pith. machine review for the scientific record. sign in

arxiv: 2602.16312 · v2 · submitted 2026-02-18 · 🧮 math.NA · cs.NA

Recognition: 2 theorem links

· Lean Theorem

An agglomeration-based multigrid solver for the discontinuous Galerkin discretization of cardiac electrophysiology

Marco Feder , Pasquale Claudio Africa
Authors on Pith no claims yet

Pith reviewed 2026-05-15 21:34 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords discontinuous Galerkinmultigridagglomerationmonodomain modelcardiac electrophysiologypreconditionerpolytopic grids
0
0 comments X

The pith

An agglomeration-based multilevel preconditioner accelerates iterative solvers for the discontinuous Galerkin discretization of cardiac electrophysiology models.

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

This paper develops an agglomeration-based multilevel preconditioner to speed up iterative solvers for the linear systems that come from using discontinuous Galerkin methods on the monodomain model of heart electrophysiology. The key is to create coarser grids by combining fine mesh elements into general polytopic shapes, forming a nested hierarchy without needing special coarse meshes. This matters because cardiac simulations on realistic 3D heart geometries involve large systems that are expensive to solve directly, and better preconditioners can make high-fidelity modeling feasible. Experiments on 2D and 3D domains with various ionic models demonstrate that the method scales well as the polynomial degree rises and as more multigrid levels are used.

Core claim

The central discovery is that agglomerating elements from a fine initial mesh produces a hierarchy of polytopic grids that supports an effective multilevel preconditioner for the DG-discretized monodomain equations. The preconditioner leverages these general coarse elements to achieve robust convergence of iterative solvers. Tests confirm strong performance and favorable scaling with respect to polynomial degree and number of levels on unstructured geometries and different ionic models.

What carries the argument

The agglomeration strategy for constructing nested hierarchies of polytopic grids from a fine mesh, which supports the multilevel preconditioner while preserving necessary stability properties.

If this is right

  • The solver remains effective across increasing polynomial degrees in the discontinuous Galerkin discretization.
  • Favorable scalability holds with respect to the number of levels in the multigrid preconditioner.
  • The approach applies successfully to realistic 3D unstructured geometries and multiple ionic models.
  • General polytopic elements at coarser levels maintain the approximation properties required for the preconditioner.

Where Pith is reading between the lines

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

  • Such preconditioners could support larger-scale simulations of cardiac wave propagation by reducing the computational cost of each solve.
  • The agglomeration technique might generalize to other partial differential equations discretized with discontinuous Galerkin methods on complex domains.
  • Integrating this with adaptive refinement could further improve efficiency in modeling heterogeneous heart tissue.

Load-bearing premise

The agglomeration procedure must produce polytopic elements regular enough to preserve the stability and approximation properties of the discontinuous Galerkin method across levels.

What would settle it

If numerical experiments on a realistic 3D heart geometry show iteration counts growing rapidly or divergence when using high polynomial degrees or many multigrid levels, that would indicate the claim does not hold.

Figures

Figures reproduced from arXiv: 2602.16312 by Marco Feder, Pasquale Claudio Africa.

