TY - JOUR

T1 - PARAMETER-ROBUST STOCHASTIC GALERKIN MIXED APPROXIMATION FOR LINEAR POROELASTICITY WITH UNCERTAIN INPUTS

AU - Khan, Arbaz

AU - Powell, Catherine

PY - 2021/3/18

Y1 - 2021/3/18

N2 - Linear poroelasticity models have important applications in biology and geophysics. In particular, the well-known Biot consolidation model describes the coupled interaction between the linear response of a porous elastic medium saturated with fluid and a diffusive fluid flow within it, assuming small deformations. Although deterministic linear models and finite element methods for solving them numerically have been well studied, there is little work to date on robust algorithms for solving poroelasticity models with uncertain inputs and for performing uncertainty quantification (UQ). The Biot model has a number of important physical parameters and inputs whose precise values are often uncertain in real world scenarios. In this work, we introduce and analyse the well-posedness of a new five-field model with uncertain and spatially varying Young's modulus and hydraulic conductivity field. By working with a properly weighted norm, we establish that the weak solution is stable with respect to variations in key physical parameters, including the Poisson ratio. We then introduce a novel locking-free stochastic Galerkin mixed finite element method that is robust in the nearly incompressible case. Armed with an appropriate norm, we construct a parameter-robust preconditioner for the associated discrete systems. Our new method facilitates forward UQ, allowing efficient calculation of statistical quantities of interest and is provably robust with respect to variations in the Poisson ratio, the Biot-Willis constant and the storage coefficient, as well as the discretization parameters.

AB - Linear poroelasticity models have important applications in biology and geophysics. In particular, the well-known Biot consolidation model describes the coupled interaction between the linear response of a porous elastic medium saturated with fluid and a diffusive fluid flow within it, assuming small deformations. Although deterministic linear models and finite element methods for solving them numerically have been well studied, there is little work to date on robust algorithms for solving poroelasticity models with uncertain inputs and for performing uncertainty quantification (UQ). The Biot model has a number of important physical parameters and inputs whose precise values are often uncertain in real world scenarios. In this work, we introduce and analyse the well-posedness of a new five-field model with uncertain and spatially varying Young's modulus and hydraulic conductivity field. By working with a properly weighted norm, we establish that the weak solution is stable with respect to variations in key physical parameters, including the Poisson ratio. We then introduce a novel locking-free stochastic Galerkin mixed finite element method that is robust in the nearly incompressible case. Armed with an appropriate norm, we construct a parameter-robust preconditioner for the associated discrete systems. Our new method facilitates forward UQ, allowing efficient calculation of statistical quantities of interest and is provably robust with respect to variations in the Poisson ratio, the Biot-Willis constant and the storage coefficient, as well as the discretization parameters.

M3 - Article

SN - 1064-8275

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

ER -