Nonparametric face-based finite element methods on complex hybrid three-dimensional meshes

Aubin Brunel; J. -C. Latch{é}

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Nonparametric face-based finite element methods on complex hybrid three-dimensional meshes

Aubin Brunel, J. -C. Latch{é}

arXiv:2610.06208v1Oct 5, 2026math.NAcs.NA

TL;DR

Curved faces can make the standard finite element scheme's consistency error uncontrolled and prevent convergence. The paper constructs nonparametric face-based spaces and uses a DG-like formulation for curved-face meshes; the schemes achieve optimal rates on planar-face meshes, while the DG-like terms restore convergence with curved faces.

Problem
On meshes with curved faces, the standard finite element scheme's consistency terms cannot be properly controlled, and experiments show nonconvergence as the mesh is refined.
Source: §2 · p. 6, §2.2 · p. 10
Method
The paper builds nonparametric local approximation spaces with face-based degrees of freedom and adds DG-like consistency and stabilization terms for curved-face meshes.
Source: §2 · p. 6, §5 · p. 18
Results
The schemes achieve optimal convergence rates on planar-face meshes, while DG-like consistency and stabilization terms restore convergence for curved-face meshes.
Source: §5 · p. 18
Takeaways & Limitations
The Poisson schemes are shown to work on arbitrary polytopal meshes, with curved faces requiring the DG-like formulation for convergence.
Source: §5 · p. 18
For parametric counterparts on prismatic and pyramidal cells, the stated analysis requires P1(K) inclusion, which holds only for affine mappings and thus severely restricts those mappings.
Source: §3.3 · p. 15

Abstract

from arXiv · show

In this work, we propose and analyze low-order nonconforming face-based finite element methods for elliptic problems on complex hybrid three-dimensional meshes, composed of arbitrary polytopal cells. The focus is on scalar problems, but the approach can be extended in a tensor manner to design generalizations of the classical Crouzeix--Raviart and Rannacher--Turek methods for mixed formulations. We adopt a nonparametric construction of the local approximation spaces. We demonstrate that it avoids the degradation of approximation properties commonly induced by parametric mappings on distorted meshes. Using these approximation spaces, we define finite element schemes for very general cells. We further demonstrate that, on strongly degraded meshes, an approach inspired by discontinuous Galerkin methods enables removing consistency errors, even in the presence of potentially curved faces. Explicit examples of the construction of the polynomial space are given for prismatic (of triangular basis) and pyramidal (of quadrilateral basis) cells. The resulting schemes are shown theoretically and numerically to be stable and convergent, with an optimal convergence rate.

1. Space discretization and nonconforming finite element space

The section defines the mesh geometry and regularity assumptions, then constructs a nonconforming finite element space using cellwise polynomials and face-mean degrees of freedom. It also establishes broken norms and jump notation for describing functions across cells.

1.1.1. Definition

A mesh is a finite partition of Ω into convex polygonal or polyhedral cells; in 3D, a quadrilateral face may be curved when its four vertices are not coplanar.Faces are internal when shared by two cells and external when they lie on the domain boundary.Source: §1.1.1 · p. 3

1.1.2. Mesh regularity

The mesh assumptions control cell and face distortion, while star-shapedness supports trace and interpolation estimates.The star-shapedness condition may be relaxed with additional regularity assumptions and slightly less sharp estimates.Source: §1.1.2 · pp. 3–4

1.2. Continuous and discrete functional spaces

Each cell uses a unisolvent polynomial space between P1(K) and P2(K), with one degree of freedom per face given by its mean value.The global space enforces shared face means but is generally not pointwise continuous; in 3D, the P2(K) restriction limits cells to at most 10 faces.Source: §1.2 · pp. 4–5

1.3. Discrete semi-norms and other notations

Broken Sobolev seminorms measure cellwise regularity, while the jump and face-mean notations describe interface behavior.For globally H^k functions, the broken seminorm agrees with the usual H^k seminorm.Source: §1.3 · p. 5

2. Discrete schemes for the Poisson problem

The paper presents a nonconforming finite-element scheme for meshes without curved faces and a DG-like modification for meshes with curved faces. Under mesh regularity assumptions, the analyses establish optimal convergence for the respective schemes.

