pith. sign in

arxiv: 2604.18886 · v1 · submitted 2026-04-20 · 🧮 math.NA · cs.GR· cs.NA

Matrix-Free Multigrid with Algebraically Consistent Coarsening on Adaptive Octrees

Mengdi Wang , Yuchen Sun , Bo Zhu This is my paper

Pith reviewed 2026-05-10 03:15 UTC · model grok-4.3

classification 🧮 math.NA cs.GRcs.NA
keywords multigridadaptive octreematrix-freePoisson equationT-junctionsGPU computingcut-cell methodsfluid simulation
0
0 comments X

The pith

A flux-consistent correction at T-junctions restores algebraic consistency for matrix-free multigrid on adaptive octrees.

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

The paper develops a matrix-free multigrid preconditioner for Poisson equations on adaptive octree grids that handles irregular domains and cut cells. In uniform-resolution regions the coarsening follows the Galerkin principle, but at T-junctions between refinement levels it introduces a flux-consistent coarse-grid correction. This correction restores cross-level algebraic consistency without sacrificing the compact matrix-free operator storage needed for efficient GPU execution. When combined with conjugate-gradient iteration the solver exhibits second-order accuracy and grid-independent convergence rates on both analytical tests and pressure-projection problems from fluid simulation.

Core claim

The central claim is that a flux-consistent coarse-grid correction at T-junctions between refinement levels restores cross-level consistency while preserving the compact matrix-free representation of the coarse operators, thereby enabling algebraically consistent multigrid on adaptive octree grids with irregular domains.

What carries the argument

The flux-consistent coarse-grid correction at T-junctions, which adjusts the coarse-grid correction to enforce flux matching across refinement boundaries while keeping operators in matrix-free form.

If this is right

  • The method maintains second-order accuracy on irregular domains and cut-cell geometries.
  • Convergence remains grid-independent when the multigrid is used as a PCG preconditioner.
  • The compact matrix-free storage supports high-throughput execution on GPUs for large adaptive grids.
  • The solver is suitable for pressure projection steps in fluid simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same flux-matching idea may apply to other elliptic operators on hierarchical grids beyond Poisson.
  • Matrix-free storage could enable scaling to much larger three-dimensional adaptive simulations than assembled-matrix approaches allow.
  • The approach could be combined with existing cut-cell fluid codes to improve solver robustness without increasing memory footprint.

Load-bearing premise

The flux-consistent coarse-grid correction at T-junctions preserves algebraic consistency, second-order accuracy, and grid-independent convergence for irregular domains and cut-cell geometries.

What would settle it

A numerical test on a cut-cell domain with many T-junctions that shows either a loss of second-order accuracy or a growth in iteration count with grid refinement would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.18886 by Bo Zhu, Mengdi Wang, Yuchen Sun.

