An Algebraic Dynamic Multilevel (ADM) method [1, 2] for fully implicit simulations of multiphase flow in heterogeneous fractured natural porous media is presented. The fine-scale fully-implicit (FIM) system is obtained following the Embedded Discrete Fracture Modelling (EDFM)  approach. A set of nested coarse grids at different resolutions (or levels) is constructed independently for each medium. At each time-step the fine-scale FIM system is mapped to a dynamically selected grid formed by grid-blocks of the previously defined grids. The grid resolution is chosen, independently for each medium, based on a front- tracking criterion that ensures that fine-scale resolution is employed only where most physical interactions take place (i.e., moving saturation front). Mapping between different grid resolutions is performed by employing sequences of restriction and prolongation (interpolation) operators. Finite-volume restriction  operators are employed to ensure mass conservation whereas different interpolation strategies are employed for each variable. In particular, multiscale basis functions  are considered as pressure interpolators to ensure an accurate interpolation of the pressure field inside coarse blocks.
The efficiency and accuracy of the proposed algorithm is shown through a set of challenging 2D and 3D test cases involving various non-linear physics. The sensitivity of the method to the choice of the coarsening and interpolation strategy is also presented.
 Matteo Cusini, Cor van Kruijsdijk, and Hadi Hajibeygi. Algebraic dynamic multilevel (adm) method for fully implicit simulations of multiphase flow in porous media. J. Comput. Phys., 314:60–79, 2016.
 Matteo Cusini, Barnaby Fryer, Cor van Kruijsdijk, and Hadi Hajibeygi. Algebraic dynamic multilevel method for compositional flow in heterogeneous porous media. J. Comput. Phys., 354:593-612, 2018.
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