2. Discrete schemes for the Poisson problem

The paper uses the standard FE scheme when faces are non-curved and adds consistency and stabilization terms in a DG-like formulation when curved faces are present.For the curved-face formulation, the consistency term is symmetrized and a stabilization term is added to recover coercivity.Source: §2 · p. 6, §2.2 · p. 10

2.1. A FE scheme for meshes with non-curved faces

On meshes with non-curved faces, the finite-element scheme achieves optimal convergence under the stated mesh-regularity assumptions.The error estimate applies to an H2 weak solution and its discrete solution.Source: §2.1 · p. 6

2.1.1. Interpolation error estimate

An interpolation operator Π from H1(Ω) to XM provides an interpolation-error estimate for H2 functions, with its constant depending on mesh regularity.Source: §2.1.1 · p. 7

2.1.2. Consistency error estimate

Although XM is nonconforming and the FE scheme is inconsistent, the consistency error can be controlled on meshes without curved faces.The analysis uses trace and inverse estimates whose constants depend on mesh regularity.Source: §2.1.2 · pp. 7–8

2.2. A DG-like scheme for meshes with curved faces

On meshes with curved faces, the DG-like scheme is consistent and recovers the optimal order of convergence.The error estimate assumes an H2 weak solution and a mesh satisfying the stated regularity assumptions.Source: §2.2 · p. 11

3. A methodology to build a finite element on different types of cells

The section constructs nonparametric, face-based polynomial spaces for general cells by designing them on reference cells and mapping the basis to real cells. It gives explicit constructions for hexahedra, quadrilateral pyramids, and triangular prisms.

3. A methodology to build a finite element on different types of cells

The proposed additional polynomial sets are non-unique, and their conditioning was checked numerically rather than established theoretically.The selection was guided by the numerical conditioning of the associated rigidity matrices.Source: §3 · p. 11
The method chooses a reference-cell polynomial space, maps its basis to the real cell, and computes shape functions there from face-based degrees of freedom.The space is designed to contain P1(K), remain within P2(K), and have dimension equal to the number of cell faces, supporting unisolvency.Source: §3 · pp. 11–12
A reference-cell unisolvent element motivates the construction; when the affine approximation is sufficiently close to the real cell, unisolvency there is expected but not formally proved.The authors describe the real-cell system as a perturbation of the affine-approximation system and cite numerical experiments as evidence of robustness.Source: §3 · p. 12
The construction differs from parametric methods by building shape functions on each real cell, adding a per-cell linear-system solve whose cost is described as usually negligible.For time-dependent problems, the computed shape functions can be stored after the initial time step; computations are independent across cells.Source: §3 · pp. 12–13

3.1. Hexahedral case

For hexahedra, the Rannacher–Turek polynomial space is unisolvent on the reference cube, with the real-cell coordinate map determined from the ordered vertices.The map's matrix columns are defined from the vertices of a similarly ordered hexahedron.Source: §3.1 · p. 13

3.2. Quadrilateral pyramidal case

For quadrilateral pyramids, the authors specify an unisolvent reference-pyramid polynomial space and a linear coordinate map for the five-vertex cell.The reference pyramid has a quadrilateral base and an apex.Source: §3.2 · p. 13

3.3. Triangular prismatic case

For triangular prisms, the authors specify an unisolvent reference-prism polynomial space and a linear coordinate map for the six-vertex cell.The reference prism's vertices are given explicitly.Source: §3.3 · p. 14
Parametric counterparts retain P1(K) only under affine mappings, restricting pyramids to parallelogram bases and prisms to extruded triangles; non-affine mappings are associated with approximation degradation.The paper reports the degradation theoretically and numerically.Source: §3.3 · p. 15

4. Numerical results

The numerical tests recover the theoretical optimal orders on meshes with non-curved faces, while curved faces expose a consistency failure in the classical scheme. A DG-like formulation restores convergence on several curved-face mesh families, though hybrid meshes retain a slower H1 rate.

4. Numerical results

The Poisson convergence test uses the domain Ω=[0,1]3 with homogeneous Dirichlet boundary conditions and right-hand side f := −∆u.Source: §4 · p. 15

4.1. Test on meshes with non-curved faces

The tests compare regular and distorted mesh sequences across hexahedral, prismatic, pyramidal, and hybrid cells; hybrid meshes combine these cell families in separate extruded subdomains.Source: §4.1 · pp. 15–16
On hexahedral, prismatic, pyramidal, and hybrid meshes with non-curved faces, both the H1 semi-norm and L2 norm recover the optimal theoretical orders.Source: §4.1 · p. 16

4.2. Test on meshes with curved faces

On curved-face hexahedral, prismatic, and pyramidal meshes the classical scheme does not converge; the DG-like formulation recovers optimal orders, but hybrid meshes require γ=10 and retain a slightly subunit H1 rate.The hybrid H1 rate also saturates slightly on refined meshes.Source: §4.2 · p. 17

5. Conclusion

The paper extends nonparametric face-based finite elements to arbitrary polytopal meshes and analyzes Poisson schemes, including a DG-like correction for curved-face inconsistency. It reports optimal rates for planar faces and identifies challenges for future staggered Stokes schemes.