Figure 1
Figure 1. Figure 1: (a) MPI partitioning of the realistic hexahedral ventricle mesh into 128 subdomains. Each color corresponds to a different MPI process. Bottom row: detailed views of an agglomerate on the boundary of the domain (b) and its 8 sub-agglomerates (c) and (d), each displayed in a different color. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) The computational mesh partitioned using parmetis among five MPI processes. To build the multigrid hierarchy, aggregates are generated using the locally owned cells within each MPI rank. Snapshots of the transmembrane potential U at (b) t = 0.04s and (c) t = 0.16s after the external application of the current Iapp(x, t) in Equation (14). as shown in Fig. 2a. As previously mentioned in Section 4.1, we g… view at source ↗
Figure 3
Figure 3. Figure 3: Number of PCG iterations per time step for Problem ( [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Idealized left ventricle test case: (a) hexahedral mesh of the ellipsoid and (b) time evolution of the transmembrane potential at a selected point. Idealized ventricle (5 processes) Level index l Card(Tl) l = 0 407,904 l = 1 50,990 l = 2 6,375 l = 3 800 l = 4 100 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Fiber field computed using the Laplace-Dirichlet Rule-Based Methods [ [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) Hexahedral mesh Th representing a realistic left ventricle. (b) Propagation of the trans￾membrane potential U. range from 8 to 13, far below the counts of AMG, which reach up to 21 iterations, and of Block-Jacobi, which reach up to 68 iterations. 6.4. Performance results on realistic three-dimensional test We now examine the performance of all the considered preconditioners for the realistic three-dime… view at source ↗
Figure 7
Figure 7. Figure 7: Snapshots of the transmembrane potential [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Number of PCG iterations per time step for Problem ( [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: PCG iteration time per time step for Problem ( [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of average wall-clock time (measured in seconds) to perform one time step for [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Breakdown of the setup phase cost for the agglomeration-based multigrid preconditioner on [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Scalability analysis with respect to the number of MPI processes for the three-dimensional [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Strong scaling results for the realistic ventricle test case with [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
read the original abstract

This work presents a novel agglomeration-based multilevel preconditioner designed to accelerate the convergence of iterative solvers for linear systems arising from the discontinuous Galerkin discretization of the monodomain model in cardiac electrophysiology. The proposed approach exploits general polytopic grids at coarser levels, obtained through the agglomeration of elements from an initial, potentially fine, mesh. By leveraging a robust and efficient agglomeration strategy, we construct a nested hierarchy of grids suitable for multilevel solver frameworks. The effectiveness and performance of the methodology are assessed through a series of numerical experiments on two- and three-dimensional domains, involving different ionic models and realistic unstructured geometries. The results demonstrate strong solver effectiveness and favorable scalability with respect to both the polynomial degree of the discretization and the number of levels selected in the multigrid preconditioner.

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

1 major / 2 minor

Summary. The paper proposes an agglomeration-based multilevel preconditioner to accelerate iterative solvers for linear systems arising from discontinuous Galerkin discretizations of the monodomain model in cardiac electrophysiology. It constructs nested hierarchies of polytopic grids via element agglomeration from an initial mesh and evaluates performance through numerical experiments on 2D and 3D domains with different ionic models and realistic unstructured geometries, claiming strong solver effectiveness together with favorable scalability in both polynomial degree and number of multigrid levels.

Significance. If the agglomeration procedure reliably preserves the shape-regularity constants required for uniform DG stability and approximation properties, the method would supply a practical, scalable solver for high-order simulations of cardiac electrical propagation on complex geometries, addressing an important computational bottleneck in electrophysiology modeling.

major comments (1)
  1. [Abstract] Abstract: the claim of favorable scalability with respect to polynomial degree and number of levels rests on the unverified assertion that the agglomeration strategy produces polytopic coarse elements whose shape-regularity constants (minimum angles, diameter ratios) remain controlled. No a priori bound, quality metric, or analysis is supplied showing that the coercivity/continuity constants of the DG bilinear form for the monodomain operator stay uniform; numerical results alone therefore leave the central scalability claim dependent on an unproven assumption about the coarse-grid geometry.
minor comments (2)
  1. [Numerical experiments] Add a table or figure panel that explicitly reports iteration counts, convergence factors, and effective condition-number estimates versus polynomial degree and level count to make the scalability statements quantitatively transparent.
  2. Clarify the precise definition of the agglomeration criterion (e.g., any angle or aspect-ratio threshold) and state whether it is applied uniformly across all tested geometries and ionic models.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address the major comment below and have revised the manuscript to clarify the nature of our scalability claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of favorable scalability with respect to polynomial degree and number of levels rests on the unverified assertion that the agglomeration strategy produces polytopic coarse elements whose shape-regularity constants (minimum angles, diameter ratios) remain controlled. No a priori bound, quality metric, or analysis is supplied showing that the coercivity/continuity constants of the DG bilinear form for the monodomain operator stay uniform; numerical results alone therefore leave the central scalability claim dependent on an unproven assumption about the coarse-grid geometry.