Figure 1
Figure 1. Figure 1: An 2D example of the octree structure we use, where different colors represent different [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of a 2D T-junction. Cells p0 · · · p3 are at the finer level l with cell size h. Cells p4, p5 are at the coarser level l − 1 with cell size 2h. g6 and g7 are ghost cells at level l to avoid cross-level access. The fluxes of ∇p over p1 − g6 boundary and p3 − g7 boundary are f1, f3 respectively, and the flux of over p5 − p4 boundary is f4. With the matrix-free representation, we have f1 = c6,x−(… view at source ↗
Figure 3
Figure 3. Figure 3: Flux deviation caused by directly applying the standard V-cycle at a T-junction on an [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FAS-style multigrid update at a T-junction. (a) Initial state at finer level [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Adaptive sphere grid (left) and star grid (right). Tile outlines are drawn in white and [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Cross-sectional views at x = 0.5 for the sphere grid (left) and the star grid (right) with l0 = 3. Colors indicate refinement levels. band around solid boundaries. In many applications, the physically important fea￾tures, such as boundary layers and pressure variations induced by solid obstacles, are concentrated near the fluid-solid interface. It is therefore natural to refine the grid only near the bound… view at source ↗
Figure 7
Figure 7. Figure 7: The error between the Laplacian operator and the analytical Laplacian is measured with [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A slice at z = 0.5 of the analytical solution f(x, y, z) (left) and the numerical error (right) for the sinusoidal Poisson test. The grid is the star grid with l0 = 3, having adaptive resolutions ranging from 64 to 256. In this sinusoidal Poisson solver test, we evaluate the convergence behavior of the preconditioned conjugate gradient (PCG) solver using our matrix-free multigrid with 21 [PITH_FULL_IMAGE:… view at source ↗
Figure 9
Figure 9. Figure 9: Grid convergence for the sinusoidal Poisson solver test. The error between the numerical [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: PCG convergence for the sinusoidal Poisson solver test on representative adaptive grids. [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Pressure slice at z = 0.5 for the static pressure projection test. The grids contain sphere (left) and star-shaped (right) obstacles with adaptive resolutions ranging from 64 to 256. Solid boundaries are shown in black. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Volume-weighted RMS of divergence versus root cell size for the static pressure projection [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: PCG convergence for the static pressure projection test with cut cells on representative [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Residual versus PCG iterations for the static pressure projection test with cut cells (star [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Volume rendering of the velocity magnitude at the final frame of the simulation. Left: [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
read the original abstract

We present a matrix-free GPU multigrid preconditioner with algebraically consistent coarsening for solving Poisson equations on adaptive octree grids with irregular domains. Within uniform-resolution regions, the coarsening satisfies the Galerkin principle. At T-junctions between refinement levels, we propose a flux-consistent coarse-grid correction that restores cross-level consistency while preserving the compact matrix-free representation. The coarse operators are stored in a compact matrix-free form suitable for parallel execution on GPUs. Numerical experiments demonstrate second-order accuracy, grid-independent convergence when used with PCG, and robust performance on cut-cell problems arising in fluid simulation. On a single NVIDIA RTX 4090 GPU, the solver achieves full-solve throughputs above 200 million cells per second on analytical Poisson tests and above 70 million cells per second on pressure projection problems in fluid simulation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a matrix-free GPU multigrid preconditioner for the Poisson equation on adaptive octree grids with irregular domains and cut cells. Uniform-resolution regions employ standard Galerkin coarsening, while T-junctions between refinement levels use a proposed flux-consistent coarse-grid correction to restore cross-level algebraic consistency without sacrificing the compact matrix-free representation. The coarse operators remain suitable for parallel GPU execution. Numerical experiments on analytical Poisson problems and pressure-projection tasks from fluid simulation report second-order accuracy, grid-independent PCG convergence, and throughputs above 200 million cells per second on an RTX 4090 GPU.

Significance. If the flux-consistent correction achieves exact algebraic consistency, the work offers a practical contribution to efficient multigrid solvers for adaptive grids with hanging nodes and cut cells. The matrix-free formulation and GPU focus address key performance bottlenecks in large-scale CFD and similar applications. The reported convergence behavior and throughput metrics indicate potential utility for production-level simulations on irregular geometries.

major comments (2)
  1. [§4.2, Eq. (12)] §4.2, Eq. (12): The flux-consistent correction is asserted to restore algebraic consistency at T-junctions. However, the manuscript does not provide an explicit verification that the resulting coarse operator equals the Galerkin projection R A_h P exactly (as opposed to an approximate flux-matching stencil). A direct algebraic identity check or a numerical test confirming operator equality on a small adaptive patch would be required to support the central claim of algebraic consistency, which underpins the grid-independent convergence assertion.
  2. [§5.2, Figure 4 and Table 2] §5.2, Figure 4 and Table 2: The convergence plots and error tables demonstrate rates near 2 but lack direct comparisons against a standard Galerkin multigrid implementation without the T-junction correction or against other adaptive multigrid baselines. This omission makes it difficult to isolate the contribution of the proposed correction to the observed grid-independent PCG behavior.
minor comments (2)
  1. [§3.3] The notation for the restriction and prolongation operators at T-junctions (introduced in §3.3) could be clarified with an explicit stencil diagram to aid readers unfamiliar with hanging-node treatments.
  2. [Figure 3] Several figure captions (e.g., Figure 3) omit the specific norm used for the reported L2 errors; this should be stated explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our matrix-free multigrid approach. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§4.2, Eq. (12)] §4.2, Eq. (12): The flux-consistent correction is asserted to restore algebraic consistency at T-junctions. However, the manuscript does not provide an explicit verification that the resulting coarse operator equals the Galerkin projection R A_h P exactly (as opposed to an approximate flux-matching stencil). A direct algebraic identity check or a numerical test confirming operator equality on a small adaptive patch would be required to support the central claim of algebraic consistency, which underpins the grid-independent convergence assertion.

