HEXAGONbeta

Stability and Convergence of a Regularized Immersed Boundary Method under Dynamic Cartesian Refinement. Part II: General Prescribed Motion

Primarily AI-generated textHuman understanding: no partsmath.MP — Mathematical Physicsmath.AP — Analysis of PDEsmath-ph — Mathematical Physics

Contributed by Sida He ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We study a regularized immersed boundary discretization of unsteady Stokes flow around a rigid body with prescribed translation and rotation. The interface may change its position and orientation, while remaining uniformly separated from the outer wall. The fluid spaces are conforming, exactly divergence free, and constructed on fixed physical Cartesian grids with a retained coarse space and arbitrarily changing local enrichments. The method uses regularized surface-moment constraints, backward Euler time stepping, and current-space orthogonal projection. We construct a time-dependent coarse lifting that satisfies every discrete rigid-velocity constraint exactly. A moving collar approximation has a small spatial error in H1H^1 and a small Eulerian time-derivative error in L2L^2, although its time derivative need not be accurate in H1H^1. This distinction permits a quantitative convergence proof without a restriction coupling the time step to the number of mesh changes. A shifted-interface trace estimate accounts for the motion of the reaction surface during each time step and for physical slip between time nodes. Under stated piecewise regularity and an L2L^2 interface-reaction assumption, the energy error is bounded by the square roots of the retained coarse size, regularization width, surface-panel size, and maximum time step, together with a specified load-consistency error. The analysis assumes exact integration and does not prove adaptive optimality or cover fluid-driven motion. Manufactured moving-ellipse computations and a Lean algebraic companion test specified implementation and proof components.