    Authors: We agree with the referee that the manuscript provides no a priori analysis or bounds establishing that the agglomeration procedure preserves shape-regularity constants uniformly, and therefore does not prove that the DG coercivity and continuity constants remain independent of the number of levels or polynomial degree. The scalability statements in the original abstract were based solely on the numerical experiments reported in Sections 4 and 5. To correct this, we have revised the abstract to state that the observed scalability is empirical: “The results demonstrate strong solver effectiveness and favorable scalability with respect to both the polynomial degree of the discretization and the number of levels selected in the multigrid preconditioner, as observed across the numerical experiments.” We have also added a brief remark in the introduction and conclusion noting that a theoretical analysis of the agglomeration’s effect on mesh-regularity constants lies outside the scope of the present work. These changes remove the implication of a proven uniform bound while preserving the paper’s focus on practical performance. revision: yes

Circularity Check

0 steps flagged

No circularity: solver construction and numerical validation remain independent of fitted self-references

full rationale

The paper introduces an agglomeration-based multilevel preconditioner for DG discretizations of the monodomain equation. The core construction relies on standard agglomeration operators to generate coarse polytopic grids and on established multigrid theory for the hierarchy; neither the bilinear-form stability constants nor the reported p- and level-scalability are obtained by fitting parameters to the target performance metrics themselves. Numerical experiments on 2-D/3-D cardiac geometries and ionic models supply external evidence that the agglomeration preserves sufficient shape-regularity for the observed convergence rates. No self-definitional loop, fitted-input-renamed-as-prediction, or load-bearing self-citation chain appears in the derivation or the claimed results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from numerical PDE theory and cardiac modeling; no new free parameters or invented entities are introduced in the abstract description.

axioms (2)
  • domain assumption The monodomain model is a sufficient approximation for the electrical activity in cardiac tissue under the tested conditions.
    Invoked implicitly as the target PDE for discretization and solver testing.
  • domain assumption Agglomerated polytopic elements maintain the necessary approximation and stability properties for the DG method and multilevel preconditioner.
    Required for the hierarchy to function effectively but not proven in the abstract.

pith-pipeline@v0.9.0 · 5424 in / 1260 out tokens · 28215 ms · 2026-05-15T21:34:52.019404+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

84 extracted references · 84 canonical work pages · 1 internal anchor

  1. [1]

    Abdelhamid, N

    T. Abdelhamid, N. M. M. Huynh, S. Zampini, R. Chen, L. F. Pavarino, and S. Scac- chi. Adaptive BDDC preconditioners for the bidomain model on unstructured ven- tricular finite element meshes.Computer Methods in Applied Mechanics and Engi- neering, 447:118366, 2025

    work page 2025

  2. [2]

    Adams, M

    M. Adams, M. Brezina, J. Hu, and R. Tuminaro. Parallel multigrid smoothing: polynomial versus Gauss-Seidel.Journal of Computational Physics, 188(2):593–610, 2003

    work page 2003

  3. [3]

    lifex-fiber: anopen tool for myofibers generation in cardiac computational models.BMC Bioinformatics, 24(1):143, April 2023

    P.C.Africa, R.Piersanti, M.Fedele, L.Dede’, andA.Quarteroni. lifex-fiber: anopen tool for myofibers generation in cardiac computational models.BMC Bioinformatics, 24(1):143, April 2023

    work page 2023

  4. [4]

    P. C. Africa, R. Piersanti, F. Regazzoni, M. Bucelli, M. Salvador, M. Fedele, S. Pa- gani, L. Dede, and A. Quarteroni. lifex-ep: a robust and efficient software for cardiac electrophysiology simulations.BMC Bioinformatics, 24, 10 2023

    work page 2023

  5. [5]

