Computational multiscale methods for linear poroelasticity with high contrast

Abstract

In this work, we employ the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) to solve the problem of linear heterogeneous poroelasticity with coefficients of high contrast. The proposed method makes use of the idea of energy minimization with suitable constraints in order to generate efficient basis functions for the displacement and the pressure. These basis functions are constructed by solving a class of local auxiliary optimization problems based on eigenfunctions containing local information on the heterogeneity. Techniques of oversampling are adapted to enhance the computational performance. Convergence of first order is shown and illustrated by a number of numerical tests.

Constraint energy minimizing generalized multiscale finite element method for nonlinear poroelasticity and elasticity

“…Remark 3.1. Theoretically, as in [17], combining Korn's first inequality ( [22]) and the Poincaré inequality as well as recalling Remark 2.4, we obtain…”

“…We Finally, we derive some convergence result of the CEM-GMsFEM for this dynamic case. At the time step s (1 ≤ s ≤ S) defined in (3.6)-(3.7), within the nth Picard iteration (n ≥ 1), the following result and its proof are obtained directly from [17].…”

“…Within the structure of multiscale methods, in [6], the GMsFEM was used to handle nonlinear problems in poroelasticity. Then, the idea of CEM-GMsFEM was adopted, for linear poroelasticity in [17] (thanks to [5]), to create multiscale basis functions (with locally minimal energy) for the pressure and the displacement. In this paper, for the case of poroelasticity, to deal with the nonlinearity, after the time-discretization, we employ the Picard iteration procedure; and at each iteration, the CEM-GMsFEM is applied as in [17].…”

“…Then, the idea of CEM-GMsFEM was adopted, for linear poroelasticity in [17] (thanks to [5]), to create multiscale basis functions (with locally minimal energy) for the pressure and the displacement. In this paper, for the case of poroelasticity, to deal with the nonlinearity, after the time-discretization, we employ the Picard iteration procedure; and at each iteration, the CEM-GMsFEM is applied as in [17]. The primary component of the CEM-GMsFEM is the construction of local basis functions for each coarse element (by using the GMsFEM to create the auxiliary multiscale basis functions) then for each oversampled domain (by employing the CEM to obtain the set multiscale basis functions).…”

Numerical homogenization beyond scale separation

“…Other important applications are parabolic problems (Målqvist and Persson 2018), convection-dominated problems (Li, Peterseim and Schedensack 2018), two-phase flow (Elfverson et al 2015), H(div)-problems in mixed formulation (Hellman, Henning and Målqvist 2016), elasticity (Henning and Persson 2016), thermoelasticity (Målqvist and Persson 2017), poroelasticity (Altmann et al 2020a, Fu et al 2019, fractional Laplacian equations (Brown, Gedicke and Peterseim 2018), domains with complicated boundary (Elfverson, Larson and Målqvist 2017), elliptic problems with similar and perturbed coefficients (Hellman and R. A , P. H D. P Målqvist 2019, Hellman, Keil and, discrete network models (Kettil et al 2020), non-linear monotone problems (Verfürth 2019b), problems with sign-changing coefficients (Chaumont-Frelet and Verfürth 2020), diffuse interface models (Hennig et al 2020), heterogeneous bulk-surface coupling (Altmann and Verfürth 2021) and, last but not least, time-dependent non-linear Schrödinger equations where the approach turned out to be game-changing (Henning and Wärnegård 2020).…”