Pith. sign in

REVIEW 3 major objections 5 minor 78 references

Convergence of a CutFEM for fluid--structure interaction with a deforming interface

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A rigorous error bound is proved for a cut finite element method applied to fully coupled fluid–structure interaction with a moving interface.

desk verdict First convergence proof for unfitted FSI with a deforming interface; serious and likely correct, but the central Ritz projection lemma is sketched and the claimed rate is not yet fully supported. read the letter →

arxiv 2608.08140 v1 pith:QLVTC5ZC submitted 2026-08-08 math.NA cs.NA

Miguel A. Fernández , Buyang Li , Maxim A. Olshanskii This is my paper · ORCID
classification math.NAcs.NA MSC 65M6065M1574F1076D05
keywords CutFEMunfittedfiniteelementmethodfluid–structureinteractionmovinginterfaceapriorierroranalysisLagrangemultiplierghostpenaltyNavier–Stokes
topics Navier-Stokes Regularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves, for the first time, a rigorous a priori error estimate for an unfitted (cut) finite element method applied to a fully coupled fluid–structure interaction problem in which the interface between the fluid and the solid is part of the unknown solution. The method couples the incompressible Navier–Stokes equations to a linearly elastic solid through a Lagrange multiplier that enforces the kinematic and traction conditions on the moving interface. Under smoothness assumptions on the exact solution, the paper shows that with finite elements of degree $k\geq 3$, the fluid velocity and solid displacement converge at order $k$ in their natural energy norms, while the pressure converges at order $k-1$. The significance is that it closes the two-way feedback loop between the error in the interface position and the error in the fluid solution, which is the main obstacle in analyzing unfitted methods for deforming interfaces.

What carries the argument

The central technical object is the interpolated configuration $X^*_H = I + d^*_H$, the finite-element interpolant of the exact solid motion, and the families of intermediate interfaces $\Gamma[X^\theta_H]$, $\Gamma[X^{*,\theta}_H]$, and $\Gamma[X^{\#,\theta}_H]$ that connect the exact, interpolated, and numerical interfaces. Transport formulas with respect to the auxiliary parameter $\theta$ turn geometric perturbations into integrals involving the displacement error $e_d = d_H - d^*_H$, which is the key to quantifying two-way coupling. Equally important is the projected kinematic mismatch $P_{\Gamma[X_H]} e_u - \partial_t e_d \circ X_H^{-1}$ on the numerical interface, where $P_{\Gamma[X_H]}$ is the $L^2$ projection onto the interface multiplier space; estimating this object transfers control between the fluid velocity error and the solid velocity error. Uniform ghost-penalty, trace, inverse, and inf-sup estimates on the moving cut domains supply the mesh-robustness needed throughout, and a continuation-in-time argument converts the required Lipschitz bounds on the numerical deformation into consequences of the energy estimates themselves.

What would settle it

Run the Section 10 radially oscillating annulus test with $k=2$ on a sequence of refined meshes: if the $H^1$ velocity error decays strictly slower than $h^2$, the theorem's conclusion (or its conjectured extension to $k=2$) would be false. More generally, construct an FSI solution whose fluid domain develops a re-entrant corner at the interface at some time, so the uniform Stokes regularity in Assumption (C1) fails; the bound (3.14) should then break, showing the smoothness hypothesis is load-bearing.

Watch

Extended reading notes

Core claim

Theorem 3.1 is the paper's central claim. For a semidiscrete CutFEM with ghost-penalty stabilization, a Lagrange multiplier on the reference interface, and finite elements of degree $k\geq 3$, it establishes the error bound $$\max_{t\in[0,T]}\big(\|u_h-u\|^2_{$L^{2}$(\Omega_f[X_H])}+\|d_H-d\circ\Phi_H\|^2_{$H^{1}$(\hat\$\Omega$^s_H)}+\|\partial_t d_H-\partial_t d\circ\Phi_H\|^2_{$L^{2}$(\hat\$\Omega$^s_H)}\big)+\int_0^T\|u_h-u\|^2_{$H^{1}$(\Omega_f[X_H])}\,dt+$H^{2}$\int_0^T\big(\|p_h-p\|^2_{$L^{2}$(\Omega_f[X_H])}+\|\hat\sigma_H-\hat\$\sigma$\circ\Phi_H\|^2_{$H^{{-1/2}}$(\hat\Gamma_H)}\big)\,dt\le C($h^{{2k}}$+$H^{{2k}}$),$$ provided the exact solution is smooth and the moving-domain Stokes problem has uniform $H^2\times H^1$ regularity. The proof treats the fluid equations on the numerically deformed domain, so the discrete interface, its normal, and the pulled-back traction all depend on the solid displacement error; conversely, the solid error is driven by the fluid traction and kinematic mismatch. The authors close this loop by introducing an interpolated configuration, establishing uniform norm equivalences and trace/inverse estimates on the moving cut domains, and using a continuation-in-time argument to upgrade a priori smallness assumptions into unconditional convergence on the whole smoothness interval.

