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

This paper presents the first rigorous error analysis for a semidiscrete immersed CutFEM applied to fully coupled fluid-structure interaction problems with a deforming interface, proving optimal convergence rates for fluid velocity and solid displacement, as well as near-optimal rates for fluid pressure, under sufficient solution regularity and finite element degrees of at least three.

Original authors: Miguel A. Fernández, Buyang Li, Maxim A. Olshanskii

Published 2026-08-11

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CC BY 4.0

Original authors: Miguel A. Fernández, Buyang Li, Maxim A. Olshanskii

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to predict how a jellyfish swims through the ocean or how a heart valve flutters with every beat. These are problems of Fluid-Structure Interaction (FSI), a branch of science where two very different worlds collide: the fluid (like water or blood) and the solid (like a fish's fin or a heart valve). The fluid is messy, flowing, and constantly changing shape, while the solid is rigid or elastic, trying to hold its form. The tricky part is that they are glued together at a moving boundary. As the solid moves, it pushes the fluid; as the fluid pushes back, the solid deforms. It's a chaotic dance where the stage itself is constantly being rebuilt.

To solve these puzzles on a computer, scientists usually build a digital grid, like a mesh of tiny Lego bricks, to represent the space. In the old days, if the solid moved, the Lego bricks had to stretch and squish to fit the new shape perfectly. This worked fine for small movements, but if the solid twisted or turned too much, the Lego bricks would get tangled, break, or turn into useless, distorted shapes, causing the whole simulation to crash. To fix this, researchers developed a clever trick called CutFEM (or "Cut Finite Element Method"). Instead of forcing the Lego bricks to fit the moving object, they let the object float freely through a fixed, rigid grid. When the object cuts through the bricks, the computer simply "cuts" the bricks to match the object's shape and solves the math on the remaining pieces. It's like having a fixed cookie cutter and a rolling pin of dough; you don't reshape the cutter; you just cut the dough to fit it.

However, just because a method looks like it works doesn't mean it's mathematically perfect. In the world of high-stakes engineering and medicine, you need to know exactly how close your computer guess is to the real truth. This is where the paper by Miguel A. Fernández, Buyang Li, and Maxim Olshanskii comes in. They didn't just run a simulation; they performed a rigorous mathematical proof to show that this "cutting" method actually converges to the right answer as you make the Lego bricks smaller.

The authors tackled a specific, difficult version of the problem: a fully coupled system where an incompressible (un-squishable) fluid interacts with an elastic solid that deforms significantly. They used a "Lagrange multiplier," which you can think of as a digital invisible glue, to ensure the fluid and solid stick together perfectly at the interface without slipping. Their main finding is a rigorous error analysis that proves the method works. Specifically, they showed that if you use finite elements of degree k ≥ 3 (which are like using more complex, curved Lego pieces rather than simple flat ones), the error in the fluid's speed and the solid's movement shrinks at a rate of k as the mesh gets finer. For the pressure (the force pushing on the fluid), the error shrinks at a rate of k − 1.

Crucially, the paper rules out the idea that this method is only an approximation that might fail for large deformations. The authors proved that, provided the solution is smooth enough and the mesh is fine enough, the method is stable and converges on the entire time interval. They didn't just suggest this with computer simulations; they derived a mathematical theorem that guarantees the error bounds hold true. This is a significant step forward because, prior to this work, there was no rigorous proof that this specific type of "unfitted" method (where the mesh doesn't follow the object) could handle a fully coupled, deforming interface system with such precision. It confirms that the "cutting" strategy isn't just a handy hack, but a mathematically sound way to simulate complex, moving interactions in the real world.

Technical Summary: Convergence of a CutFEM for Fluid–Structure Interaction with a Deforming Interface

Problem Setting
This work addresses the numerical approximation of Fluid–Structure Interaction (FSI) problems involving an incompressible viscous fluid and a linearly elastic solid. The system is characterized by a moving interface where the fluid domain is defined by the deformation of the solid. The governing equations consist of the incompressible Navier–Stokes equations in the fluid domain (Eulerian description) and linear elasticity equations in the solid domain (Lagrangian description). The coupling is enforced through kinematic continuity (velocity matching) and dynamic balance of stresses on the evolving interface.

The primary challenge in numerically solving such problems, particularly with large deformations, is the movement of the interface relative to the computational mesh. Traditional fitted mesh methods (e.g., Arbitrary Lagrangian–Eulerian or ALE) suffer from mesh degradation under large rotations or deformations. Unfitted methods, such as Immersed Boundary or Fictitious Domain methods, avoid mesh regeneration but often struggle with accuracy near the interface due to the lack of resolution of discontinuities or boundary conditions.

