2020/03/14 by Arbaz Khan, Khan, Arbaz, Catherine E. Powell +1
Decision Sciences · Environmental Science · #35R60 #65F08 #65N30 #FOS: Mathematics #Groundwater flow and contamination studies #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2003.06628
openalex publication_date 2020/03/14 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Linear poroelasticity models have a number of important applications in\nbiology and geophysics. In particular, Biot's consolidation model is a\nwell-known model that describes the coupled interaction between the linear\nresponse of a porous elastic medium and a diffusive fluid flow within it,\nassuming small deformations. Although deterministic linear poroelasticity\nmodels and finite element methods for solving them numerically have been well\nstudied, there is little work to date on robust algorithms for solving\nporoelasticity models with uncertain inputs and for performing uncertainty\nquantification (UQ). The Biot model has a number of important physical\nparameters and inputs whose precise values are often uncertain in real world\nscenarios. In this work, we introduce and analyse the well-posedness of a new\nfive-field model with uncertain and spatially varying Young's modulus and\nhydraulic conductivity field. By working with a properly weighted norm, we\nestablish that the weak solution is stable with respect to variations in key\nphysical parameters, including the Poisson ratio. We then introduce a novel\nlocking-free stochastic Galerkin mixed finite element method that is robust in\nthe incompressible limit. Armed with the `right' norm, we construct a\nparameter-robust preconditioner for the associated discrete systems. Our new\nmethod facilitates forward UQ, allowing efficient calculation of statistical\nquantities of interest and is provably robust with respect to variations in the\nPoisson ratio, the Biot--Willis constant and the storage coefficient, as well\nas the discretization parameters.\n