Load-bearing premise

The proof requires the exact fluid–structure solution to be smooth on the entire time interval and the moving-domain Stokes problem at each time to have uniform $H^2\times H^1$ regularity; for the full incompressible Navier–Stokes/elasticity system, such global regularity is not known and is only expected for short times and restricted data.

Editorial extensions

If this is right

  • For any FSI solution satisfying the smoothness assumptions and for polynomial degree $k\ge 3$, the semidiscrete CutFEM converges at the optimal rate $O(h^k+H^k)$ in the energy norms, and this rate is achieved uniformly on the whole time interval $[0,T]$.
  • The error constant is independent of the stabilization parameter $\epsilon_h\in(0,h]$, so the analysis covers the limit $\epsilon_h\to 0$ and hence the pure ghost-penalty formulation without time-derivative stabilization.
  • The theorem is the first rigorous convergence result for an unfitted FEM with a genuinely deforming interface; it opens the door to analyzing other immersed FSI formulations, including Nitsche-type and distributed-Lagrange-multiplier variants, by the same interpolated-configuration strategy.
  • The restriction $k\ge 3$ is technical: the numerical experiments in Section 10 show clean second-order convergence for $k=2$, suggesting the theorem should extend once a direct $W^{1,p}$ error estimate for the cut-domain Ritz projection is available.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The interpolated-configuration and transport-formula machinery is likely to transfer to other moving-interface coupled problems—two-phase flow, free-boundary problems, or FSI with hyperelastic solids—where the same type of geometry-error feedback appears, provided the corresponding regularity assumptions hold.
  • Because the theorem is conditional on global smoothness of the FSI solution, and such smoothness is only known locally in time for the Navier–Stokes–elasticity system, the practical reach of the result is a short-time error bound; extending it to long-time or nonsmooth regimes would require either global regularity results or a different (e.g., weak-solution) error framework.
  • The observed $(k+1)$-th order convergence in $L^\infty(L^2)$-type norms in the numerical tests is not covered by the analysis; a sharper duality argument might recover these rates and would be a natural next step.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a semidiscrete immersed CutFEM for a fully coupled fluid--structure interaction problem with a deforming interface, using a Lagrange multiplier to enforce the kinematic coupling and a global ghost-penalty stabilization. The main result, Theorem 3.1, claims an a priori error bound of order k for the fluid velocity and solid displacement and order k-1 for the pressure, for finite elements of degree k>=3, under smoothness assumptions on the exact solution and a mesh proportionality condition. The proof is organized around an interpolated exact configuration, geometric transport identities, consistency estimates, and a continuation argument. The paper also reports numerical experiments with k=2 and k=3.

Significance. If the stated convergence result is correct, this is a significant contribution: it would be the first rigorous error analysis of an unfitted finite element method for a fully coupled FSI system with a genuinely deforming interface. The paper introduces a systematic framework for handling the coupling between interface geometry error and fluid/solid errors, and it explicitly formulates the required uniform geometric, trace, inverse, and inf-sup estimates on moving cut domains. The claimed result is concrete and falsifiable, and the numerical experiments, although not covered by the theorem for k=2, are consistent with the stated rates. However, the proof contains load-bearing gaps that, as written, prevent the claimed rates from being fully established.

