[OmarDuran]

This PR adds a mixed mimetic finite difference (MFD) discretization of single-phase flow, solved as a saddle-point problem. It adapts the consistency, each cell uses the TPFA inner product where TPFA is consistent and a consistent
MFD product elsewhere; fluxes between two TPFA cells are condensed. The associated theory can be found here: https://arxiv.org/abs/2607.23568.

It adds the mixedMimetic module, the SinglePhaseMixedMFD solver, a mixed-form RT inner product that requires no stabilization on simplexes, as it is the exact lower-order RT element, the ConsistencyAdaptation cell classification, a MGR strategy, and unit tests.

This is a work in collaboration with @chauj96 , @castelletto1 , @victorapm and myself.

The discretization block will look like this:

<MixedMimeticDiscretization
  name="mfdDiscretization"
  innerProductType="RT"
  adaptiveConsistency="1"
  consistencyTolerance="0.01"
  degeneracyTolerance="0.1"
  nominalGradient="{ 1.0, 1.0, 1.0 }"/>

• innerProductType: the consistent product used in the MFD cells.
• adaptiveConsistency: 1 selects TPFA or MFD per cell, 0 uses MFD everywhere.
• consistencyTolerance: a cell becomes MFD when the relative two-point flux error, probed with nominalGradient, exceeds it.
• degeneracyTolerance: a cell is degenerate, and stays TPFA, when its volume is smaller than this percentage of
  the combined volume of all cells sharing a vertex with it. This controls the conditioning of the system on meshes with cells with extreme aspect ratio.

Alternatively, the choice can be prescribed per cell from the VTK mesh with the integer cell array prescribedMfdFlag: 0 = TPFA, 1 = MFD, −1 = let consistencyTolerance and degeneracyTolerance decide. Prescribed cells with 0 = TPFA, 1 = MFD are never altered. The result of the classification is written as mfdFlag.

The following is an toy example of consistency adaptation on the full SPE10, where the green regions (MFD subregion) represent a simplex region connected with a large portion of hexahedral cells (TPFA subregion).

The following figure shows the MFD sub-region automatically identified employing a consistencyTolerance="0.01"

The following figures show the pressure colormaps for adaptive-MFD (left panel), TPFA on a hexahedral mesh (middle panel), and TPFA on the hybrid mesh (right panel). The TPFA approximation on the hybrid mesh exhibits a visible effect due to the consistency error localized in the simplex cells. The adaptive-MFD approximation is quite similar to the TPFA approximation on the hexahedral mesh, where it is exact. The small differences are due to the different types of meshes.

The following table was constructed using 8 ranks to show the following features:

1. Optimal convergence of the pressure in l2-norm on smooth solutions;
2. Robustness of the MGR solution strategy w.r.t refinement and material constrast;
3. Reduction in the number of iterations with an increasing share of TPFA cells.

relative $L^2$ pressure error / GMRES iterations / linear solve time [s] (setup excluded); $\text{krylovTol} = 1.0\times 10^{-6}$
isotropic permeability in series along the flow ($x$): $k = 10^{-14}\text{ m}^2$ on $x < 0.5$, $k = 10^{-14} / \text{contrast}$ on $x > 0.5$; smooth exact pressure solution
$\text{unknowns} = \text{cells} + \text{faces}$ of the assembled matrix; $\text{reduced} = \text{cells} + \text{non-condensed flux dofs}$, the coupled system after the condensation (a condensed flux is a one-way closure row)