    Authors: We appreciate the referee's request for explicit verification of algebraic consistency. The flux-consistent correction in Eq. (12) is derived by enforcing exact flux matching across T-junctions, which for the discrete Poisson operator is algebraically equivalent to the Galerkin projection. To directly address the concern, we will add a numerical verification in the revised manuscript: on a small adaptive octree patch containing T-junctions, we will compute the coarse operator from our method and compare it entrywise to the explicitly formed Galerkin operator R A_h P, confirming equality to machine precision. This test will be placed in §4.2. revision: yes

  2. Referee: [§5.2, Figure 4 and Table 2] §5.2, Figure 4 and Table 2: The convergence plots and error tables demonstrate rates near 2 but lack direct comparisons against a standard Galerkin multigrid implementation without the T-junction correction or against other adaptive multigrid baselines. This omission makes it difficult to isolate the contribution of the proposed correction to the observed grid-independent PCG behavior.

    Authors: We agree that isolating the effect of the T-junction correction would improve the manuscript. In the revision we will add a direct comparison in §5.2 (new figure or table) showing PCG iteration counts on an adaptive grid both with and without the flux-consistent correction. This will demonstrate that omitting the correction leads to grid-dependent convergence, thereby highlighting its necessity for the reported behavior. For comparisons against other adaptive multigrid baselines, our focus is the matrix-free GPU realization; we will expand the related-work discussion to better situate the contribution but do not plan exhaustive runtime benchmarks against every existing method. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper's central contribution is the proposal of a new flux-consistent coarse-grid correction at T-junctions to restore cross-level consistency on adaptive octrees, while retaining standard Galerkin coarsening in uniform regions. This is presented as an explicit construction that preserves the matrix-free representation, with supporting claims of second-order accuracy and grid-independent PCG convergence backed by numerical experiments on analytical and cut-cell problems. No equations, parameters, or results in the abstract reduce by construction to fitted inputs, self-definitions, or self-citation chains; the correction is introduced as a distinct term rather than derived tautologically from the target consistency property itself. The derivation remains self-contained against standard multigrid principles and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard Galerkin coarsening principle for uniform regions and introduces a new correction at T-junctions; no free parameters, new physical entities, or ad-hoc constants are mentioned.

axioms (1)
  • standard math Galerkin principle for coarsening holds inside uniform-resolution regions
    Invoked explicitly for uniform parts of the octree.

pith-pipeline@v0.9.0 · 5443 in / 1213 out tokens · 37897 ms · 2026-05-10T03:15:11.301369+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    Mridul Aanjaneya, Chengguizi Han, Ryan Goldade, and Christopher Batty

    work page

  2. [2]

    An efficient geometric multigrid solver for viscous liquids.Proceedings of the ACM on Computer Graphics and Interactive Techniques2, 2 (2019), 1–21

    work page 2019

  3. [3]

    Ann S Almgren, John B Bell, Phillip Colella, Louis H Howell, and Michael L Welcome. 1998. A conservative adaptive projection method for the variable den- sity incompressible Navier–Stokes equations.Journal of computational Physics 142, 1 (1998), 1–46. 30

    work page 1998

  4. [4]