major comments (3)
  1. [Section 9, Lemma 9.1, Eqs. (9.2a)-(9.2b)] The Ritz projection estimates (9.2a) and (9.2b) are central to the proof: they are used in Lemma 9.2, in the estimates of (9.15a), (9.22), (9.39)-(9.40), and in the construction of the extension w in Part 8. The proof of Lemma 9.1, however, states 'The details are omitted here' for both estimates. Since this is a cut-domain Ritz projection on a moving interface with ghost penalties, these estimates are not routine off-the-shelf results, and the time-derivative bound (9.2b) only gives h^{k-1} in the H^1-type norm. A loss of one order in h at this stage would propagate through (9.30)-(9.36) and (8.132) and degrade the final bound in Theorem 3.1 from O(h^{2k}) to O(h^{2k-2}). The authors must supply the full proof of (9.2a)-(9.2b) or otherwise justify that no order loss occurs.
  2. [Section 8.5 and Section 8.6, Eqs. (8.94), (8.95), (8.121), (8.132)] The intermediate energy estimates contain lower-order terms that appear incompatible with the claimed O(h^{2k}) rate. For example, (8.94)-(8.95) include C(h^{2k-2}+H^{-1}h^{2k}+h^{-1}H^{2k}+H^{2k-1}), which with h~H is dominated by h^{2k-2}. This lower-order term is carried into the estimate of (4.14h) in (8.121), which introduces h^{2k-3} coefficients. The final estimate (8.132), however, states C(h^{2k}+H^{2k}) without explaining how these lower-order terms are absorbed or cancelled. Since h^{2k-2} > h^{2k} for small h, the written estimates do not establish the convergence rate claimed in Theorem 3.1. The authors should either revise the estimates to obtain the optimal rate or state the weaker rate that the current estimates actually yield.
  3. [Section 5 and Section 8.6, Eqs. (5.2)-(5.9), (8.132)-(8.134)] The continuation argument in Section 5 recovers the smallness assumptions (5.2) from the 'final' estimates (8.133)-(8.134). If, as noted above, the actual estimates only give O(h^{2k-2}) for the H^1-type errors, then the inverse-inequality step in (5.5) would need to be revisited. For k=3, the L-infinity bound on e_d would be O(h^{3/2}) rather than O(h^{5/2}), which is still small enough for (5.2a), but the W^{1,∞} bound in (5.6) would be O(h^{1/2}) in three dimensions instead of O(h^{3/2}); this would violate the required O(H^{1/4}) bound only for very small h, so the continuation may still close, but the stated rates in Theorem 3.1 would not follow. The proof needs to make the dependence of the constants on the final rate explicit.
minor comments (5)
  1. [Title] The title contains a typo: 'INTERF ACE' should be 'INTERFACE'.
  2. [Section 3, Remark 3.4] The passage to the limit epsilon_h -> 0 is deferred with 'the details are omitted'; since this is not central to the main convergence claim, it is acceptable, but the omission should be noted more explicitly in the text.
  3. [Section 4, Eq. (4.8)] The numbering (4.8a)-(4.8l) is dense and makes the consistency remainder difficult to follow; a table or a more descriptive decomposition would improve readability.
  4. [Section 10] The numerical experiment uses E_sigma^{2,0} in the L^2(bGamma_H) norm rather than the H^{-1/2} norm of Theorem 3.1; the text acknowledges this, but the reader should be reminded that the displayed rates for the traction are not directly comparable to the theorem.
  5. [Appendix A] The local well-posedness argument relies on the positivity of epsilon_h to turn the system into ODEs; the role of epsilon_h in the time-derivative ghost penalty should be stated even more explicitly in the main text, as it is a nonstandard term.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence theorem is derived from the continuous FSI equations and stated regularity assumptions, with self-citations used only for standard technical tools.

full rationale

Theorem 3.1 is derived from the continuous FSI equations (2.2)-(2.4) and the explicit assumptions (C1)-(C4); the error bound (3.14) is a statement about the differences u_h-u, d_H-d∘Φ_H, p_h-p, and σ̂_H-σ̂∘Φ_H, none of which is fitted or defined in terms of the claimed rate. The discrete initial data are chosen by interpolation and by the Ritz projection (3.8), and the consistency remainder R*_{h,H} is explicitly defined in (4.8) and then estimated independently in Lemmas 7.1 and 9.2. The continuation argument in Section 5 assumes the smallness bounds (5.2) only as a bootstrap hypothesis, proves (8.133)-(8.134) under that hypothesis, and then recovers (5.2) via finite-element inverse estimates; this is a standard nonlinear continuation argument, not a circular reduction. The paper cites prior work for technical tools such as trace and inverse inequalities on cut domains, uniform inf-sup conditions, transport formulas, and Lp stability of interface L2 projections; those tools do not encode the target convergence rate and are not the load-bearing content of the convergence claim. Lemma 9.1 does contain proof sketches with omitted details, most notably for (9.2a) and (9.2b), but this is an incompleteness in the written proof rather than circularity: the estimates are statements about the cut-domain Ritz projection under the stated H2×H1 Stokes regularity assumption (C1), and they do not assume the error bound of Theorem 3.1. The numerical experiments use an external manufactured solution from [73] and are not used to define or prove the theorem. No equation in the paper is shown to reduce to its own input by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard a priori regularity assumptions for the continuous FSI solution, on a moving-domain Stokes regularity assumption, on a comparability condition between fluid and solid mesh sizes, and on a continuation hypothesis that is later recovered from the energy estimates. No parameters are fitted to data and no new physical entities are introduced.