Methodology
The authors propose and analyze a semi-discrete immersed CutFEM (Cut Finite Element Method) for this fully coupled FSI system. The method employs the following key components:

Unfitted Mesh: The fluid is discretized on a fixed background mesh that does not conform to the solid boundary. The solid is discretized using an isoparametric fitted mesh on its reference domain.
Lagrange Multiplier Formulation: The fluid–solid coupling conditions are enforced weakly using a Lagrange multiplier. This multiplier represents the fluid traction pulled back to the reference solid interface. The kinematic condition ( $u = \partial_t d \circ X^{-1}$ ) is enforced on the current interface, while the traction balance is formulated via an integral relation involving the reference configuration.
Ghost Penalty Stabilization: To ensure stability and robustness against arbitrarily small cut elements, global ghost-penalty terms are added to the velocity and pressure formulations. These terms penalize jumps in normal derivatives across interior faces of the background mesh, effectively extending the stability of the finite element space to an extended fluid domain.
Semi-Discrete Scheme: The method is spatially semi-discrete, meaning time is continuous. The fluid velocity, pressure, solid displacement, and the Lagrange multiplier are approximated using finite element spaces of degree $k$ (for velocity and displacement) and $k-1$ (for pressure).

Key Contributions and Theoretical Framework
The central contribution of this paper is the first rigorous a priori error analysis for an unfitted finite element method applied to a fully coupled FSI system with a genuinely deforming interface. Previous rigorous analyses were largely limited to linearized models, small-displacement approximations, or fitted mesh approaches.

The proof of convergence relies on several novel analytical ingredients:

Interpolated Configuration: The authors introduce an interpolated configuration $X^*_H$ of the exact solution to compare the exact interface $\Gamma[X]$ , the interpolated interface $\Gamma[X^*_H]$ , and the numerical interface $\Gamma[X_H]$ . This allows for the systematic quantification of errors arising from the displacement of the interface.
Uniform Estimates: The analysis establishes uniform norm equivalences, trace inequalities, inverse estimates, and inf–sup conditions on the moving cut domains. Crucially, these estimates are uniform with respect to the position of the interface within the background mesh and with respect to time, provided the numerical deformation remains sufficiently close to the exact deformation.
Coupling of Errors: A significant analytical difficulty addressed is the two-way dependence between the fluid error and the geometry error (solid displacement). The error in the solid displacement alters the numerical fluid domain and interface, which in turn affects the fluid solution. The authors close this loop by deriving stability estimates for the projected kinematic mismatch on the interface and utilizing a continuation-in-time argument.
Continuation Argument: The proof assumes a priori bounds on the error (specifically $W^{1,\infty}$ and $L^\infty$ bounds) up to a maximal time $t^*$ . By proving energy estimates with constants independent of $t^*$ , and showing that for sufficiently small mesh sizes and $k \ge 3$ , the derived convergence rates imply the assumed bounds, the authors extend the validity of the estimates to the entire time interval $[0, T]$ .

Main Results
Under assumptions of sufficient regularity for the exact solution and mesh sizes satisfying $c_1 H \le h \le c_2 H$ (where $h$ and $H$ are the fluid and solid mesh sizes, respectively), the authors prove the following convergence rates for finite elements of degree $k \ge 3$ :

Fluid Velocity ( $u_h$ ) and Solid Displacement ( $d_H$ ): Convergence of order $k$ in the natural energy norms.
Fluid Pressure ( $p_h$ ): Convergence of order $k-1$ in the $L^2$ norm (associated with the energy norm).
Lagrange Multiplier (Interface Traction): Convergence of order $k-1$ in the $H^{-1/2}$ norm.

Specifically, the error bound is given by:
$\max_{t \in [0,T]} \left( \|u_h - u\|_{L^2}^2 + \|d_H - d \circ \Phi_H\|_{H^1}^2 + \|\partial_t d_H - \partial_t d \circ \Phi_H\|_{L^2}^2 \right) + \int_0^T \|u_h - u\|_{H^1}^2 dt + H^2 \int_0^T (\|p_h - p\|_{L^2}^2 + \|\sigma_H - \sigma \circ \Phi_H\|_{H^{-1/2}}^2) dt \le C(h^{2k} + H^{2k})$

Significance
The paper claims that this work represents the first rigorous convergence analysis of an unfitted finite element method for a fully coupled FSI system with a deforming interface. While previous works have analyzed unfitted methods for prescribed moving domains or simplified coupled problems, the nonlinearity and the feedback loop between the fluid solution and the moving geometry in a fully coupled FSI system had not been rigorously treated in the unfitted context. The results provide a theoretical foundation for the use of CutFEM in complex FSI applications involving large deformations, validating the method's accuracy and stability without relying on small-displacement or fixed-domain approximations. The restriction $k \ge 3$ is noted as technical, related to specific inverse estimates used in the proof, and the authors suggest that the result may extend to $k=2$ with future refinements.

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