Conservative methods for poroelasticity

本项目与数值分析领域有关，该领域考虑到科学问题的计算算法的数学特性，例如代表物理，医学或工程现象的方程的解。特别是，在这里，我们将专注于开发新的数值方法，用于有效和准确地模拟涉及多孔弹性材料的自然和工业过程。常见的例子出现在不同的应用中，包括海绵、电池、软活组织和裂缝地下的流动。设计这种方法的主要目标之一是，所产生的近似解继承了它们所复制的物理现象的一些关键固有特性。例如，这些寻求的功能可能是当地的质量守恒，一个显着的多尺度性质，或异质性。我们提出了一个形式主义的基础上家庭的混合离散化，这是能够恢复的现象学行为的兴趣。在这些方法被构造之后，将对其数学和数值性质进行彻底的分析（涉及生成的解的存在性、唯一性和正则性，以及离散化的收敛性和稳定性以及所实现的计算算法的性能），这将代表在匹配数值方法领域的理论和应用知识方面向前迈出的重要一步，科学计算和应用孔隙力学。

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.