assumptions (6)
  • domain assumption The moving-domain Stokes problem (3.11) satisfies H2 x H1 regularity with a uniform constant for every t in [0,T].
    Assumption C1. Used in the duality argument for the Ritz projection estimates in Lemma 9.1 and in the inf-sup discussion of Appendix H. Not proven for arbitrary moving interfaces.
  • domain assumption The exact solid displacement is smooth, the map X(t)=I+d(t) is an orientation-preserving diffeomorphism, and the inverse deformation gradient is uniformly bounded.
    Assumption C2. Needed for norm equivalences between moving interfaces, for the validity of the interpolated configuration, and for the continuation argument.
  • domain assumption The exact fluid solution (u,p) is smooth in the space-time fluid domain and admits a smooth extension to the whole background domain.
    Assumption C3. Used throughout for approximation properties of the Lagrange interpolants and for consistency estimates. Global-in-time smoothness of FSI solutions is not known in general.
  • domain assumption The fluid and solid mesh sizes satisfy c1 H <= h <= c2 H, with sufficiently small h0 and H0.
    Assumption C4. Required for inverse estimates, trace inequalities, ghost-penalty extension estimates, and closing the continuation argument with finite element inverse inequalities.
  • ad hoc to paper The continuation hypothesis (5.2) holds: on [0,t*], the discrete errors satisfy small W1,infty and H1 bounds, and eu is small in L infinity on the extended fluid region.
    The proof first assumes these bounds, then proves energy estimates, and finally uses inverse inequalities to show the bounds persist past t*. This is a standard continuation argument but relies on the smallness conditions being satisfiable for k>=3 and sufficiently small meshes.
  • ad hoc to paper The finite element degree satisfies k>=3.
    Remark 3.3 states that k>=3 is used only in estimate (9.22), where an inverse estimate controls the W1,4 error between the Lagrange interpolant and the Ritz projection by the H1 error. The numerical tests suggest k=2 also converges, but the proof does not cover it.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Convergence of a CutFEM for fluid--structure interaction with a deforming interface." pith.science (2026). https://pith.science/paper/QLVTC5ZC

@misc{pith2026260808140,
  author       = {Pith},
  title        = {Pith review of: Convergence of a CutFEM for fluid--structure interaction with a deforming interface},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLVTC5ZC}},
  note         = {Machine review of arXiv:2608.08140}
}
abstract

We present a rigorous error analysis for a semidiscrete immersed CutFEM applied to fluid--structure interaction (FSI) problems involving an incompressible viscous fluid and an elastic solid. The method employs a Lagrange multiplier to enforce the fluid-solid coupling conditions while fully accounting for the effect of the structural deformation on the fluid domain. For sufficiently regular FSI solutions and finite elements of degree $k\geq3$, we prove convergence of order $k$ for the fluid velocity and solid displacement and convergence of order $k-1$ for the fluid pressure in the energy norms associated with the FSI problem. To the best of our knowledge, this is the first rigorous convergence analysis of an unfitted finite element method for a fully coupled FSI system with a deforming interface.

Figures

Figures reproduced from arXiv: 2608.08140 by the authors.

Figure 1
Figure 1. Fluid, solid and the solid reference domains, with Ω = Ωf (t) ∪ Ωs(t) [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The computational domain with the background mesh, the em￾bedded solid domain, its triangulation, and the fluid-solid interfaces. spaces for the fluid velocity and pressure, the solid displacement, and the pull-back normal stress (pulled back to the reference interface Γ): b V f h def =  vh ∈ H1 0 (Ω)d : vh|K ∈ Pk(K) d ∀K ∈ Th [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Overlapping meshes of the fluid and solid computational domains. Γ(t) = {x ∈ R 2 : |x| = rΓ(t)}, rΓ(t) def = R1 + ¯d sin(ωt), and the fluid occupies the exterior domain Ω f (t) = {x ∈ Ω : |x| > rΓ(t)}. The interpolated exact velocity is imposed strongly on the outer boundary of the box ∂Ω, while the discrete initial data are defined as in Section 3.3. The expressions of the exact solutions can be found in [73], wher… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Spatial convergence with finite elements of degree k = 2. Left: errors in the natural norms of Theorem 3.1, showing 2nd-order convergence. Right: L∞(0, T;L 2 )-type errors, showing an observed 3rd-order convergence [PITH_FULL_IMAGE:figures/full_fig_p080_4.png]
Figure 5
Figure 5. Figure 5: Spatial convergence with finite elements of degree k = 3. Left: errors in the natural norms of Theorem 3.1, showing 3rd-order convergence. Right: L∞(0, T;L 2 )-type errors, showing an observed 4th-order convergence. The convergence results for k = 2 and k = 3 are shown…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

78 extracted references · 76 canonical work pages

  1. [1]

    C. Ager, B. Schott, A.-T. Vuong, A. Popp, and W. A. Wall. A consistent approach for fluid-structure- contact interaction based on a porous flow model for rough surface contact.Int. J. Num. Meth. Eng., 119(13):1345–1378, 2019

    work page 2019

  2. [2]