    P. C. Africa, M. Salvador, P. Gervasio, L. Dede’, and A. Quarteroni. A matrix-free high-order solver for the numerical solution of cardiac electrophysiology.Journal of Computational Physics, 478:111984, 2023

    work page 2023

  6. [6]

    P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet. Hybrid scheduling for the parallel solution of linear systems.Parallel Computing, 32(2):136–156, 2006. 33

    work page 2006

  7. [7]

    P. F. Antonietti, S. Berrone, M. Busetto, and M. Verani. Agglomeration-based ge- ometric multigrid schemes for the Virtual Element Method.SIAM Journal on Nu- merical Analysis, 61(1):223–249, 2023

    work page 2023

  8. [8]

    P. F. Antonietti, S. Bertoluzza, and F. Credali. The Reduced Basis Multigrid scheme for the Virtual Element Method.arXiv:2511.22219, 2026

    work page arXiv 2026

  9. [9]

    P. F. Antonietti, M. Caldana, I. Mazzieri, and A. R. Fraschini. MAGNET: an open- source library for mesh agglomeration by graph neural networks.Engineering with Computers, pages 1–26, 2025

    work page 2025

  10. [10]

    P. F. Antonietti, M. Corti, and G. Martinelli. Polytopal mesh agglomeration via geometrical deep learning for three-dimensional heterogeneous domains.Mathematics and Computers in Simulation, 241:335–353, 2026

    work page 2026

  11. [11]

    P. F. Antonietti, L. Dede’, G. Loli, M. Montardini, G. Sangalli, and P. Tesini. Space- time isogeometric analysis of cardiac electrophysiology.Computer Methods in Applied Mechanics and Engineering, 441:117957, 2025

    work page 2025

  12. [12]

    P. F. Antonietti, P. Houston, and G. Pennesi. Fast numerical integration on polytopic meshes with applications to discontinuous Galerkin finite element methods.Journal of Scientific Computing, 77:1339–1370, 2019

    work page 2019

  13. [13]

    P. F. Antonietti, P. Houston, G. Pennesi, and E. Süli. An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids.Mathematics of Computation, 89(325):2047–2083, 2020

    work page 2047

  14. [14]

    P. F. Antonietti, P. Houston, M. Sarti, and M. Verani. Multigrid algorithms forhp- version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes.Calcolo, 54:1169–1198, 2017

    work page 2017

  15. [15]

    P. F. Antonietti, L. Mascotto, and M. Verani. A multigrid algorithm for the p-version of the Virtual Element Method.ESAIM: M2AN, 52(1):337–364, 2018

    work page 2018

  16. [16]

    P. F. Antonietti and L. Melas. Algebraic multigrid schemes for high-order nodal dis- continuous Galerkin methods.SIAM Journal on Scientific Computing, 42(2):A1147– A1173, 2020

    work page 2020

  17. [17]

    P. F. Antonietti and G. Pennesi. V-cycle multigrid algorithms for discontinuous Galerkin methods on non-nested polytopic meshes.Journal of Scientific Computing, 78:625–652, 2019

    work page 2019

  18. [18]

    P. F. Antonietti, M. Sarti, and M. Verani. Multigrid algorithms forhp-discontinuous Galerkin discretizations of elliptic problems.SIAM Journal on Numerical Analysis, 53(1):598–618, 2015. 34

    work page 2015

  19. [19]

    Arndt, W

    D. Arndt, W. Bangerth, M. Bergbauer, M. Feder, M. Fehling, J. Heinz, T. Heister, L. Heltai, M. Kronbichler, M. Maier, P. Munch, J.-P. Pelteret, B. Turcksin, D. Wells, and S.Zampini. Thedeal.IIlibrary, version 9.5.Journal of Numerical Mathematics, 31(3):231–246, 2023

    work page 2023

  20. [20]

    Pelteret, B

    D.Arndt, W.Bangerth, D.Davydov, T.Heister, L.Heltai, M.Kronbichler, M.Maier, J.-P. Pelteret, B. Turcksin, and D. Wells. The deal.II finite element library: Design, features, and insights.Computers & Mathematics with Applications, 81:407–422, Jan. 2021

    work page 2021

  21. [21]

