Adaptively-refinable polar-spline discrete differential forms: hierarchical construction, exactness, and applications

1 Introduction

• The construction addresses tensor-product limitations: refinement can increase degrees of freedom across the whole domain, while regular parameterizations and multi-patch interfaces constrain geometry and smoothness.The paper also notes that many patches may be needed to construct simple geometries.Source: §1 · pp. 1–2
• Hierarchical splines combine basis functions from multiple levels, but a valid de Rham sequence requires mesh conditions; sufficient conditions are known in two dimensions and general dimensions.The cited hierarchical constructions use a hierarchical mesh and basis selected according to basis-function supports.Source: §1 · p. 2
• Polar splines provide C^k continuity at the pole, and earlier polar spline differential-form spaces were shown to have the correct cohomological structure.The earlier work also provided numerical evidence for optimal Hodge–Laplace approximation and inf-sup stability.Source: §1 · p. 2
• The paper combines polar and hierarchical splines to construct locally refined polar spaces forming a discrete de Rham complex.It proves linear independence of the active basis functions and sufficient conditions for exactness; mesh constraints need only be checked away from the pole.Source: §1 · p. 2
• The paper introduces a reduced complex at the pole to analyze exactness and reports numerical experiments to confirm the theory and illustrate the construction.The outline identifies exactness as the main theoretical result and the numerical results as a later section.Source: §1 · p. 2

2 B-spline complexes

• The section specializes the hierarchical construction to classical polynomial B-splines and uses bivariate tensor products to form cochain complexes.The authors note that the construction and many results extend straightforwardly to multi-degree or Tchebycheffian splines.Source: §2 · p. 3

2.1 Univariate B-spline complexes

• Knot multiplicity is the number of repetitions of a knot value; the defined univariate B-spline bases span piecewise-polynomial spaces with at least C1 smoothness.The knot-vector assumptions restrict interior multiplicities and allow recovery of the degree from endpoint multiplicities.Source: §2.1 · p. 3
• The periodic angular B-spline space identifies θ = 0 with θ = 1, has smoothness νper there, and dimension mθ,per = mθ − νper − 1.Source: §2.1 · p. 3

2.2 Bivariate periodic B-spline de Rham complexes

• The periodic bivariate B-spline spaces define a de Rham complex whose basis sizes and cohomology are characterized in the section.The periodic domain is formed by identifying θ = 0 and θ = 1.Source: §2.2 · p. 4

3.1 The polar geometric map

• The polar map sends the edge ρ = 0 to a pole, where its Jacobian degenerates, while remaining invertible away from the pole.The section considers one pole, with double-pole settings noted as an analogous extension.Source: §3.1 · p. 5

3.2 Jets, local closedness/exactness, and reduced complexes at the pole

• For a complex reduced at the pole, locally closed 1-forms are locally exact there; under the stated subcomplex conditions, vanishing H2 transfers to the subcomplex.In particular, every 1-form in the subcomplex is locally exact at the pole.Source: §3.2 · p. 6

3.3 B-spline complex on a polar domain

• The intermediate B-spline complex is reduced at the pole and has H0 ≅ R, H1 = H2 = 0, but its forms may lack the regularity required for the L2 de Rham complex.In particular, its 2-forms are generally not L2-integrable on the polar domain.Source: §3.3 · pp. 7–8

3.4 Polar spline complex

• The polar spline spaces form a complex with locally linearly independent bases, every 1-form is locally exact at the pole, and its cohomology is H0 ≅ R and H1 = H2 = 0.Except for basis functions near the pole, the polar basis functions coincide with tensor-product B-spline forms.Source: §3.4 · pp. 9–10
• The polar spline spaces form an exact subcomplex of the L2 de Rham complex on Ω, as shown in.The total extraction operators are identical to those defined in.Source: §3.4 · p. 10

4.1.1 A general construction of hierarchical spline spaces

• The hierarchical k-form space is the span of generators selected from nested spline spaces according to the domain hierarchy and active index sets.The construction applies to both hierarchical B-spline and hierarchical polar-spline spaces.Source: §4.1.1 · p. 11, §4.1.2 · p. 13