    Alauzet, B

    F. Alauzet, B. Fabr` eges, M.A. Fern´ andez, and M. Landajuela. Nitsche-XFEM for the coupling of an incompressible fluid with immersed thin-walled structures.Comput. Methods Appl. Mech. Eng., 301:300– 335, 2016

    work page 2016

  3. [3]

    Annese, M.A

    M. Annese, M.A. Fern´ andez, and L. Gastaldi. Splitting schemes for a Lagrange multiplier formulation of FSI with immersed thin-walled structure: stability and convergence analysis.IMA J. Numer. Anal., 43(2):881–919, 2023

    work page 2023

  4. [4]

    Astorino, J.-F

    M. Astorino, J.-F. Gerbeau, O. Pantz, and K.-F. Traore. Fluid–structure interaction and multi-body contact: application to aortic valves.Comput. Methods Appl. Mech. Eng., 198(45-46):3603–3612, 2009

    work page 2009

  5. [5]

    Astorino and C

    M. Astorino and C. Grandmont. Convergence analysis of a projection semi-implicit coupling scheme for fluid-structure interaction problems.Numer. Math., 116:721–767, 2010

    work page 2010

  6. [6]

    F. P. T. Baaijens. A fictitious domain/mortar element method for fluid–structure interaction.Int. J. Numer. Methods Fluids, 35(7):743–761, 2001

    work page 2001

  7. [7]

    Bazilevs, K

    Y. Bazilevs, K. Takizawa, and T. E. Tezduyar.Computational Fluid–Structure Interaction: Methods and Applications. John Wiley & Sons, Chichester, UK, 2013

    work page 2013

  8. [8]

    Bergmann, A

    M. Bergmann, A. Fondan` eche, and A. Iollo. An Eulerian finite-volume approach of fluid-structure inter- action problems on quadtree meshes.J. Comput. Phys., 471:111647, 2022

    work page 2022

Show all 78 references
  1. [9]

    Bernardi, M

    C. Bernardi, M. Costabel, M. Dauge, and V. Girault. Continuity properties of the inf-sup constant for the divergence.SIAM J. Math. Anal., 48(2):1250–1271, 2016

    2016

  2. [10]

    Boffi, N

    D. Boffi, N. Cavallini, and L. Gastaldi. Finite element approach to immersed boundary method with different fluid and solid densities.Math. Models Methods Appl. Sci., 21(12):2523–2550, 2011

    2011

  3. [11]

    Boffi, N

    D. Boffi, N. Cavallini, and L. Gastaldi. The finite element immersed boundary method with distributed Lagrange multiplier.SIAM J. Numer. Anal., 53(6):2584–2604, 2015

    2015

  4. [12]

    Boffi and L

    D. Boffi and L. Gastaldi. A fictitious domain approach with Lagrange multiplier for fluid-structure inter- actions.Numer. Math., 135(3):711–732, 2017

    2017

  5. [13]

    Boilevin-Kayl, M.A

    L. Boilevin-Kayl, M.A. Fern´ andez, and J.-F. Gerbeau. Numerical methods for immersed FSI with thin- walled structures.Computers&Fluids, 179:744–763, 2019

    2019

  6. [14]

    Bouck, R

    L. Bouck, R. H. Nochetto, M. Shakipov, and V. Yushutin. Inf-sup stability of parabolic TraceFEM.Found. Comput. Math., 2026, https://doi.org/10.1007/s10208-026-09757-7

    2026 doi

  7. [15]

    Bukaˇ c and B

    M. Bukaˇ c and B. Muha. Stability and convergence analysis of the extensions of the kinematically coupled scheme for the fluid-structure interaction.SIAM J. Numer. Anal., 54(5):3032–3061, 2016

    2016

  8. [16]

    Bukaˇ c and S

    M. Bukaˇ c and S. ˇCani´ c. A partitioned numerical scheme for fluid–structure interaction with slip.Math. Model. Nat. Phenom., 16:8, 2021

    2021

  9. [17]

    E. Burman. Ghost penalty.Comptes Rendus Mathematique, 348(21–22):1217 – 1220, 2010

    2010

  10. [18]

    Burman, S

    E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: Discretizing geometry and partial differential equations.Int. J. Numer. Methods Eng., 104(7):472–501, 2015

    2015

  11. [19]

    Burman, S

    E. Burman, S. Claus, and A. Massing. A stabilized cut finite element method for the three field Stokes problem.SIAM J. Sci. Comput., 37(4):A1705–A1726, 2015

    2015

  12. [20]

    Burman, R

    E. Burman, R. Durst, M.A. Fern´ andez, and J. Guzm´ an. Fully discrete loosely coupled Robin-Robin scheme for incompressible fluid–structure interaction: stability and error analysis.Numerische Mathe- matik, 151(4):807–840, 2022