    D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontin- uous Galerkin methods for elliptic problems.SIAM Journal on Numerical Analysis, 39(5):1749–1779, 2002

    work page 2002

  22. [22]

    Bangerth, C

    W. Bangerth, C. Burstedde, T. Heister, and M. Kronbichler. Algorithms and data structures for massively parallel generic adaptive finite element codes.ACM Trans. Math. Softw., 38(2), Jan. 2012

    work page 2012

  23. [23]

    Bassi, L

    F. Bassi, L. Botti, A. Colombo, D. Di Pietro, and P. Tesini. On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations.Journal of Computational Physics, 231(1):45–65, 2012

    work page 2012

  24. [24]

    Bastian, M

    P. Bastian, M. Blatt, and R. Scheichl. Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems.Numerical Linear Algebra with Applications, 19(2):367–388, 2012

    work page 2012

  25. [25]

    Beckmann, H.-P

    N. Beckmann, H.-P. Kriegel, R. Schneider, and B. Seeger. The R*-tree: an efficient and robust access method for points and rectangles. InACM SIGMOD Conference, 1990

    work page 1990

  26. [26]

    Benzi, G

    M. Benzi, G. H. Golub, and J. Liesen. Numerical solution of saddle point problems. Acta Numerica, 14:1–137, 2005

    work page 2005

  27. [27]

    M. L. Bittencourt, C. C. Douglas, and R. A. Feijóo. Nonnested multigrid methods for linear problems.Numerical Methods for Partial Differential Equations, 17(4):313– 331, 2001

    work page 2001

  28. [28]

    Boost C++ Libraries.http://www.boost.org/, 2015

    Boost. Boost C++ Libraries.http://www.boost.org/, 2015

    work page 2015

  29. [29]

    Botta, M

    F. Botta, M. Calafà, P. C. Africa, C. Vergara, and P. F. Antonietti. High-order discontinuous Galerkin methods for the monodomain and bidomain models.Mathe- matics in Engineering, 6(6):726–741, 2024

    work page 2024

  30. [30]

    Botti, A

    L. Botti, A. Colombo, and F. Bassi. h-multigrid agglomeration based solution strate- gies for discontinuous Galerkin discretizations of incompressible flow problems.Jour- nal of Computational Physics, 347:382–415, 2017. 35

    work page 2017

  31. [31]

    Bueno-Orovio, E

    A. Bueno-Orovio, E. M. Cherry, and F. H. Fenton. Minimal model for human ven- tricular action potentials in tissue.Journal of Theoretical Biology, 253(3):544–560, 2008

    work page 2008

  32. [32]

    Burstedde, L

    C. Burstedde, L. C. Wilcox, and O. Ghattas.p4est: Scalable algorithms for par- allel adaptive mesh refinement on forests of octrees.SIAM Journal on Scientific Computing, 33(3):1103–1133, 2011

    work page 2011

  33. [33]

    Cangiani, Z

    A. Cangiani, Z. Dong, and E. Georgoulis.hp-version discontinuous Galerkin methods on essentially arbitrarily-shaped elements.Mathematics of Computation, 91(333):1– 35, 2022

    work page 2022

  34. [34]

    Cangiani, Z

    A. Cangiani, Z. Dong, E. H. Georgoulis, and P. Houston.hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes. SpringerBriefs in Mathemat- ics. Springer, Cham, 2017

    work page 2017

  35. [35]

    T. F. Chan, J. Xu, and L. T. Zikatanov. An agglomeration multigrid method for unstructured grids.Contemporary mathematics, 1998

    work page 1998

  36. [36]

    Chen and G

    Y. Chen and G. N. Wells. Multigrid on unstructured meshes with regions of low quality cells.arXiv preprint arXiv:2402.12947, 2024

    work page arXiv 2024

  37. [37]

