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POLYDIM: A C++ library for POLYtopal DIscretization Methods

T0 review · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read PolyDiM is a new open-source C++ library that implements Virtual Element and other polytopal discretizations for PDEs in 2D and 3D on complex meshes.

desk verdict Software announcement with solid math but missing repository details; worth refereeing if the authors can make the artifact verifiable. read the letter →

arxiv 2505.14063 v1 pith:JVTEH6ZS submitted 2025-05-20 math.NA cs.NA

Stefano Berrone , Andrea Borio , Gioana Teora , Fabio Vicini This is my paper · ORCID
classification math.NAcs.NA
keywords librarypolydimadvanceddiscretizationmethodspolytopalsettingssupport
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

PolyDiM is a C++ library for solving partial differential equations on meshes made of polygons or polyhedra, not just standard triangles or cubes. The star of the library is the Virtual Element Method (VEM), which lets each mesh element have almost any shape, including shapes with many sides, concave corners, or hanging nodes. The paper describes the mathematical spaces the library uses, how it computes integrals over arbitrary polytopes by cutting them into triangles or tetrahedra, how it assembles the global linear system, and what solvers it can talk to (Eigen and PETSc).

The library is built on top of an existing geometry library called GeDiM, also written by the same team. PolyDiM provides several choices for the polynomial bases inside each element, including a monomial basis, an orthonormal basis for better conditioning at high order, and an inertial basis that is robust on badly-shaped elements. It also supports three flavors of VEM: the primal (single unknown), the mixed (velocity and pressure), and a divergence-free version that is useful for fluid problems like Stokes and Brinkman.

The paper shows many past and new examples: elliptic problems on concave meshes, elasticity (Cook's membrane), Navier-Stokes, Brinkman, mesh refinement and agglomeration, discrete fracture networks, and even 3D-1D coupling for root water uptake. Most of these tests come from the authors' earlier papers that used PolyDiM. The key practical point is that all examples are claimed to be available on GitHub, and there are Python and MATLAB interfaces, so the library is meant to be used by other researchers.

Extended reading notes

Core claim

The central claim is that PolyDiM is an open-source C++ library that implements polytopal discretization methods, especially VEM in 2D and 3D, with support for non-convex geometries, hanging nodes, Discrete Fracture Networks, and mixed-dimensional coupling (abstract and Sections 3 to 5). If true, it provides a reusable research and production tool for solving PDEs on complex polytopal meshes.

Load-bearing premise

The paper's claims depend on the actual availability and correctness of the released code. Specifically, the statement in Section 5 that 'All examples are fully implemented and publicly available on GitHub' must hold, and the version of the code must be capable of reproducing the convergence results shown in Figures 5, 9, 10, and similar. If the public repository is incomplete, outdated relative to the paper, or contains implementation bugs in the VEM projections, the central claim that PolyDiM is a robust tool is not established.

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Editorial analysis

A structured set of objections, weighed in public.

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

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper adds no fitted constants or invented entities. It depends on established VEM theory, which is cited rather than proved.

assumptions (4)
  • domain assumption Polytopal mesh decompositions consist of star-shaped polytopes satisfying standard regularity assumptions (Section 2).
    Invoked for the VEM error estimates and projection properties stated in Sections 3.1.2, 3.2.2, and 3.3.2.
  • standard math The local degrees of freedom are unisolvent and allow exact computation of the projections in Propositions 1, 2, and 3.
    The proofs are cited to [25,26,27,31,32,34], not given in the paper.
  • standard math The discrete bilinear forms satisfy the consistency and stability properties (Section 3.1.2) that imply the error bounds (21), (34), and (35).
    This is standard VEM theory from [26] and related references.
  • standard math The divergence-free VEM spaces yield pointwise or projected divergence-free solutions as in Remark 2.
    Based on [5,24,34].

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Pith. "Pith review of POLYDIM: A C++ library for POLYtopal DIscretization Methods." pith.science (2026). https://pith.science/paper/JVTEH6ZS

@misc{pith2026250514063,
  author       = {Pith},
  title        = {Pith review of: POLYDIM: A C++ library for POLYtopal DIscretization Methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JVTEH6ZS}},
  note         = {Machine review of arXiv:2505.14063}
}
read the original abstract

This paper introduces PolyDiM, an open-source C++ library tailored for the development and implementation of polytopal discretization methods for partial differential equations. The library provides robust and modular tools to support advanced numerical techniques, with a focus on the Virtual Element Method in both 2D and 3D settings. PolyDiM is designed to address a wide range of challenging problems, including those involving non-convex geometries, Discrete Fracture Networks, and mixed-dimensional coupling. It is integrated with the geometry library GeDiM, and offers interfaces for MATLAB and Python to enhance accessibility. Distinguishing features include support for multiple polynomial bases, advanced stabilization strategies, and efficient local-to-global assembly procedures. PolyDiM aims to serve both as a research tool and a foundation for scalable scientific computing in complex geometrical settings.

Figures

Figures reproduced from arXiv: 2505.14063 by the authors.

Figure 1
Figure 1. Degrees of freedom for k = 3 and d = 2. Left: DOFs associated with the local primal space V d h,k defined in (10). Center: Degrees of freedom for the local mixed spaces. Circles refer to the velocity space V d h,k(E), defined in (27), whereas squares refer to the pressure space Qd h,k(E). Right: Degrees of freedom for the local divergence-free spaces. Circles refer to the velocity space V d h,k(E), defined in (39), … view at source ↗
Figure 2
Figure 2. Representation of Polygons and Polyhedrons. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Sub-triangulation for a polygon (Left) and a polyhedron (right). green dots represent quadrature points related [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Left: Mesh cells coloured by mesh markers. Right: DOF cells coloured by problem markers. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Left: H1 -error e1 for the primal version of the method as the method order k varies in two-dimensions for the different approaches proposed in Section 3.1. Center: H1 -error for the primal version of the method as k varies in three-dimensions for the different approac…
Figure 6
Figure 6. Figure 6: Meshes characterized by concave elements and a copious number of hanging nodes taken from Discrete [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: A SUPG-stabilized VEM solution related to advection-dominated problem. Test described in [40]. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Test 2: Cook’s membrane. Undeformed (gray) and deformed (red) configurations. [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Left: Domain and the finest mesh used to solve the Navier-Stokes problem. Center: Behaviour of errors [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: Left: Velocity field computed by solving the Brinkman equation related to a coarse square mesh obtained [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Test described in [9]. Left: Mesh agglomeration in DFN. Right: DFN coloured by the numerical pressure. [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: Test described in [42]. Left: Mesh agglomeration performed in a mechanical component. Right: Computa￾tional domain coloured by the numerical pressure. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Refinement of the L-shaped domain using two different polygonal meshes. The example is detailed in [7]. [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: These examples are detailed in [43]. Left: Refinement of a three-dimensional L-shaped domain. Right: Refinement of a mechanical component. Different colours represent polyhedra with a different number of faces. It is well-known that using generic polytopal discrete sp…
Figure 15
Figure 15. Figure 15: The root water pressure across multiple phases of root growth. Left: After 80 days elapsed. Right: 160 days [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]

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