    2022

  13. [21]

    Burman and M.A

    E. Burman and M.A. Fern´ andez. An unfitted Nitsche method for incompressible fluid-structure interaction using overlapping meshes.Comput. Methods Appl. Mech. Eng., 279:497–514, 2014

    2014

  14. [22]

    Burman, M.A

    E. Burman, M.A. Fern´ andez, and F. M. Gerosa. Convergence analysis of an unfitted mesh semi-implicit coupling scheme for incompressible fluid-structure interaction.Vietnam J. Math., 51(1):37–69, 2023. 100 M.A. FERN ´ANDEZ, B. LI, AND M. OLSHANSKII

    2023

  15. [23]

    Burman and P

    E. Burman and P. Hansbo. Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method.Appl. Numer. Math., 62(4):328–341, 2012

    2012

  16. [24]

    Burman and P

    E. Burman and P. Hansbo. Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem.M2AN Math. Model. Numer. Anal., 48:859–874, 2014

    2014

  17. [25]

    Burman, P

    E. Burman, P. Hansbo, M.G. Larson, and S. Zahedi. Cut finite element methods.Acta Numer., 34:1–121, 2025

    2025

  18. [26]

    P. G. Ciarlet.Mathematical Elasticity. Vol. I, volume 20 ofStudies in Mathematics and its Applications. North-Holland, 1988

    1988

  19. [27]

    Cottet, E

    G.-H. Cottet, E. Maitre, and T. Milcent. Eulerian formulation and level set models for incompressible fluid-structure interaction.M2AN Math. Model. Numer. Anal., 42(3):471–492, 2008

    2008

  20. [28]

    Coutand and S

    D. Coutand and S. Shkoller. Motion of an elastic solid inside an incompressible viscous fluid.Arch. Rational Mech. Anal., 176:25–102, 2005

    2005

  21. [29]

    De Hart, G

    J. De Hart, G. W. M. Peters, P. J. G. Schreurs, and F. P. T. Baaijens. A three-dimensional computational analysis of fluid–structure interaction in the aortic valve.J. Biomech., 36(1):103–112, 2003

    2003

  22. [30]

    Donea, S

    J. Donea, S. Giuliani, and J.-P. Halleux. An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions.Comput. Methods Appl. Mech. Eng., 33(1-3):689–723, 1982

    1982

  23. [31]

    N. D. Dos Santos, J.-F. Gerbeau, and J.-F. Bourgat. A partitioned fluid–structure algorithm for elastic thin valves with contact.Comput. Methods Appl. Mech. Eng., 197(19):1750–1761, 2008

    2008

  24. [32]

    Q. Du, M. D. Gunzburger, L. S. Hou, and J. Lee. Semidiscrete finite element approximations of a linear fluid-structure interaction problem.SIAM J. Numer. Anal., 42(1):1–29, 2004

    2004

  25. [33]

    Dziuk and C

    G. Dziuk and C. M. Elliott. Finite element methods for surface PDEs.Acta Numer., 22:289–396, 2013

    2013

  26. [34]

    C. M. Elliott and T. Ranner. A unified theory for continuous-in-time evolving finite element space ap- proximations to partial differential equations in evolving domains.IMA J. Numer. Anal., 41:1696–1845, 2021

    2021

  27. [35]

    Fern´ andez

    M.A. Fern´ andez. Incremental displacement-correction schemes for incompressible fluid-structure interac- tion: stability and convergence analysis.Numer. Math., 123(1):21–65, 2013

    2013

  28. [36]

    Fern´ andez and J

    M.A. Fern´ andez and J. Mullaert. Convergence and error analysis for a class of splitting schemes in incompressible fluid-structure interaction.IMA J. Numer. Anal., 36(4):1748–1782, 2016

    2016

  29. [37]

    G. Gao, B. Li, and T. Tang. Convergent finite element approximations of surface evolution with relaxed minimal deformation.Numer. Math., 158:671–714, 2026

    2026

  30. [38]

    Gonz´ alez and V

    M. Gonz´ alez and V. Selgas. Analysis of a velocity–stress–pressure formulation for a fluid–structure inter- action problem.Appl. Numer. Math., 123:275–299, 2018

    2018

  31. [39]

    Grisvard.Elliptic Problems in Nonsmooth Domains

    P. Grisvard.Elliptic Problems in Nonsmooth Domains. Classics in Applied Mathematics. SIAM, 2011

    2011

  32. [40]

    Guzm´ an and M

    J. Guzm´ an and M. Olshanskii. Inf-sup stability of geometrically unfitted Stokes finite elements.Math. Comp., 87(313):2091–2112, 2018

