Semi-explicit discretization schemes for weakly coupled elliptic-parabolic problems

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

“…We consider a 3 × 3 system in order to test the semi-explicit discretization for a nonlinear system, which is motivated by the nonlinear permeability considered in [4]. In the linear setting, we consider finite-dimensional spaces This example satisfies the weak coupling condition proposed in [2] and we refer to this paper for a numerical study. In the linear case, the semi-explicit Euler schemes approximates the true solution well with the same order as the fully implicit scheme.…”

“…Numerical approximations of (1) usually consider an implicit Euler scheme combined with finite elements, leading to a large (nonlinear) system that has to be solved in each time step. In the linear case, the coupled system can be decoupled with a semi-explicit approach as proposed in [2] (based on observations made in [1]) without decreasing the convergence order in time. To ensure convergence, however, one needs a weak coupling condition in the sense that the continuity constant of d is small compared to the ellipticity constants of a and c. This contribution aims to sound the applicability of semi-explicit discretizations also for the nonlinear model (1) since this would improve the computational efficiency dramatically.…”

A decoupling and linearizing discretization for weakly coupled poroelasticity with nonlinear permeability

“…In order to prove convergence of the semi-explicit scheme, we follow the idea first presented in [AMU21a] for the linear case and consider a delay system, which is closely related to the original system (2.4). This is the subject of the following subsection.…”

“…Proof of Proposition 3.5. The proof is based on [AMU21a] and follows the ideas of [EM09]. We set η n u := ūn − u n ∈ V and η n p := pn − p n ∈ Q as well as…”

“…We refer to this particular configuration as weak coupling, which is satisfied in many applications. The advantage of a semi-explicit discretization for the classical (linear) poroelastic model was studied in [AMU21a]. Therein it was shown that a semi-explicit Euler discretization maintains the first-order convergence and at the same time decouples the system equations.…”