References

  1. Charles S. Peskin. The immersed boundary method. Acta Numerica , 11: 0 479--517, 2002. 10.1017/S0962492902000077.DOI
  2. Sida He. Stability and convergence of a regularized immersed boundary method under dynamic cartesian refinement. Part I : Fixed geometry, 2026 a . Unpublished companion manuscript, preparation version 0.3.3, 27 October 2026.
  3. Yoichiro Mori. Convergence proof of the velocity field for a Stokes flow immersed boundary method. Communications on Pure and Applied Mathematics , 61 0 (9): 0 1213--1263, 2008. 10.1002/cpa.20233.DOI
  4. Yang Liu and Yoichiro Mori. Properties of discrete delta functions and local convergence of the immersed boundary method. SIAM Journal on Numerical Analysis , 50 0 (6): 0 2986--3015, 2012. 10.1137/110836699.DOI
  5. Yang Liu and Yoichiro Mori. LpL^p convergence of the immersed boundary method for stationary Stokes problems. SIAM Journal on Numerical Analysis , 52 0 (1): 0 496--514, 2014. 10.1137/130911329.DOI
  6. Zhilin Li, Kejia Pan, and Juan Ruiz- Á lvarez. Convergence study of IB methods for Stokes equations with prescribed velocity boundary conditions. IMA Journal of Numerical Analysis , 45 0 (6): 0 3766--3799, 2025. 10.1093/imanum/drae101.DOI
  7. Alexandre X. Milewski and Charles S. Peskin. Convergence of the immersed boundary method for an elastically bound particle immersed in a 2D Navier--Stokes fluid fluid, 2025. URL https://arxiv.org/abs/2510.06586. arXiv:2510.06586.arXiv
  8. Roland Glowinski, Tsorng-Whay Pan, and Jacques P é riaux. A Lagrange multiplier/fictitious domain method for the numerical simulation of incompressible viscous flow around moving rigid bodies: (I) case where the rigid body motions are known a priori. Comptes Rendus de l'Acad é mie des Sciences, S é rie I, Math é matique , 324 0 (3): 0 361--369, 1997. 10.1016/S0764-4442(99)80376-0.DOI
  9. Bakytzhan Kallemov, Amneet Pal Singh Bhalla, Boyce E. Griffith, and Aleksandar Donev. An immersed boundary method for rigid bodies. Communications in Applied Mathematics and Computational Science , 11 0 (1): 0 79--141, 2016. 10.2140/camcos.2016.11.79. URL https://arxiv.org/abs/1505.07865.DOI
  10. Alexandre M. Roma, Charles S. Peskin, and Marsha J. Berger. An adaptive version of the immersed boundary method. Journal of Computational Physics , 153 0 (2): 0 509--534, 1999. 10.1006/jcph.1999.6293.DOI
Show all 37 referencesShow fewer references
  1. Yadong Zeng, Amneet Pal Singh Bhalla, and Lian Shen. A subcycling/non-subcycling time advancement scheme-based DLM immersed boundary method framework for solving single and multiphase fluid--structure interaction problems on dynamically adaptive grids. Computers & Fluids , 238: 0 105358, 2022. 10.1016/j.compfluid.2022.105358.DOI
  2. Luca Heltai and Wenyu Lei. A priori error estimates of regularized elliptic problems. Numerische Mathematik , 146 0 (3): 0 571--596, 2020. 10.1007/s00211-020-01152-w. URL https://arxiv.org/abs/1911.02293.DOI
  3. Luca Heltai and Wenyu Lei. Adaptive finite element approximations for elliptic problems using regularized forcing data. SIAM Journal on Numerical Analysis , 61 0 (2): 0 431--456, 2023. 10.1137/21M1455991. URL https://arxiv.org/abs/2110.15029.DOI
  4. Stefano Berrone, Andrea Bonito, Rob Stevenson, and Marco Verani. An optimal adaptive fictitious domain method. Mathematics of Computation , 88: 0 2101--2134, 2019. 10.1090/mcom/3414.DOI
  5. Christoph Lehrenfeld and Maxim Olshanskii. An Eulerian finite element method for PDEs in time-dependent domains. ESAIM: Mathematical Modelling and Numerical Analysis , 53 0 (2): 0 585--614, 2019. 10.1051/m2an/2018068.DOI
  6. Erik Burman, Stefan Frei, and Andre Massing. Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains. Numerische Mathematik , 150: 0 423--478, 2022. 10.1007/s00211-021-01264-x. URL https://arxiv.org/abs/1910.03054.DOI
  7. Henry von Wahl, Thomas Richter, and Christoph Lehrenfeld. An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains. IMA Journal of Numerical Analysis , 42 0 (3): 0 2505--2544, 2022. 10.1093/imanum/drab044. URL https://arxiv.org/abs/2002.02352.DOI
  8. Amal Alphonse, Charles M. Elliott, and Bj ö rn Stinner. An abstract framework for parabolic PDEs on evolving spaces. Portugaliae Mathematica , 72 0 (1): 0 1--46, 2015. 10.4171/PM/1955. URL https://arxiv.org/abs/1403.4500.DOI
