Conservative methods for poroelasticity

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The present project is relevant to the field of numerical analysis, which takes into account the mathematical properties of computational algorithms for scientific problems, such as the solution of equations representing physical, medical, or engineering phenomena. In particular, here we will focus on the development of novel numerical methods for the efficient and accurate simulation of natural and industrial processes involving poroelastic materials. Common examples arise in diverse applications and include sponges, batteries, soft living tissues, and flow in fractured subsurface. One of the main goals in the design of such methods, is that the produced approximate solutions inherit some of the key intrinsic properties of the physical phenomenon they replicate. For instance, these sought features might be local mass conservation, a marked multiscale nature, or an heterogeneous character. We propose a formalism based on a family of hybrid discretisations, which are able to recover the phenomenological behaviour of interest. After these methods are constructed, a thorough analysis of their mathematical and numerical properties will be carried out (involving existence, uniqueness, and regularity of the generated solutions together with convergence and stability of the discretisations and performance of the implemented computational algorithms), and this will represent an important step forward in matching theoretical and applicative knowledge in the fields of numerical methods, scientific computing, and applied poromechanics.