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

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R. Altmann, R. Maier and B. Unger
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## Abstract:

We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. The presented convergence proof is based on an interpretation of the scheme as an implicit method applied to a constrained partial differential equation with delay term. Here, the delay time equals the used step size. This connection also allows a deeper understanding of the weak coupling condition, which we accomplish to quantify explicitly.## References

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## Additional Information

**R. Altmann**- Affiliation: Department of Mathematics, University of Augsburg, Universitätsstr. 14, 86159 Augsburg, Germany
- MR Author ID: 977251
- ORCID: 0000-0002-4161-6704
- Email: robert.altmann@math.uni-augsburg.de
**R. Maier**- Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96 Göteborg, Sweden
- MR Author ID: 1329180
- Email: roland.maier@chalmers.se
**B. Unger**- Affiliation: Institute of Mathematics MA 4-5, Technical University Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
- Address at time of publication: Stuttgart Center for Simulation Science (SC Sim Tech), University of Stuttgart, Universitätsstr. 32, 70569 Stuttgart, Germany
- MR Author ID: 1187453
- ORCID: 0000-0003-4272-1079
- Email: benjamin.unger@simtech.uni-stuttgart.de
- Received by editor(s): September 9, 2019
- Received by editor(s) in revised form: July 11, 2020
- Published electronically: March 3, 2021
- Additional Notes: The second author gratefully acknowledges support by the German Research Foundation (DFG) in the Priority Program 1748
*Reliable simulation techniques in solid mechanics*(PE2143/2-2). The work of the third author was supported by the German Research Foundation (DFG) Collaborative Research Center 910*Control of self-organizing nonlinear systems: Theoretical methods and concepts of application*, project number 163436311. - © Copyright 2021 American Mathematical Society
- Journal: Math. Comp.
**90**(2021), 1089-1118 - MSC (2020): Primary 65M12, 65L80, 65M60, 76S05
- DOI: https://doi.org/10.1090/mcom/3608
- MathSciNet review: 4232218