DG heat equation tutorial

[AbdAlazezAhmed]

Tested convergence using an edited version of #640 and it seems to converge * currently doing some edits to optimize it and see where the results stagnate. Will leave the script in a resolved comment once finished so it doesn't take much space *

These plots are in log-log, and N is the number of elements in one of the dimensions (so for quad the number of elements is N^2)

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[termi-official]

Is the continuous below order 2 or the discontinuous above order 2?

[AbdAlazezAhmed]

Is the continuous below order 2 or the discontinuous above order 2?

I think you mean what the two lines at the bottom are, right?
The two lines are the bottom are order 2, the lower one is discontinuous and the above one is continuous

[termi-official]

But they differ in slope, right? What are their exact orders?

[termi-official]

I think we must clarify some details here.

[AbdAlazezAhmed]

Some convergence script results as graphs can be harder to read (all for quadrilateral):

• First order continuous

 Row │ N      Linf         L2           L2_expected_next  H1         H1_expected_next
     │ Int64  Float64      Float64      Float64           Float64    Float64
─────┼────────────────────────────────────────────────────────────────────────────────
   1 │     2  0.130196     0.202511          0.0506276    1.00388           0.50194
   2 │     4  0.0460564    0.0510494         0.0127623    0.502255          0.251128
   3 │     8  0.012506     0.0128262         0.00320656   0.251625          0.125813
   4 │    16  0.00319102   0.00321117        0.000802793  0.125888          0.0629439
   5 │    32  0.000801829  0.00080309        0.000200773  0.0629537         0.0314769
   6 │    64  0.000200712  0.000200791       5.01978e-5   0.0314781         0.015739

• First order discontinuous

 Row │ N      Linf         L2           L2_expected_next  H1         H1_expected_next
     │ Int64  Float64      Float64      Float64           Float64    Float64
─────┼────────────────────────────────────────────────────────────────────────────────
   1 │     2  0.130196     0.202511          0.0506276    1.00388           0.50194
   2 │     4  0.0453722    0.0501761         0.012544     0.502248          0.251124
   3 │     8  0.0124713    0.0127358         0.00318395   0.251674          0.125837
   4 │    16  0.00318923   0.00320436        0.000801089  0.125898          0.0629488
   5 │    32  0.000801729  0.000802629       0.000200657  0.0629551         0.0314776
   6 │    64  0.000200707  0.000200761       5.01903e-5   0.0314783         0.0157391

• Second order continuous

 Row │ N      Linf         L2           L2_expected_next  H1           H1_expected_next
     │ Int64  Float64      Float64      Float64           Float64      Float64
─────┼──────────────────────────────────────────────────────────────────────────────────
   1 │     2  0.0247816    0.028836          0.0036045    0.204098          0.0510244
   2 │     4  0.00307976   0.00386503        0.000483128  0.0511228         0.0127807
   3 │     8  0.000388316  0.000490246       6.12807e-5   0.0127715         0.00319286
   4 │    16  4.81335e-5   6.14925e-5        7.68657e-6   0.00319204        0.00079801
   5 │    32  5.97622e-6   7.6931e-6         9.61637e-7   0.000797955       0.000199489
   6 │    64  7.44099e-7   9.61841e-7        1.2023e-7    0.000199485       4.98713e-5

• Second order discontinuous

 Row │ N      Linf         L2           L2_expected_next  H1           H1_expected_next
     │ Int64  Float64      Float64      Float64           Float64      Float64
─────┼──────────────────────────────────────────────────────────────────────────────────
   1 │     2  0.0247816    0.028836          0.0036045    0.204098          0.0510244
   2 │     4  0.00283828   0.00320227        0.000400284  0.0531917         0.0132979
   3 │     8  0.000371974  0.000358141       4.47676e-5   0.0133091         0.00332728
   4 │    16  4.69751e-5   4.18634e-5        5.23293e-6   0.00332116        0.00083029
   5 │    32  5.88702e-6   5.04621e-6        6.30776e-7   0.000829385       0.000207346
   6 │    64  7.36358e-7   6.18999e-7        7.73749e-8   0.000207235       5.18088e-5