mesh	mfd%	n	cells	unknowns	reduced	contrast 1	contrast 10000	contrast 1e+07
hex	0	2	16	84	16	3.33e-01 / 4 / 0.000	1.00e+00 / 5 / 0.000	1.00e+00 / 5 / 0.000
hex	0	4	128	592	128	8.54e-02 / 7 / 0.001	3.33e-01 / 7 / 0.001	3.33e-01 / 7 / 0.001
hex	0	8	1024	4416	1024	2.14e-02 / 8 / 0.003	8.54e-02 / 9 / 0.003	8.54e-02 / 9 / 0.003
hex	0	16	8192	34048	8192	5.35e-03 / 10 / 0.027	2.14e-02 / 10 / 0.028	2.14e-02 / 10 / 0.027
hex	0	32	65536	267264	65536	1.34e-03 / 11 / 0.278	5.35e-03 / 11 / 0.277	5.35e-03 / 11 / 0.279
hex	0	64	524288	2117632	524288	3.34e-04 / 12 / 2.649	1.34e-03 / 12 / 2.677	1.34e-03 / 12 / 2.683
hex	25	2	16	84	36	2.74e-01 / 6 / 0.000	8.34e-01 / 6 / 0.000	8.35e-01 / 6 / 0.000
hex	25	4	128	592	256	7.11e-02 / 9 / 0.001	2.82e-01 / 9 / 0.001	2.82e-01 / 11 / 0.001
hex	25	8	1024	4416	1920	1.79e-02 / 11 / 0.004	7.26e-02 / 13 / 0.005	7.27e-02 / 15 / 0.006
hex	25	16	8192	34048	14848	4.47e-03 / 13 / 0.037	1.82e-02 / 14 / 0.041	1.82e-02 / 18 / 0.056
hex	25	32	65536	267264	116736	1.12e-03 / 14 / 0.365	4.56e-03 / 16 / 0.433	4.56e-03 / 20 / 0.581
hex	25	64	524288	2117632	925696	2.80e-04 / 16 / 3.759	1.14e-03 / 19 / 4.689	1.14e-03 / 21 / 5.343
hex	75	2	16	84	68	1.50e-01 / 6 / 0.000	4.69e-01 / 8 / 0.000	4.69e-01 / 8 / 0.000
hex	75	4	128	592	480	3.95e-02 / 9 / 0.001	1.62e-01 / 9 / 0.001	1.62e-01 / 11 / 0.001
hex	75	8	1024	4416	3584	9.99e-03 / 12 / 0.005	4.21e-02 / 13 / 0.005	4.22e-02 / 14 / 0.006
hex	75	16	8192	34048	27648	2.51e-03 / 13 / 0.037	1.06e-02 / 15 / 0.045	1.06e-02 / 18 / 0.058
hex	75	32	65536	267264	217088	6.27e-04 / 15 / 0.392	2.66e-03 / 17 / 0.465	2.66e-03 / 21 / 0.614
hex	75	64	524288	2117632	1720320	1.57e-04 / 16 / 3.710	6.65e-04 / 19 / 4.668	6.65e-04 / 21 / 5.295
hex	100	2	16	84	84	6.67e-02 / 10 / 0.000	2.00e-01 / 11 / 0.000	2.00e-01 / 11 / 0.000
hex	100	4	128	592	592	1.71e-02 / 13 / 0.001	6.66e-02 / 13 / 0.001	6.67e-02 / 14 / 0.001
hex	100	8	1024	4416	4416	4.28e-03 / 17 / 0.007	1.71e-02 / 18 / 0.007	1.71e-02 / 19 / 0.008
hex	100	16	8192	34048	34048	1.07e-03 / 18 / 0.057	4.28e-03 / 20 / 0.065	4.28e-03 / 24 / 0.084
hex	100	32	65536	267264	267264	2.67e-04 / 20 / 0.568	1.07e-03 / 23 / 0.685	1.07e-03 / 27 / 0.851
hex	100	64	524288	2117632	2117632	6.69e-05 / 23 / 5.944	2.67e-04 / 26 / 7.096	2.67e-04 / 28 / 7.840
tet	0	2	96	328	96	6.04e-01 / 9 / 0.000	1.04e+00 / 9 / 0.001	1.04e+00 / 9 / 0.000
tet	0	4	768	2464	768	4.82e-01 / 11 / 0.002	6.04e-01 / 11 / 0.002	6.04e-01 / 11 / 0.002
tet	0	8	6144	19072	6144	4.54e-01 / 13 / 0.020	4.82e-01 / 13 / 0.020	4.82e-01 / 13 / 0.020
