Ensuring local mass conservation in reduced order models by using exact discrete complexes

  • Boon, Wietse (Politecnico di Milano)
  • Fumagalli, Alessio (Politecnico di Milano)

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Constructing fast solution schemes often involves deciding which errors are acceptable and which approximations can be made for the sake of computational efficiency. Herein, we consider mixed formulations of flow systems and take the perspective that the physical law of mass conservation is significantly more important than the constitutive relationship, e.g. Darcy's law. Within this point of view, we propose a three-step solution technique \cite{Boon} that guarantees local mass conservation. In the first step, an initial flux field is obtained by using a locally conservative method such as the TPFA Finite Volume Method. Although this scheme is computationally efficient, it lacks consistency and therefore requires a suitable correction. Since this correction is divergence-free, the Helmholtz decomposition ensures us that it is given by the curl of a potential field. The second step therefore employs an H(curl)-conforming discretization to compute the correction potential and update the flux field. Third, the pressure field is post-computed by using the same TPFA system from the first step. The procedure guarantees local mass conservation regardless of the quality of the computed correction. Thus, we relax this computation using tools from reduced order modeling and novel techniques based on deep learning. We introduce a surrogate model that is capable of rapidly producing a potential field for given permeability fields. By applying the curl to this field, we ensure that the correction is divergence-free and mass conservation is not impacted, independent of the accuracy of the surrogate model. Finally, we show performance of the method in the case of flows in fractured porous media. We rewrite the equations in terms of semi-discrete differential operators and identify the problem as a mixed-dimensional Darcy flow system. In turn, the proposed three-step solution procedure is directly applicable, using the mixed-dimensional curl to ensure local mass conservation.