A space-time isogeometric method for the partial differential-algebraic system of Biot's poroelasticity model

Abstract

Biot's equations of poroelasticity contain a parabolic system for the evolution of the pressure, which is coupled with a quasi-stationary equation for the stress tensor. Thus, it is natural to extend the existing work on isogeometric space-time methods to this more advanced framework of a partial differential-algebraic equation (PDAE). A space-time approach based on finite elements has already been introduced. We present a new weak formulation in space and time that is appropriate for an isogeometric discretization and analyze its convergence properties. Our approach is based on a single variational problem and hence differs from the iterative space-time schemes considered so far. Further, it enables high-order convergence. Numerical experiments that have been carried out confirm the theoretical findings.

A semi‐explicit integration scheme for weakly‐coupled poroelasticity with nonlinear permeability

“…For the respective spatial error analysis, we refer to the aforementioned sources. We would also like to mention the possibility of space-time approaches as considered in [BRK17,AS22]. These, however, are not compatible with the proposed time steppe scheme.…”

A novel iterative time integration scheme for linear poroelasticity

“…For details on the spatial discretization we refer to [15,20,23,24,25] and the references therein. Related space-time approaches were considered in [5,7]. A direct application of the implicit Euler scheme results in an unconditionally stable first-order method [13].…”