Computational Multiscale Methods for Linear Heterogeneous Poroelasticity

Abstract

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling. This poroelasticity problem suffers from rapidly oscillating material parameters, which calls for a thorough numerical treatment. In this paper, we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity. Therein, local corrector problems are constructed in line with the static equations, whereas we propose to consider the full system. This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure. We prove the optimal first-order convergence of this method and verify the result by numerical experiments.

Computational multiscale methods for linear poroelasticity with high contrast

“…Adopting the idea of CEM-GMsFEM, we propose a multiscale method for the problem of linear poroelasticity with high contrast and construct multiscale spaces for both, the pressure and the displacement. Based on the previous work [1], we prove the first-order convergence of the implicit Euler scheme combined with CEM-GMsFEM for the spatial discretization. Numerical results are provided to demonstrate the efficiency of the proposed method.…”

“…Proof. The proof mainly follows the lines of the proof of [1,Thm. 3.7] and makes use of the results of Lemma 4.4.…”

“…This method was applied in [22] to linear heterogeneous thermoelasticity and optimal first-order convergence of the fully discretized system based on LOD and an implicit Euler discretization in time was proven. This approach was transferred to the present poroelastic setting in [1].…”

Efficient implementation of the localized orthogonal decomposition method

“…The final fine scale approximation is then given by u h := U h + g h ∈ V h . Now observe that problem (27) is basically of the same structure as the homogenous problem (24). This suggest to apply the same methodology as before.…”

“…So far it has been successfully applied to linear elliptic multiscale problems in the context of continuous finite elements [1,2,3], discontinuous finite elements [4,5,6], mixed finite elements [7,8], partition of unity methods [9] and reduced basis simulations [10]. The range of applications covers linear and quadratic eigenvalue problems [11,12], problems in perforated domains [13] and high-contrast media [14,15], stochastic homogenization [16,17], semilinear elliptic problems [18], the wave equation [19,20,21], parabolic and coupled problems [22,23,24], the Buckley-Leverett equation [25], fractional diffusion problems [26], Helmholtz problems [27,28,29,30] and the simulation of Bose-Einstein condensates [31]. An introductionary general overview is given in [32].…”