    2018

  33. [41]

    Hansbo and P

    A. Hansbo and P. Hansbo. An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems.Comput. Methods Appl. Mech. Eng., 191(47–48):5537 – 5552, 2002

    2002

  34. [42]

    Hansbo and P

    A. Hansbo and P. Hansbo. A finite element method for the simulation of strong and weak discontinuities in solid mechanics.Comput. Methods Appl. Mech. Eng., 193(33-35):3523–3540, 2004

    2004

  35. [43]

    Kamensky, M.-C

    D. Kamensky, M.-C. Hsu, D. Schillinger, J. Evans, A. Aggarwal, Y. Bazilevs, M. S. Sacks, and T. J. R. Hughes. An immersogeometric variational framework for fluid–structure interaction: Application to bio- prosthetic heart valves.Comput. Methods Appl. Mech. Eng., 284:1005–1053, 2015

    2015

  36. [44]

    Kov´ acs, B

    B. Kov´ acs, B. Li, C. Lubich, and C. A. Power Guerra. Convergence of finite elements on an evolving surface driven by diffusion on the surface.Numer. Math., 137(3):643–689, 2017

    2017

  37. [45]

    Landajuela, M

    M. Landajuela, M. Vidrascu, D. Chapelle, and M.A. Fern´ andez. Coupling schemes for the FSI forward pre- diction challenge: comparative study and validation.Int. J. Numer. Methods Biomed. Eng., 33(4):e2813, 2017

    2017

  38. [46]

    Le Tallec and S

    P. Le Tallec and S. Mani. Numerical analysis of a linearised fluid-structure interaction problem.Numer. Math., 87(2):317–354, 2000

    2000

  39. [47]

    Le Tallec and J

    P. Le Tallec and J. Mouro. Fluid structure interaction with large structural displacements.Comput. Methods Appl. Mech. Eng., 190:3039–3067, 2001

    2001

  40. [48]

    Legendre and T

    G. Legendre and T. Takahashi. Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation.M2AN Math. Model. Numer. Anal., 42(4):609–644, 2008

    2008

  41. [49]

    Lehrenfeld and M

    C. Lehrenfeld and M. Olshanskii. An Eulerian finite element method for PDEs in time-dependent domains. M2AN Math. Model. Numer. Anal., 53(2):585–614, 2019. 101

    2019

  42. [50]

    M. Lenoir. Optimal isoparametric finite elements and error estimates for domains involving curved bound- aries.SIAM J. Numer. Anal., 23(3):562–580, 1986

    1986

  43. [51]

    B. Li, Q. Rao, and P. Sun. Convergence of ALE interface tracking methods for fluid–structure interaction with a moving interface. In review, 2025

    2025

  44. [52]

    B. Li, W. Sun, Y. Xie, and W. Yu. OptimalL 2 error analysis of a loosely coupled finite element scheme for thin-structure interactions.SIAM J. Numer. Anal., 62(4):1782–1813, 2024

    2024

  45. [53]

    Li and T

    B. Li and T. Tang. Dynamic Ritz projection of mean curvature flow and optimalL 2 convergence of parametric FEM.SIAM J. Numer. Anal., 63:1454–1481, 2025

    2025

  46. [54]

    Lidstr¨ om

    P. Lidstr¨ om. Moving regions in Euclidean space and Reynolds’ transport theorem.Math. Mech. Solids, 16(4):366–380, 2011

    2011

  47. [55]

    Massing, M

    A. Massing, M. G. Larson, A. Logg, and M. E. Rognes. A stabilized Nitsche fictitious domain method for the Stokes problem.J. Sci. Comput., 61(3):604–628, 2014

    2014

  48. [56]

    Massing, B

    A. Massing, B. Schott, and W. A. Wall. A stabilized Nitsche cut finite element method for the Oseen problem.Comput. Methods Appl. Mech. Eng., 328:262–300, 2018

    2018

  49. [57]

    U. M. Mayer, A. Gerstenberger, and W. A. Wall. Interface handling for three-dimensional higher-order XFEM-computations in fluid–structure interaction.Int. J. Num. Meth. Eng., 79(7):846–869, 2009

    2009

  50. [58]

    U. M. Mayer, A. Popp, A. Gerstenberger, and W. A. Wall. 3D fluid–structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach.Comput. Mech., 46(1):53–67, 2010

    2010

  51. [59]

    Mittal and G

    R. Mittal and G. Iaccarino. Immersed boundary methods.Annu. Rev. Fluid Mech., 37(1):239–261, 2005

    2005

  52. [60]

    U. K. M¨ uller, A. Wasim, E. Fontaine, O. Berg, Y. Cao, D. Lentink, S. Kranenbarg, and J. L. van Leeuwen. Fish and Flag – Exploring Fluid-Structure Interaction during Undulatory Swimming in Fish. In C. T. Lim and J. C. H. Goh, editors,6th World Congress of Biomechanics (WCB 20...

