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We analyse three time integration schemes for unfitted methods in fluidstructure interaction. In Alghorithm 1 we propose a fully discrete monolithicalgorithm with P1 P1 stabilized finite elements for the fluid problem; for thisalghorithm we prove well-posedness, unconditional stability and convergence inthe case of linearized problem (see Propositions 2.4.2, 2.4.3 and Theorem3.3.7, respectively). The analysis optimal convergence rates as expected fromthe Euler scheme, and the supposed regularity of the solution to the continuousproblem. Moreover we introduce two algorithms that allow for a partitioning ofthe coupled problem by exploiting an explicit-implicit treatment of thetransmission conditions. Algorithm 2 represents, essentially, a simplificationof Algorithm 1 since it simply treat the solid elastic forces in explicit formusing the displacement and velocities of the structure evaluated in theprevious time steps. Instead, Algorithm 3, is really a splitting algorithm thatinvolves the solution of two staggered problems. It splits the forces thatsolid transfers to fluid in two contributions: the inertial contribution thatis treated in implicit form and the elastic contribution that is treated inexplicit form. We perform the stability analysis for both the schemes inTheorems 4.3.1 and 4.3.3. Algorithm 2 results conditionally stable for all theextrapolations considered, instead Algorithm 3 is unconditionally stable, forextrapolations of order zero and one, and conditionally stable for theextrapolation of order two. Since Algorithm 3 is the most promising, we performthe convergence analysis in the linearized case (see Theorem 4.4.2) obtainingresults in line with those of the monolithic case, in particular the splittingintroduced preserves optimal conevegence rates.

Time integration schemes for fluid-structure interaction problems: non fitted FEMs for immersed thin structures