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


Authors: R. Altmann, R. Maier and B. Unger
Journal: Math. Comp. 90 (2021), 1089-1118
MSC (2020): Primary 65M12, 65L80, 65M60, 76S05
DOI: https://doi.org/10.1090/mcom/3608
Published electronically: March 3, 2021
MathSciNet review: 4232218
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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.


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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

Keywords: Elliptic-parabolic problem, semi-explicit time discretization, delay, poroelasticity, multiple-network
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.
Article copyright: © Copyright 2021 American Mathematical Society