• Third order continuous

Row │ N      Linf         L2           L2_expected_next  H1           H1_expected_next
   │ Int64  Float64      Float64      Float64           Float64      Float64
─────┼──────────────────────────────────────────────────────────────────────────────────
 1 │     2  0.00287087   0.00271881        0.000169926  0.0268203         0.00335254
 2 │     4  0.000242052  0.000176249       1.10156e-5   0.00338103        0.000422628
 3 │     8  1.61882e-5   1.11276e-5        6.95476e-7   0.000423456       5.2932e-5
 4 │    16  1.03618e-6   6.97278e-7        4.35799e-8   5.29573e-5        6.61966e-6
 5 │    32  6.51477e-8   4.36082e-8        2.72552e-9   6.62044e-6        8.27556e-7
 6 │    64  4.07782e-9   2.72595e-9        1.70372e-10  8.2758e-7         1.03448e-7

• Third order discontinuous

 Row │ N      Linf         L2           L2_expected_next  H1           H1_expected_next
     │ Int64  Float64      Float64      Float64           Float64      Float64
─────┼──────────────────────────────────────────────────────────────────────────────────
 1 │     2  0.00287087   0.00271881        0.000169926  0.0268203         0.00335254
 2 │     4  0.000228848  0.000172388       1.07742e-5   0.00351791        0.000439739
 3 │     8  1.60126e-5   1.10347e-5        6.89667e-7   0.000436887       5.46109e-5
 4 │    16  1.03317e-6   6.95645e-7        4.34778e-8   5.36008e-5        6.7001e-6
 5 │    32  6.50996e-8   4.35815e-8        2.72385e-9   6.64427e-6        8.30534e-7
 6 │    64  4.07685e-9   2.72552e-9        1.70345e-10  8.28381e-7        1.03548e-7

The expected norm is the current norm * 1/2^(order) for H1 and norm * 1/2^(order + 1) for L2
Looks like I misread the data at first, this one looks like it tends to follow the expected behavior?
Since we're halving the element size each time then $\Delta Log_2(L2) \approx P+1$ and $\Delta Log_2(H1) \approx P $

[ Info: order = 1
[ Info: slopes of L2 vs N = [-2.0129235398778857, -1.9781123405090923, -1.9907819594249005, -1.9972307802964462, -1.9992507830984803]
[ Info: slopes of H1 vs N = [-0.9991152976325772, -0.9968411962760191, -0.999308479759299, -0.9998539901566406, -0.9999670340139399]

[ Info: order = 2
[ Info: slopes of L2 vs N = [-3.1707031297879107, -3.1604949308839316, -3.096765601242465, -3.052417650566909, -3.027191020911541]
[ Info: slopes of H1 vs N = [-1.939987107438954, -1.99878412103502, -2.0026566568598767, -2.001573208098428, -2.000772421073222]

[ Info: order = 3
[ Info: slopes of L2 vs N = [-3.979248450192239, -3.9655415209701683, -3.9875483912823775, -3.9965627848873275, -3.999110899899076]
[ Info: slopes of H1 vs N = [-2.9305366294295183, -3.0093852067151214, -3.0269354855259873, -3.012071199303561, -3.0037441172379964]

[termi-official]

LGTM

[KnutAM]

Very nice work!
I just had some time to read and give some thoughts and comments (this is how far I got now)

Overall, I think it would be nice to avoid repeating too many things from the basic heat equation tutorial and focus on what is new here. IMO this is an advanced tutorial and readers should be familiar with the basic Ferrite syntax.

[KnutAM]

This is looking very good!
Mostly formatting (boldface) comments.

[KnutAM]

After the suggested fine-tuning, it is good to go from my perspective regarding formatting and clarity. (But I haven't checked the correctness wrt. the literature. I leave this to @termi-official :))

Nice work @AbdAlazezAhmed!

[KristofferC]

🎉