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Decoupling multistep schemes for elliptic–parabolic problems
Robert Altmann1; Abdullah Mujahid2; Benjamin Unger3
1Institute of Analysis and Numerics, Otto von Guericke University Magdeburg, Magdeburg, Germany
2Stuttgart Center for Simulation Science (SC SimTech), University of Stuttgart, Stuttgart, Germany
3Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology, Karlsruhe, Germany
The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 371-391

We study the construction and convergence of decoupling multistep schemes of higher order using the backward differentiation formulae for an elliptic–parabolic problem, which includes multiple-network poroelasticity as a special case. These schemes were first introduced in [Altmann, Maier, Unger, BIT Numer. Math., 64:20, 2024], where a convergence proof for the second-order case is presented. Here, we present a slightly modified version of these schemes using a different construction of related time delay systems. We present a novel convergence proof relying on concepts from G-stability applicable for any order and providing a sharper characterization of the required weak coupling condition. The key tool for the convergence analysis is the construction of a weighted norm enabling a telescoping argument for the sum of the errors.

Published online:
DOI: 10.5802/smai-jcm.152
Classification: 65M12, 65J10, 76S05
Keywords: poroelasticity, decoupling, higher-order discretization, backward differentiation formulae

Robert Altmann  1 ; Abdullah Mujahid  2 ; Benjamin Unger  3

1 Institute of Analysis and Numerics, Otto von Guericke University Magdeburg, Magdeburg, Germany
2 Stuttgart Center for Simulation Science (SC SimTech), University of Stuttgart, Stuttgart, Germany
3 Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology, Karlsruhe, Germany 
Robert Altmann; Abdullah Mujahid; Benjamin Unger. Decoupling multistep schemes for elliptic–parabolic problems. The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 371-391. doi: 10.5802/smai-jcm.152
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