Convergence and error analysis for a class of splitting schemes in incompressible fluid-structure interaction

Miguel Angel Fernández 1, * Jimmy Mullaert 1
* Corresponding author
1 REO - Numerical simulation of biological flows
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt, UPMC - Université Pierre et Marie Curie - Paris 6
Abstract : This paper is devoted to the convergence analysis of the generalized Robin-Neumann schemes introduced in [Internat. J. Numer. Methods Engrg., 101(3):199–229, 2015] for the coupling of a viscous incompressible fluid with a thick-walled elastic or viscoelastic structure. To this purpose, a representative linearized setting is considered. The methods are formulated within a class of operator splitting schemes which treat implicitly the coupling between the fluid and the solid inertia contributions. This guarantees energy stability. A priori error estimates are derived for all the explicit and semi-implicit variants. The analysis predicts a non-uniformity in space of the splitting error, hence confirming the numerical evidence of [Internat. J. Numer. Methods Engrg., 101(3):199–229, 2015] for the explicit variants. Besides, the analysis demonstrates that the genesis of this accuracy loss is the spatial non-uniformity of the discrete elastic or viscoelastic solid operator. The theoretical findings are illustrated via a numerical study which shows, in particular, that alternative splitting schemes recently reported in the literature also suffer from these accuracy issues.
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Miguel Angel Fernández, Jimmy Mullaert. Convergence and error analysis for a class of splitting schemes in incompressible fluid-structure interaction. IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2016, 36 (4), pp.1748-1782. ⟨10.1093/imanum/drv055⟩. ⟨hal-01102975v3⟩

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