5. Conclusion

The framework extends Crouzeix–Raviart and Rannacher–Turek elements to arbitrary polytopal cells using face degrees of freedom defined solely by face means.Source: §5 · p. 18
Local approximation spaces are built directly on real cells to include low-order polynomials, recovering optimal interpolation error estimates under minimal geometric assumptions.Source: §5 · p. 18
For Poisson problems on arbitrary polytopal meshes, the scheme has optimal rates with planar faces, while DG-like consistency and stabilization terms restore convergence for curved faces.The classical scheme is non-convergent with curved faces because of its lack of consistency.Source: §5 · p. 18
Explicit local-space constructions include triangular prismatic and quadrilateral pyramidal cells.Source: §5 · p. 18
Staggered Stokes extensions remain future work, requiring an inf-sup condition and careful velocity-pressure coupling; curved faces and possible checkerboard modes pose additional design challenges.Source: §5 · pp. 19–20

Appendix A. Building three-dimensional cells

The appendix constructs vertex-determined mappings from geometric reference cells to physical three-dimensional cells. It specifies conformity requirements for triangular and quadrilateral faces and the basic ingredients used to build the mappings.

Appendix A. Building three-dimensional cells

The mapping from a reference cell to a physical cell is determined entirely by the physical cell’s vertices, which are typically the geometric data supplied by mesh-generation tools.Source: Appendix A · p. 21
Mappings should be affine on regular cells and define shared faces consistently across neighboring cells, regardless of their cell types.Source: Appendix A · p. 21
A triangular face is represented as an affine triangle, while a quadrilateral face is represented as a bilinear surface independent of its neighboring cell types.Source: Appendix A · p. 21
The mapping construction uses a simple geometric reference cell, a low-order function space when possible, and vertex-associated degrees of freedom with shape functions taking Kronecker-delta values at reference vertices.Source: Appendix A · pp. 21–22

A.1. Mapping for tetrahedal cells

The tetrahedral construction uses the classical P1 space, and its mapping satisfies Criterion A.1 and is affine by definition.

A.1. Mapping for tetrahedal cells

The tetrahedral construction adopts the classical P1 construction.Source: §A.1 · p. 22
Its geometric polynomial space is span {1, bx, by, bz}.Source: §A.1 · p. 22
The mapping satisfies Criterion A.1 and is affine by definition.Source: §A.1 · p. 22

A.2. Mapping for hexahedral cells

The reference-cube table presents its vertices and shape functions.

A.2. Mapping for hexahedral cells

The table lists vertices and shape functions for the reference cube.Source: Table 2 · p. 22

A.3. Mapping for quadrilateral pyramid cells

For quadrilateral pyramids, the construction uses a nonpolynomial Bedrosian-function space and a mapping derived from the reference cube.

A.3. Mapping for quadrilateral pyramid cells

The pyramid construction adopts the nonpolynomial Bedrosian-function space, although alternatives exist.Source: §A.3 · p. 22
The pyramid shape functions are derived from reference-cube Q1 functions through a non-bijective mapping to the reference pyramid.Source: §A.3 · p. 22
The mapping satisfies Criteria A.1 and A.2, and is affine when the quadrilateral face is a parallelogram.Source: §A.3 · p. 22

A.4. Mapping for prismatic cells

The prismatic construction uses a six-function geometric space and a mapping that satisfies Criteria A.1 and A.2.

A.4. Mapping for prismatic cells

The prismatic geometric space is span {1, bx, by, bz, bxby, bxbz}.Source: §A.4 · p. 23
The prism mapping is built using a non-bijective mapping from the reference cube to the reference prism.Source: §A.4 · p. 23
The mapping satisfies Criteria A.1 and A.2, and is affine when the triangular faces lie in parallel planes.Source: §A.4 · p. 23

Appendix B. A trace lemma on possibly curved faces

Appendix B adapts a trace-lemma proof to curved faces using truncated cones, without requiring cells to be star shaped. The resulting bound relies on geometric regularity conditions involving cell anisotropy and face curvature.

Appendix B. A trace lemma on possibly curved faces

The curved-face trace lemma adapts the proof in without assuming star-shaped cells, using truncated cones instead.The full cone need not lie inside the cell; the truncated cone must be contained in it.Source: Appendix B · pp. 23, 25
For α in (0, 1), the lemma applies when a truncated cone from the face to an exterior point lies inside the cell, yielding an inequality for H1 functions and ε > 0.The face is represented through a C1 mapping in the lemma's stated setup.Source: Appendix B · p. 23
The proof uses a density argument and maps face parameters and the cone parameter onto the truncated cone before applying a Jacobian bound and Cauchy–Schwarz.The derivation then bounds the resulting terms using the cone's geometric quantities.Source: Appendix B · pp. 24–25
Obtaining a comparable trace bound requires fixing α and ε and bounding d, D, and a; the regularity constants control cell anisotropy and face curvature.Star-shapedness with respect to a point in the cell is sufficient for the cone condition, but is not necessary.Source: Appendix B · p. 25