6.562605e-02    -    - 1.735667e-01    -    -
  3072 2.289416e-03 3.54 1.82 2.693394e-02 2.44 1.28 9.615581e-02 1.81 0.85
 24576 5.896877e-04 3.88 1.96 7.780330e-03 3.46 1.79 4.927053e-02 1.95 0.96
196608 1.485159e-04 3.97 1.99 2.081313e-03 3.74 1.90 2.478494e-02 1.99 0.99

P2 on tets
Number of dofs: 729
Number of dofs: 4913
Number of dofs: 35937
Number of dofs: 274625
 nElm           L2_u                   Li_u                   H1_u          
   384 1.118739e-03    -    - 1.184543e-02    -    - 2.361912e-02    -    -
  3072 1.637572e-04 6.83 2.77 3.376606e-03 3.51 1.81 6.731956e-03 3.51 1.81
 24576 2.117353e-05 7.73 2.95 4.535571e-04 7.44 2.90 1.732030e-03 3.89 1.96
196608 2.668408e-06 7.93 2.99 5.866221e-05 7.73 2.95 4.360374e-04 3.97 1.99

P3 on tets
Number of dofs: 2197
Number of dofs: 15625
Number of dofs: 117649
Number of dofs: 912673
 nElm           L2_u                    Li_u                    H1_u          
   384 1.074553e-04     -    - 7.422949e-04     -    - 2.346755e-03    -    -
  3072 8.111057e-06 13.25 3.73 1.300853e-04  5.71 2.51 3.362943e-04 6.98 2.80
 24576 5.264880e-07 15.41 3.95 8.130328e-06 16.00 4.00 4.327503e-05 7.77 2.96
196608 3.320390e-08 15.86 3.99 5.081455e-07 16.00 4.00 5.447386e-06 7.94 2.99

Is there something that I'm doing wrong, or is it just a bug somewhere in FE_SimplexP?

I have joined a MWE with the 2D and 3D meshes (in .msh4 format).

Thank you for your help, and thanks a lot for developing the library!
Arthur Bawin

[blais...@gmail.com]

Dear Arthur,

Thank you for the very thorough and detailed question. I'll look into and try to get back to you (or at least try to reach out to people that might have a clearer idea). If that can reassure you, I have tried to reproduce your results using Lethe and I get the same conclusion to you regarding convergence of P3 triangles.

I'll look into it and ask around. Might take a small while, but I am on it :)

[Wells, David]

Hi Arthur,

Thank you for reporting this! This is another simplex bug inside deal.II. From 1999ish until 2021 we assumed 2d grids had consistently oriented lines, but this cannot be enforced in for triangles in 2d, and over the last two years I've been fixing a long tail of minor bugs related to this assumption. I have a patch which fixes this here:

https://github.com/dealii/dealii/pull/18832

I gave you credit for figuring this one out in the changelog.

Best,

David Wells

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From: dea...@googlegroups.com <dea...@googlegroups.com> on behalf of blais...@gmail.com <blais...@gmail.com>
Sent: Wednesday, August 20, 2025 9:56 AM
To: deal.II User Group <dea...@googlegroups.com>
Subject: [deal.II] Re: Convergence issues for cubic FE_SimplexP on triangles

 

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[Arthur Bawin]

Hello Bruno and David,

Thank you for the quick reply and the quick fix! I'll give it a try.

Cheers,

Arthur