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% TITLE: replace text with your abstract title WITHOUT full stop
\title{On monolithic and Chorin-Temam schemes for incompressible flows in moving domains.}
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% 2- Multiple authors with multiple affiliations. Complete as follows: author[i] <-> affiliation[i]  
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%\author[1]{Jerem\'ias Garay}
%\author[2]{Second B. Author}
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% \author[1]{Jerem\'ias Garay}
\author[1]{Reidmen Ar\'ostica}
% \author[1]{David Nolte}
\author[1]{Crist\'obal Bertoglio}

\affil[1]{{Bernoulli Institute}, {University of Groningen},
{Groningen}, The Netherlands}
%\affil[cmm]{{Center for Mathematical Modeling}, {Universidad de Chile}, {Santiago}, Chile}
%\affil[tub]{{Department of Fluid Dynamics}, {Technische Universit\"at Berlin}, {Berlin}, Germany}

%\affil[2]{{Bernoulli Institute}, {University of Groningen},
%{Groningen}, The Netherlands}

%\affil[2]{{Biomedical Imaging Center}, {Pontificia Universidad Cat\'olica de Chile},
%{Santiago}, Chile}


%\affil[3]{{School of Biomedical Engineering}, {Universidad  de Valparaiso},
%{Valparaiso}, Chile}
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%\affil[4]{{Department of Mathematical Engineering}, {Universidad de Concepci\'on},
%{Concepci\'on}, Chile}
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%\affil[5]{{Department of Mechanical Engineering}, {Universidad T\'ecnica Federico Santa Mar\'ia},
%{Santiago}, Chile}
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%\affil[6]{{Joint last authors}, {in alphabetical order}}




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\summary{Several time discretized domain for the incompressible Navier-Stokes equations (iNSE) in moving domains have been proposed in literature. Here, we introduce a unified formulation that combines different approaches found in literature, allowing a common well posedness and time stability analysis. It can be therefore shown that only a particular choice of numerical schemes ensure such properties under some restrictions. The analysis will be shown for Chorin-Temam schemes using the insight found in the monolithic case. Results are supported from numerical simulations and its usage in fluid-solid interaction problems will be presented.}

% KEYWORDS: replace text with 2-4 keywords, not capitalised, separated by comma, and without a full stop at the end.
\keywords{numerical schemes, stability analysis, incompressible flows, fluid-structure interaction}


\begin{document}
Estimating parameters from heart contraction data, using e.g. magnetic resonance imaging \cite{imperiale2021tagged, markl20124d, marchesseau2013fast, marchesseau2013regionalvols}, requires several techniques such as the estimation algorithms, e.g. \cite{pfaller2020pod}, fluid-solid coupling schemes modeling the physics itself e.g. \cite{astorino-chouly-fernandez-09, bertoglio2013sisc} and in particular, the discretization of fluid problem arising from the blood movement e.g. \cite{bevan2016comparative}.

In such a context, several works have been reported dealing with numerical solutions of the iNSE in moving domains within the Arbitrary Lagrangian Eulerian formulation (ALE). Different choices of time discretization have been reported , e.g. \cite{Basting2017}, \cite{Hessenthaler2017}. To the best of the authors knowledge, only a few monolithic schemes have been throughly analyzed, e.g. \cite{Lozovskiy2018, smaldone2014, le-tallec-mouro-01, Burtschell2017} while no analysis has been reported for Chorin-Temam (CT) methods, being an alternative option when requirements such a low time computations are needed, specially for future industrial applications.

The goal of this talk is to present the finding of well-posedness and unconditional energy balance of the iNSE-ALE for several reported CT discretization schemes within a single formulation, published in \cite{arostica2021monolithic}. It will be supplemented with an usage case for fluid-solid interaction problems.


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