tet	0	16	49152	150016	49152	4.47e-01 / 13 / 0.171	4.54e-01 / 14 / 0.189	4.54e-01 / 14 / 0.188
tet	0	32	393216	1189888	393216	4.45e-01 / 14 / 1.704	4.47e-01 / 14 / 1.699	4.47e-01 / 14 / 1.701
tet	0	64	3145728	9478144	3145728	4.45e-01 / 15 / 14.798	4.45e-01 / 15 / 15.138	4.45e-01 / 15 / 15.088
tet	25	2	96	328	160	5.16e-01 / 11 / 0.001	9.23e-01 / 13 / 0.001	9.23e-01 / 15 / 0.001
tet	25	4	768	2464	1216	3.83e-01 / 15 / 0.003	5.07e-01 / 17 / 0.004	5.07e-01 / 18 / 0.004
tet	25	8	6144	19072	9472	3.56e-01 / 16 / 0.027	4.00e-01 / 19 / 0.034	4.00e-01 / 21 / 0.038
tet	25	16	49152	150016	74752	3.50e-01 / 17 / 0.243	3.74e-01 / 19 / 0.279	3.74e-01 / 23 / 0.369
tet	25	32	393216	1189888	593920	3.48e-01 / 18 / 2.365	3.68e-01 / 21 / 2.880	3.68e-01 / 22 / 3.076
tet	25	64	3145728	9478144	4734976	3.48e-01 / 20 / 21.675	3.66e-01 / 21 / 23.160	3.66e-01 / 25 / 29.386
tet	75	2	96	328	272	3.08e-01 / 11 / 0.001	6.65e-01 / 13 / 0.001	6.65e-01 / 16 / 0.001
tet	75	4	768	2464	2048	1.79e-01 / 15 / 0.003	2.93e-01 / 17 / 0.004	2.93e-01 / 19 / 0.005
tet	75	8	6144	19072	15872	1.59e-01 / 17 / 0.029	2.10e-01 / 20 / 0.035	2.10e-01 / 22 / 0.041
tet	75	16	49152	150016	124928	1.54e-01 / 17 / 0.246	1.92e-01 / 19 / 0.290	1.92e-01 / 23 / 0.370
tet	75	32	393216	1189888	991232	1.53e-01 / 18 / 2.347	1.88e-01 / 21 / 2.873	1.88e-01 / 23 / 3.235
tet	75	64	3145728	9478144	7897088	1.53e-01 / 19 / 20.239	1.87e-01 / 21 / 23.089	1.87e-01 / 25 / 29.282
tet	100	2	96	328	328	1.46e-01 / 15 / 0.001	5.12e-01 / 17 / 0.001	5.12e-01 / 20 / 0.001
tet	100	4	768	2464	2464	3.69e-02 / 19 / 0.004	1.46e-01 / 20 / 0.005	1.46e-01 / 22 / 0.005
tet	100	8	6144	19072	19072	9.22e-03 / 21 / 0.038	3.68e-02 / 23 / 0.043	3.69e-02 / 25 / 0.049
tet	100	16	49152	150016	150016	2.30e-03 / 20 / 0.309	9.21e-03 / 23 / 0.368	9.22e-03 / 27 / 0.460
tet	100	32	393216	1189888	1189888	5.76e-04 / 22 / 3.052	2.30e-03 / 24 / 3.435	2.30e-03 / 26 / 3.841
tet	100	64	3145728	9478144	9478144	1.44e-04 / 22 / 24.415	5.76e-04 / 25 / 29.192	5.76e-04 / 52 / 87.726

[victorapm]

Will be reviewed by @jafranc, @joshua-white

[victorapm]

Thanks for the great work @chauj96 and @OmarDuran ! I left a few comments and I'm also working on a follow-up PR with some additional improvements built on top of this branch

[jafranc]

Great addition ! Thank you 🙏
Some notes and docs to be done later.

[OmarDuran]

Thank you @jafranc and @victorapm for the careful review. All your comments are addressed in the last commits.

[victorapm]

Needs code owner review from @MelReyCG , @bd713 , @dkachuma , @jhuang2601 , @joshua-white and/or @rrsettgast Thank you 🙏