    Colli Franzone, L

    P. Colli Franzone, L. Pavarino, and B. Taccardi. Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models.Mathematical Biosciences, 197(1):35–66, 2005

    work page 2005

  38. [38]

    Colli Franzone and L

    P. Colli Franzone and L. F. Pavarino. A parallel solver for reaction-diffusion systems in computational electrocardiology.Mathematical Models and Methods in Applied Sciences, 14(06):883–911, 2004

    work page 2004

  39. [39]

    D’Ambra, F

    P. D’Ambra, F. Durastante, and S. Filippone. AMG preconditioners for linear solvers towards extreme scale.SIAM Journal on Scientific Computing, 43(5):S679–S703, 2021

    work page 2021

  40. [40]

    D’Ambra, F

    P. D’Ambra, F. Durastante, and S. Filippone. Parallel Sparse Computation Toolkit. Software Impacts, 15:100463, 2023

    work page 2023

  41. [41]

    Dargaville, A

    S. Dargaville, A. Buchan, R. Smedley-Stevenson, P. Smith, and C. Pain. A com- parison of element agglomeration algorithms for unstructured geometric multigrid. Journal of Computational and Applied Mathematics, 390:113379, 2021

    work page 2021

  42. [42]

    D.A.DiPietro, F.Hülsemann, P.Matalon, P.Mycek, U.Rüde, andD.Ruiz. Towards robust, fast solutions of elliptic equations on complex domains through hybrid high- order discretizations and non-nested multigrid methods.International Journal for Numerical Methods in Engineering, 122(22):6576–6595, 2021. 36

    work page 2021

  43. [43]

    D. A. Di Pietro, P. Matalon, P. Mycek, and U. Rüde. High-order multigrid strate- gies for HHO discretizations of elliptic equations.Numerical Linear Algebra with Applications, 30(1):e2456, 2023

    work page 2023

  44. [44]

    C. R. Dohrmann. A preconditioner for substructuring based on constrained energy minimization.SIAM Journal on Scientific Computing, 25(1):246–258, 2003

    work page 2003

  45. [45]

    R. D. Falgout. An Introduction to Algebraic Multigrid.Computing in Science and Engg., 8(6):24–33, Nov. 2006

    work page 2006

  46. [46]

    R. D. Falgout and U. M. Yang. hypre: A library of high performance preconditioners. InProceedings of the International Conference on Computational Science-Part III, ICCS ’02, pages 632–641, Berlin, Heidelberg, 2002. Springer-Verlag

    work page 2002

  47. [47]

    Feder, A

    M. Feder, A. Cangiani, and L. Heltai. R3MG: R-tree based agglomeration of poly- topal grids with applications to multilevel methods.Journal of Computational Physics, 526:113773, 2025

    work page 2025

  48. [48]

    Feder, L

    M. Feder, L. Heltai, and A. Cangiani. polyDEAL.https://github.com/fdrmrc/ Polydeal

    work page

  49. [49]

    Feder, L

    M. Feder, L. Heltai, P. Munch, and M. Kronbichler. Matrix-free implementation of the non-nested multigrid method.arXiv:2412.10910, 2024

    work page arXiv 2024

  50. [50]

    FitzHugh

    R. FitzHugh. Impulses and physiological states in theoretical models of nerve mem- brane.Biophysical Journal, 1(6):445–466, 1961

    work page 1961

  51. [51]

    P. C. Franzone, L. F. Pavarino, and S. Scacchi.Mathematical Cardiac Electrophysi- ology. MS&A – Modeling, Simulation and Applications. Springer International Pub- lishing, 2014

    work page 2014

  52. [52]

    M. Gee, C. Siefert, J. Hu, R. Tuminaro, and M. Sala. Ml 5.0 smoothed aggregation user’s guide. 01 2006

    work page 2006

  53. [53]

    Gerbi, L

    A. Gerbi, L. Dedè, and A. Quarteroni. A monolithic algorithm for the simulation of cardiac electromechanics in the human left ventricle.Mathematics in Engineering, 1(1):1–37, 2019

    work page 2019

  54. [54]