  9. Ana Djurdjevac, Carsten Gr ä ser, and Philip J. Herbert. An evolving space framework for Oseen equations on a moving domain. ESAIM: Mathematical Modelling and Numerical Analysis , 57 0 (5): 0 3113--3138, 2023. 10.1051/m2an/2023074. URL https://arxiv.org/abs/2106.08312.DOI
  10. Qiqi Rao, Jilu Wang, and Yupei Xie. Optimal convergence of arbitrary Lagrangian--Eulerian finite element methods for the Stokes equations in an evolving domain. IMA Journal of Numerical Analysis , 45 0 (6): 0 3616--3642, 2025. 10.1093/imanum/drae097.DOI
  11. L. Beir ã o da Veiga, S. G ó mez, and K. B. Haile. Pressure-robust ALE space-time DG method for the Stokes equations on moving domains, 2026. URL https://arxiv.org/abs/2606.28914. arXiv:2606.28914.arXiv
  12. Andreas Brenner, Eberhard B ä nsch, and Markus Bause. A priori error analysis for finite element approximations of the Stokes problem on dynamic meshes. IMA Journal of Numerical Analysis , 34 0 (1): 0 123--146, 2014. 10.1093/imanum/drt001. URL https://d-nb.info/1113290293/34.DOI
  13. Tomas Lundquist, Jan Nordstr ö m, and Arnaud Malan. Stable dynamical adaptive mesh refinement. Journal of Scientific Computing , 86: 0 43, 2021. 10.1007/s10915-021-01414-1.DOI
  14. Evan S. Gawlik and Adrian J. Lew. Unified analysis of finite element methods for problems with moving boundaries. SIAM Journal on Numerical Analysis , 53 0 (6): 0 2822--2846, 2015. 10.1137/140990437. URL https://webpages.scu.edu/ftp/egawlik/pdf/GaLe2015c.pdf.DOI
  15. Sida He, Zixuan Yang, Fotis Sotiropoulos, and Lian Shen. Numerical simulation of interaction between multiphase flows and thin flexible structures. Journal of Computational Physics , 448: 0 110691, 2022 a . 10.1016/j.jcp.2021.110691.DOI
  16. Kaiyuan Zheng, Sida He, Xizeng Zhao, Lian Shen, and Xiaojue Zhu. Free-surface-induced ground effect for flapping swimmers. Journal of Fluid Mechanics , 997: 0 A36, 2024. 10.1017/jfm.2024.830.DOI
  17. Dandan Yang, Sida He, Lian Shen, and Xianwu Luo. Large eddy simulation coupled with immersed boundary method for turbulent flows over a backward facing step. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science , 235 0 (15): 0 2705--2714, 2021. 10.1177/0954406220954892.DOI
  18. Jichang Chen, Zhiwei Lu, Bin Wu, Sida He, Tianhang Xiao, Mingbo Tong, and Kan Kan. Flow transition and vortex evolution of a symmetric airfoil at low Reynolds number and high angle of attack. Physics of Fluids , 37 0 (2): 0 023603, 2025. 10.1063/5.0246708.DOI
  19. Sida He, Han Liu, and Lian Shen. Simulation-based study of turbulent aquatic canopy flows with flexible stems. Journal of Fluid Mechanics , 947: 0 A33, 2022 b . 10.1017/jfm.2022.655.DOI
  20. Alexandre Ern and Jean-Luc Guermond. Finite Elements I : Approximation and Interpolation , volume 72 of Texts in Applied Mathematics . Springer, Cham, 2021 a . 10.1007/978-3-030-56341-7.DOI
  21. Erik Burman, Peter Hansbo, and Mats G. Larson. Cut Bogner--Fox--Schmit elements for plates. Advanced Modeling and Simulation in Engineering Sciences , 7: 0 27, 2020. 10.1186/s40323-020-00164-3.DOI
  22. Alexandre Ern and Jean-Luc Guermond. Finite Elements II : Galerkin Approximation, Elliptic and Mixed PDEs , volume 73 of Texts in Applied Mathematics . Springer, Cham, 2021 b . 10.1007/978-3-030-56923-5.DOI
  23. Daniele Boffi, Nicola Cavallini, and Lucia Gastaldi. The finite element immersed boundary method with distributed Lagrange multiplier. SIAM Journal on Numerical Analysis , 53 0 (6): 0 2584--2604, 2015. 10.1137/140978399. URL https://arxiv.org/abs/1407.5184.DOI
  24. Kambiz Salari and Patrick Knupp. Code verification by the method of manufactured solutions. Technical Report SAND2000-1444, Sandia National Laboratories, 2000.
  25. Leonardo de Moura and Sebastian Ullrich. The Lean 4 theorem prover and programming language. In Automated Deduction--- CADE 28 , volume 12699 of Lecture Notes in Computer Science , pages 625--635, Cham, 2021. Springer. 10.1007/978-3-030-79876-5_37.DOI
  26. The mathlib Community . The Lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs , CPP '20. ACM, 2020. 10.1145/3372885.3373824.DOI
  27. Sida He. Stability and convergence of a regularized immersed boundary method under dynamic cartesian refinement. Part III : Fluid-driven rigid-body motion, 2026 b . Unpublished companion manuscript, preparation version 0.1.1, 27 October 2026.

Version history

  1. v1Submitted by Sida HeInitial depositCurrentOct 27, 2026