A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity

Abstract

In this work we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…An alternative to GMsHDG is the HHO framework for highly oscillatory elliptic problems introduced in [73]. Moreover, in the context of coupled problems involving porous media, HHO simulations of passive transport of a solute in a fractured medium are presented in [52], whereas nonlinear poroelastic phenomena in a saturated porous medium with a slightly compressible fluid are described in [33,35].…”

An abstract analysis framework for monolithic discretisations of poroelasticity with application to Hybrid High-Order methods

“…The bound (8) on the gradient of the initial pore pressure is instrumental to proving the regularity results of Theorem 1. Under some additional assumptions on problem data, we are able to prove that p(0) solving (7) and (2c) satisfies (8). We distinguish three cases:…”

“…More recently, stable cell centered finite volume discretisations have been considered in [26]. Concerning the Hybrid High-Order (HHO) literature, we can cite the original method of [3], relying on a discretisation of the displacements in the HHO space of degree k ≥ 1 and of the pore pressure in the space of broken polynomials of total degree ≤ k, as well as its extension to nonlinear strain-stress laws considered in [8] (based in turn on the method of [7] for nonlinear elasticity). We also cite here the recent work [6] on the extension to random coefficients.…”

A Hybrid High-Order Method for Multiple-Network Poroelasticity

“…A general and up-to-date overview of HHO methods can be found in the recent monograph [22]. Concerning their applications to poroelasticity, we can cite, in particular: the HHO-Discontinuous Galerkin method for the Biot problem proposed and analysed in [8], based in turn on the methods of [23] for the mechanics and [24] for the flow; its extension to nonlinear elastic laws proposed in [14], where the mechanical term is discretised according to [13]; its application to the treatment of stochastic coefficients considered in [12] in conjunction with polynomial chaos techniques. An abstract analysis framework covering general schemes for the linear Biot problem in fully discrete formulation (cf.…”