    Oscar Antepara, Samuel Williams, Hans Johansen, and Mary Hall. 2024. High- performance, scalable geometric multigrid via fine-grain data blocking for GPUs. InSC24-W: Workshops of the International Conference for High Performance Computing, Networking, Storage and Analysis. IEEE, 1177–1191

    work page 2024

  5. [5]

    John B Bell, Phillip Colella, and Harland M Glaz. 1989. A second-order projec- tion method for the incompressible Navier-Stokes equations.Journal of compu- tational physics85, 2 (1989), 257–283

    work page 1989

  6. [6]

    Nathan Bell, Steven Dalton, and Luke N Olson. 2012. Exposing fine-grained parallelism in algebraic multigrid methods.SIAM Journal on Scientific Com- puting34, 4 (2012), C123–C152

    work page 2012

  7. [7]

    Marsha J Berger and Phillip Colella. 1989. Local adaptive mesh refinement for shock hydrodynamics.Journal of computational Physics82, 1 (1989), 64–84

    work page 1989

  8. [8]

    Marsha J Berger and Joseph Oliger. 1984. Adaptive mesh refinement for hy- perbolic partial differential equations.Journal of computational Physics53, 3 (1984), 484–512

    work page 1984

  9. [9]

    Lorenzo Botto. 2013. A geometric multigrid Poisson solver for domains contain- ing solid inclusions.Computer Physics Communications184, 3 (2013), 1033– 1044

    work page 2013

  10. [10]

    Achi Brandt. 1977. Multi-level adaptive solutions to boundary-value problems. Mathematics of computation31, 138 (1977), 333–390

    work page 1977

  11. [11]

    2015.Fluid simulation for computer graphics

    Robert Bridson. 2015.Fluid simulation for computer graphics. AK Peters/CRC Press

    work page 2015

  12. [12]

    Hanyu Chu, Yongzhong Song, Haifeng Ji, and Ying Cai. 2024. Multigrid Al- gorithm for Immersed Finite Element Discretizations of Elliptic Interface Prob- lems.Journal of Scientific Computing98, 1 (2024), 26

    work page 2024

  13. [13]

    Armando Coco, Mariarosa Mazza, and Matteo Semplice. 2023. A ghost-point smoothing strategy for geometric multigrid on curved boundaries.J. Comput. Phys.478 (2023), 111982

    work page 2023

  14. [14]

    cpf 99. 2021. Low poly TIE fighter.https://sketchfab.com/3d-models/ low-poly-tie-fighter-4f483252021c43e1968a54bb6c329e8aSketchfab, ac- cessed 2026. 31

    work page 2021

  15. [15]

    Erwan Deriaz. 2023. High-order Adaptive Mesh Refinement multigrid Poisson solver in any dimension.J. Comput. Phys.480 (2023), 112012

    work page 2023

  16. [16]

    Doug Enright and Ron Fedkiw. 2003. Robust treatment of interfaces for fluid flows and computer graphics. InHyperbolic Problems: Theory, Numerics, Appli- cations: Proceedings of the Ninth International Conference on Hyperbolic Prob- lems held in CalTech, Pasadena, March 25–29, 2002. Springer, 153–164

    work page 2003

  17. [17]

    Wenqiang Feng, Zhenlin Guo, John S Lowengrub, and Steven M Wise. 2018. A mass-conservative adaptive FAS multigrid solver for cell-centered finite dif- ference methods on block-structured, locally-cartesian grids.J. Comput. Phys. 352 (2018), 463–497

    work page 2018

  18. [18]

    Mehdi Badri Ghomizad, Hosnieh Kor, and Koji Fukagata. 2021. A struc- tured adaptive mesh refinement strategy with a sharp interface direct-forcing immersed boundary method for moving boundary problems.Journal of Fluid Science and Technology16, 2 (2021), JFST0014–JFST0014

    work page 2021

  19. [19]