    2010

  53. [61]

    Neilan and M

    M. Neilan and M. Olshanskii. An Eulerian finite element method for the linearized Navier-Stokes problem in an evolving domain.IMA J. Numer. Anal., 44(6):3234–3258, 2024

    2024

  54. [62]

    Nomura and T

    T. Nomura and T. J. R. Hughes. An arbitrary Lagrangian-Eulerian finite element method for interaction of fluid and rigid body.Comput. Methods Appl. Mech. Eng., 95(1):115–138, 1992

    1992

  55. [63]

    Olshanskii and H

    M. Olshanskii and H. von Wahl. Stability of instantaneous pressures in an Eulerian finite element method for moving boundary flow problems.J. Comput. Phys., 533:113983, 2025

    2025

  56. [64]

    C. S. Peskin. The immersed boundary method.Acta Numer., 11:479–517, 2002

    2002

  57. [65]

    T. Richter. A fully Eulerian formulation for fluid-structure-interaction problems.J. Comput. Phys., 233:227–240, 2013

    2013

  58. [66]

    Sawada and A

    T. Sawada and A. Tezuka. LLM and X-FEM based interface modeling of fluid-thin structure interactions on a non-interface-fitted mesh.Comput. Mech., 48(3):319–332, 2011

    2011

  59. [67]

    Singh, D

    P. Singh, D. D. Joseph, T. I. Hesla, R. Glowinski, and T.-W. Pan. A distributed Lagrange mul- tiplier/fictitious domain method for viscoelastic particulate flows.J. Non-Newtonian Fluid Mech., 91(2):165–188, 2000

    2000

  60. [68]

    Stein, T

    K. Stein, T. E. Tezduyar, and R. Benney. Mesh moving techniques for fluid-structure interactions with large displacements.J. Appl. Mech., 70(1):58–63, 2003

    2003

  61. [69]

    Takizawa, Y

    K. Takizawa, Y. Bazilevs, and T. E. Tezduyar. Space–time and ALE-VMS techniques for patient-specific cardiovascular fluid–structure interaction modeling.Arch. Comput. Methods Eng., 19(2):171–225, 2012

    2012

  62. [70]

    Takizawa and T

    K. Takizawa and T. E. Tezduyar. Computational methods for parachute fluid-structure interactions.Arch. Comput. Methods Eng., 19:125–169, 2012

    2012

  63. [71]

    F.-B. Tian, H. Dai, H. Luo, J. F. Doyle, and B. Rousseau. Fluid-structure interaction involving large deformations: 3D simulations and applications to biological systems.J. Comput. Phys., 258:451–469, 2014

    2014

  64. [72]

    von Wahl and T

    H. von Wahl and T. Richter. Error analysis for a parabolic PDE model problem on a coupled moving domain in a fully Eulerian framework.SIAM J. Numer. Anal., 61(1):286–314, 2023

    2023

  65. [73]

    Wan and O

    Y. Wan and O. Pironneau. A class of fluid-structure interaction problems with analytical solutions for the validation of numerical methods.J. Math. Fluid Mech., 28(1):15, 2026

    2026

  66. [74]

    Wang and L

    X. Wang and L. T. Zhang. Modified immersed finite element method for fully-coupled fluid–structure interactions.Comput. Methods Appl. Mech. Eng., 267:150–169, 2013

    2013

  67. [75]

    Zhang, A

    L. Zhang, A. Gerstenberger, X. Wang, and W. Liu. Immersed finite element method.Comput. Methods Appl. Mech. Eng., 193:2051–2067, 2004

    2004

  68. [76]

    Zilian and A

    A. Zilian and A. Legay. The enriched space-time finite element method (EST) for simultaneous solution of fluid-structure interaction.Int. J. Numer. Methods Eng., 75(3):305–334, 2008. 102 M.A. FERN ´ANDEZ, B. LI, AND M. OLSHANSKII

    2008

  69. [77]

    Zonca, C

    S. Zonca, C. Vergara, and L. Formaggia. An unfitted formulation for the interaction of an incompressible fluid with a thick structure via an XFEM/DG approach.SIAM J. Sci. Comput., 40(1):B59–B84, 2018

    2018

  70. [78]

    P. Zunino. Analysis of backward Euler/extended finite element discretization of parabolic problems with moving interfaces.Comput. Methods Appl. Mech. Eng., 258:152–165, 2013. M.A. Fern´andez: Sorbonne Universit ´e & CNRS, UMR 7598 LJLL – Inria, France Email address:miguel.fern...

    2013

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.