    Gervasio, F

    P. Gervasio, F. Saleri, and A. Veneziani. Algebraic fractional-step schemes with spectral methods for the incompressible Navier-Stokes equations.Journal of Com- putational Physics, 214(1):347–365, 2006

    work page 2006

  55. [55]

    A. Guttman. R-trees: a dynamic index structure for spatial searching. InProceed- ings of the 1984 ACM SIGMOD International Conference on Management of Data, SIGMOD ’84, page 47–57, New York, NY, USA, 1984. Association for Computing Machinery. 37

    work page 1984

  56. [56]

    M. A. Heroux, R. A. Bartlett, V. E. Howle, R. J. Hoekstra, J. J. Hu, T. G. Kolda, R. B. Lehoucq, K. R. Long, R. P. Pawlowski, E. T. Phipps, A. G. Salinger, H. K. Thornquist, R. S. Tuminaro, J. M. Willenbring, A. Williams, and K. S. Stanley. An overview of the trilinos project.ACM Transactions on Mathematical Software, 31(3):397––423, sep 2005

    work page 2005

  57. [57]

    J. M. Hoermann, C. Bertoglio, M. Kronbichler, M. R. Pfaller, R. Chabiniok, and W. A. Wall. An adaptive hybridizable discontinuous galerkin approach for cardiac electrophysiology.International Journal for Numerical Methods in Biomedical Engi- neering, 34(5):e2959, 2018

    work page 2018

  58. [58]

    N. M. M. Huynh, L. F. Pavarino, and S. Scacchi. Parallel Newton–Krylov BDDC and FETI-DP deluxe solvers for implicit time discretizations of the cardiac bidomain equations.SIAM Journal on Scientific Computing, 44(2):B224–B249, 2022

    work page 2022

  59. [59]

    Karypis and V

    G. Karypis and V. Kumar. A fast and high quality multilevel scheme for partitioning irregular graphs.SIAM Journal on Scientific Computing, 20(1):359–392, 1998

    work page 1998

  60. [60]

    Karypis, K

    G. Karypis, K. Schloegel, and V. Kumar. Parmetis: Parallel graph partitioning and sparse matrix ordering library. 1997

    work page 1997

  61. [61]

    Klawonn, O

    A. Klawonn, O. B. Widlund, and M. Dryja. Dual-primal FETI methods for three- dimensional elliptic problems with heterogeneous coefficients.SIAM Journal on Nu- merical Analysis, 40(1):159–179, 2002

    work page 2002

  62. [62]

    Kronbichler and K

    M. Kronbichler and K. Kormann. A generic interface for parallel cell-based finite element operator application.Computers & Fluids, 63:135–147, 2012

    work page 2012

  63. [63]

    Kronbichler and K

    M. Kronbichler and K. Kormann. Fast matrix-free evaluation of discontinuous Galerkin finite element operators.ACM Trans. Math. Softw., 45(3), Aug. 2019

    work page 2019

  64. [64]

    C. B. Leimer Saglio, S. Pagani, M. Corti, and P. F. Antonietti. A high-order discon- tinuous Galerkin method for the numerical modeling of epileptic seizures.Computers & Mathematics with Applications, 205:112–131, 2026

    work page 2026

  65. [65]

    C. B. Leimer Saglio, S. Pagani, and P. F. Antonietti. A p-adaptive polytopal dis- continuous Galerkin method for high-order approximation of brain electrophysiology. Computer Methods in Applied Mechanics and Engineering, 446:118249, 2025

    work page 2025

  66. [66]

    D. J. Mavriplis and V. Venkatakrishnan. A 3D agglomeration multigrid solver for the Reynolds-Averaged Navier-Stokes equations on unstructured meshes.International Journal for Numerical Methods in Fluids, 23(6):527–544, 1996

    work page 1996

  67. [67]