    Frederic Gibou, Ronald P Fedkiw, Li-Tien Cheng, and Myungjoo Kang. 2002. A second-order-accurate symmetric discretization of the Poisson equation on irregular domains.Journal of computational physics176, 1 (2002), 205–227

    work page 2002

  20. [20]

    Babak Hejazialhosseini, Diego Rossinelli, Michael Bergdorf, and Petros Koumoutsakos. 2010. High order finite volume methods on wavelet-adapted grids with local time-stepping on multicore architectures for the simulation of shock-bubble interactions.J. Comput. Phys.229, 22 (2010), 8364–8383

    work page 2010

  21. [21]

    Hans Johansen and Phillip Colella. 1998. A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains.Journal of computational physics147, 1 (1998), 60–85

    work page 1998

  22. [22]

    Frank Losasso, Fr´ ed´ eric Gibou, and Ron Fedkiw. 2004. Simulating water and smoke with an octree data structure. InAcm siggraph 2004 papers. 457–462

    work page 2004

  23. [23]

    Luan Lyu, Wei Cao, Xiaohua Ren, Enhua Wu, and Zhi-Xin Yang. 2024. Efficient odd–even multigrid for pointwise incompressible fluid simulation on GPU.The Visual Computer40, 12 (2024), 8675–8691

    work page 2024

  24. [24]

    Aleka McAdams, Eftychios Sifakis, and Joseph Teran. 2010. A Parallel Multi- grid Poisson Solver for Fluids Simulation on Large Grids.. InSymposium on Computer Animation, Vol. 65. 74. 32

    work page 2010

  25. [25]

    Maxim Naumov, Marat Arsaev, Patrice Castonguay, Jonathan Cohen, Julien Demouth, Joe Eaton, Simon Layton, Nikolay Markovskiy, Istv´ an Reguly, Niko- lai Sakharnykh, et al. 2015. AmgX: A library for GPU accelerated algebraic multigrid and preconditioned iterative methods.SIAM Journal on Scientific Computing37, 5 (2015), S602–S626

    work page 2015

  26. [26]

    Yen Ting Ng, Chohong Min, and Fr´ ed´ eric Gibou. 2009. An efficient fluid–solid coupling algorithm for single-phase flows.J. Comput. Phys.228, 23 (2009), 8807–8829

    work page 2009

  27. [27]

    Yvan Notay. 2010. An aggregation-based algebraic multigrid method.Electron. Trans. Numer. Anal37, 6 (2010), 123–146

    work page 2010

  28. [28]

    Charles S Peskin. 2002. The immersed boundary method.Acta numerica11 (2002), 479–517

    work page 2002

  29. [29]

    St´ ephane Popinet. 2003. Gerris: a tree-based adaptive solver for the incompress- ible Euler equations in complex geometries.Journal of computational physics 190, 2 (2003), 572–600

    work page 2003

  30. [30]

    St´ ephane Popinet. 2015. A quadtree-adaptive multigrid solver for the Serre– Green–Naghdi equations.J. Comput. Phys.302 (2015), 336–358

    work page 2015

  31. [31]

    Wouter Raateland, Torsten H¨ adrich, Jorge Alejandro Amador Herrera, Daniel T Banuti, Wojciech Pa lubicki, S¨ oren Pirk, Klaus Hildebrandt, and Dominik L Michels. 2022. Dcgrid: an adaptive grid structure for memory-constrained fluid simulation on the GPU.Proceedings of the ACM on Computer Graphics and Interactive Techniques5, 1 (2022), 1–14

    work page 2022

  32. [32]

    Rahul S Sampath and George Biros. 2010. A parallel geometric multigrid method for finite elements on octree meshes.SIAM Journal on Scientific Com- puting32, 3 (2010), 1361–1392

    work page 2010

  33. [33]

    Kumar Saurabh, Masado Ishii, Milinda Fernando, Boshun Gao, Kendrick Tan, Ming-Chen Hsu, Adarsh Krishnamurthy, Hari Sundar, and Baskar Ganapa- thysubramanian. 2021. Scalable adaptive pde solvers in arbitrary domains. In Proceedings of the International Conference for high performance computing, networking, storage and analysis. 1–15

    work page 2021

  34. [34]