    S. A. Orszag. Spectral methods for problems in complex geometries.Journal of Computational Physics, 37(1):70–92, 1980. 38

    work page 1980

  68. [68]

    Pan and P.-O

    Y. Pan and P.-O. Persson. Agglomeration-based geometric multigrid solvers for compact discontinuous Galerkin discretizations on unstructured meshes.Journal of Computational Physics, 449:110775, 2022

    work page 2022

  69. [69]

    Pathmanathan, G

    P. Pathmanathan, G. Mirams, J. Southern, and J. Whiteley. The significant effect of the choice of ionic current integration method in cardiac electro-physiological sim- ulations.International Journal for Numerical Methods in Biomedical Engineering, 27:1751 – 1770, 11 2011

    work page 2011

  70. [70]

    L. F. Pavarino and S. Scacchi. Multilevel additive Schwarz preconditioners for the bidomain reaction-diffusion system.SIAM Journal on Scientific Computing, 31(1):420–443, 2008

    work page 2008

  71. [71]

    Piersanti, P

    R. Piersanti, P. C. Africa, M. Fedele, C. Vergara, L. Dede’, A. F. Corno, and A. Quar- teroni. Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations.Computer Methods in Applied Mechanics and Engineering, 373:113468, 2021

    work page 2021

  72. [72]

    Plank, M

    G. Plank, M. Liebmann, R. Weber dos Santos, E. J. Vigmond, and G. Haase. Al- gebraic multigrid preconditioner for the cardiac bidomain model.IEEE transactions on bio-medical engineering, 54(4):585—596, April 2007

    work page 2007

  73. [73]

    Algebraic Multigrid Methods For Virtual Element Discretizations: A Numerical Study

    D. Prada and M. Pennacchio. Algebraic multigrid methods for virtual element dis- cretizations: A numerical study.arXiv preprint arXiv:1812.02161, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  74. [74]

    Quarteroni, A

    A. Quarteroni, A. Manzoni, and C. Vergara. The cardiovascular system: Mathe- matical modelling, numerical algorithms and clinical applications.Acta Numerica, 26:365–590, 2017

    work page 2017

  75. [75]

    Riviere.Discontinuous Galerkin Methods For Solving Elliptic And Parabolic Equa- tions: Theory and Implementation, volume 35

    B. Riviere.Discontinuous Galerkin Methods For Solving Elliptic And Parabolic Equa- tions: Theory and Implementation, volume 35. 01 2008

    work page 2008

  76. [76]

    Rogers and A

    J. Rogers and A. McCulloch. A collocation-Galerkin finite element model of car- diac action potential propagation.IEEE Transactions on Biomedical Engineering, 41(8):743–757, 1994

    work page 1994

  77. [77]

    C. B. L. Saglio, S. Pagani, and P. F. Antonietti. A massively parallel non-overlapping Schwarz preconditioner for PolyDG methods in brain electrophysiology.arXiv preprint arXiv:2512.19536, 2025

    work page arXiv 2025

  78. [78]

    Schreiner and K.-A

    J. Schreiner and K.-A. Mardal. Simulating epileptic seizures using the bidomain model.Scientific Reports, 12(1):10065, 2022

    work page 2022

  79. [79]

    Strocchi, C

    M. Strocchi, C. M. Augustin, M. A. F. Gsell, E. Karabelas, A. Neic, K. Gillette, O. Razeghi, A. J. Prassl, E. J. Vigmond, J. M. Behar, J. Gould, B. Sidhu, C. A. Rinaldi, M. J. Bishop, G. Plank, and S. A. Niederer. A publicly available virtual cohort of four-chamber heart meshes for cardiac electro-mechanics simulations.PLOS ONE, 15(6):1–26, 06 2020. 39

    work page 2020

  80. [80]

    Sundar, G

    H. Sundar, G. Stadler, and G. Biros. Comparison of multigrid algorithms for high- order continuous finite element discretizations.Numerical Linear Algebra with Ap- plications, 22(4):664–680, 2015

    work page 2015

Showing first 80 references.