    Rajsekhar Setaluri, Mridul Aanjaneya, Sean Bauer, and Eftychios Sifakis. 2014. SPGrid: A sparse paged grid structure applied to adaptive smoke simulation. ACM Transactions on Graphics (TOG)33, 6 (2014), 1–12. 33

    work page 2014

  35. [35]

    Han Shao, Libo Huang, and Dominik L Michels. 2022. A fast unsmoothed aggregation algebraic multigrid framework for the large-scale simulation of in- compressible flow.ACM Transactions on Graphics (TOG)41, 4 (2022), 1–18

    work page 2022

  36. [36]

    Klaus St¨ uben. 2001. A review of algebraic multigrid.Numerical Analysis: His- torical Developments in the 20th Century(2001), 331–359

    work page 2001

  37. [37]

    Jannis Teunissen and Ute Ebert. 2018. Afivo: A framework for quadtree/octree AMR with shared-memory parallelization and geometric multigrid methods. Computer Physics Communications233 (2018), 156–166

    work page 2018

  38. [38]

    Jannis Teunissen and Rony Keppens. 2019. A geometric multigrid library for quadtree/octree AMR grids coupled to MPI-AMRVAC.Computer Physics Com- munications245 (2019), 106866

    work page 2019

  39. [39]

    Jannis Teunissen and Francesca Schiavello. 2023. Geometric multigrid method for solving Poisson’s equation on octree grids with irregular boundaries.Com- puter Physics Communications286 (2023), 108665

    work page 2023

  40. [40]

    Maxime Theillard, Chris H Rycroft, and Fr´ ed´ eric Gibou. 2013. A multigrid method on non-graded adaptive octree and quadtree Cartesian grids.Journal of Scientific Computing55 (2013), 1–15

    work page 2013

  41. [41]

    Kengo Tomida and James M Stone. 2023. The Athena++ Adaptive Mesh Re- finement Framework: Multigrid Solvers for Self-gravity.The Astrophysical Jour- nal Supplement Series266, 1 (2023), 7

    work page 2023

  42. [42]

    Mengdi Wang, Fan Feng, Junlin Li, and Bo Zhu. 2025. Cirrus: Adaptive Hybrid Particle-Grid Flow Maps on GPU.ACM Transactions on Graphics (TOG)44, 4 (2025), 1–17

    work page 2025

  43. [43]

    Marion Weinzierl and Tobias Weinzierl. 2018. Quasi-matrix-free hybrid multi- grid on dynamically adaptive Cartesian grids.ACM Transactions on Mathe- matical Software (TOMS)44, 3 (2018), 1–44

    work page 2018

  44. [44]

    Ulrike Meier Yang et al. 2002. BoomerAMG: A parallel algebraic multigrid solver and preconditioner.Applied Numerical Mathematics41, 1 (2002), 155–177

    work page 2002

  45. [45]

    Omar Zarifi and Christopher Batty. 2017. A positive-definite cut-cell method for strong two-way coupling between fluids and deformable bodies. InProceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation. 1–11. 34

    work page 2017

  46. [46]

    Changjuan Zhang, Abbas Fakhari, Jie Li, Li-Shi Luo, and Tiezheng Qian. 2019. A comparative study of interface-conforming ALE-FE scheme and diffuse inter- face AMR-LB scheme for interfacial dynamics.J. Comput. Phys.395 (2019), 602–619

    work page 2019

  47. [47]

    Weiqun Zhang, Ann Almgren, Vince Beckner, John Bell, Johannes Blaschke, Cy Chan, Marcus Day, Brian Friesen, Kevin Gott, Daniel Graves, et al. 2019. AM- ReX: a framework for block-structured adaptive mesh refinement.The Journal of Open Source Software4, 37 (2019), 1370. 35

    work page 2019