4.1.2 Hierarchical B-spline and polar-spline complexes

• Under Assumption 1, the polar-spline spaces form a nested sequence for k = 0, 1, 2, supporting hierarchical refinement.The nestedness result is stated for the polar-spline spaces.Source: §4.1.2 · pp. 11–12
• Two-scale relations explicitly describe how polar-spline degrees of freedom relate across consecutive refinement levels.The construction applies the hierarchical selection mechanism to nested intermediate B-spline spaces and polar-spline spaces.Source: §4.1.2 · p. 13
• Applying the hierarchical construction to the nested spaces produces B-spline and polar-spline differential-form complexes.The hierarchical polar-spline complex is assembled from its hierarchical spaces of 0-, 1-, and 2-forms.Source: §4.1.2 · p. 13

4.2 Linear independence of the generating functions

• In HP0, only three functions, all from level ℓ, have nonzero values and first derivatives; in HP1, only two level-ℓ functions have nonzero values.All functions in HP2 have zero values. Except for the identified level-ℓ functions, every other function in HPk is identical to some tensor-product B-spline k-form in HBk,int, for k = 0, 1, 2.Source: §4.2 · p. 14
• Assumptions 1–2 ensure the hierarchical polar-spline generators are linearly independent and form bases for HPk, k = 0, 1, 2.The same assumptions establish the corresponding basis result for the hierarchical intermediate B-spline spaces.Source: §4.2 · pp. 14–15
• The linear-independence result for hierarchical polar splines relies on local linear independence of the single-level spaces.The theorem states the resulting basis property under Assumptions 1–2.Source: §4.2 · pp. 14–15

5.1 Exactness of the complex HB•,int

• The shortest-chains assumption requires a chain of inactive B-splines between minimally intersecting inactive splines, with consecutive indices differing by one.The index condition is interpreted periodically for the angular direction.Source: §5.1 · pp. 15–16
• Assumption 3 requires that, at any level, inactive B-splines sharing a minimal intersection have a shortest chain of inactive B-splines between them, with endpoint index-pair distance r − 1 and successive index-pair distance 1.Source: §5.1 · pp. 15–16
• The intermediate hierarchical B-spline complex is reduced at the pole, a property used in the proof for the polar-spline complex.The paper identifies this pole behavior as simplifying the subsequent exactness argument.Source: §5.1 · p. 16

5.2 Exactness of the complex HP•

• For HP•, the zero-th cohomology is one-dimensional and the second cohomology vanishes.Closed 0-forms are constant, accounting for the one-dimensional zero-th cohomology.Source: §5.2 · p. 17
• Given Assumptions 1–3, the hierarchical polar-spline complex HP• is exact.The intermediate hierarchical B-spline complex is part of the argument establishing this result.Source: §5 · p. 15, §5.2 · p. 17

6 Numerical results

• Across the examples, the polar domain approximates a disk; exact circular domains using generalized Tchebycheffian splines are outside this work's scope.Source: §6 · p. 18

6.1 Harmonics

• The deliberately cohomology-breaking refinement produces spurious results and visible accuracy loss, unlike the modified cohomology-preserving mesh.The test compares the two meshes using pθ = pρ = 3 and forcing f = dx + dy.Source: §6.1 · p. 18, Figure 2 · p. 19

6.2 Maxwell eigenvalue

• No spurious eigenmodes appear under local refinement, and both TE and TM modes are correctly approximated; geometric error eventually limits eigenvalue convergence.The locally refined mesh refines only some elements adjacent to the pole.Source: §6.2 · p. 20, Figure 3 · p. 20

6.3 n-Form Hodge Laplacian

• Locally refined polar splines converge faster than globally refined polar splines in the pre-asymptotic regime by concentrating degrees of freedom near the exact solution's steep gradient.Because the forcing is non-trivial near the pole, every nearby element is refined multiple times.Source: §6.3 · p. 21

7 Conclusions

• The construction combines polar spline differential forms with hierarchical selection of active basis functions to enable local refinement on polar domains.It extends the polar spline de Rham complex to the adaptively-refinable setting and accommodates the polar singularity.Source: §7 · p. 21
• Nested polar spline k-form spaces and locally linearly independent bases support linear independence of the hierarchical basis and the proof of exactness.A reduced complex isolates behavior near the polar point to establish exactness.Source: §7 · p. 22
• Elements surrounding the pole may be refined freely without breaking cohomology, provided the known tensor-product hierarchy conditions hold away from the pole.The stated conditions impose no constraints near the pole.Source: §7 · pp. 22–23
• The construction uses extraction operators on periodic tensor-product B-splines and was implemented in Mantis.jl, supporting incorporation into existing isogeometric software with little effort.Source: §7 · p. 23
• The numerical experiments confirm the theory and illustrate the construction’s practical behavior.Source: §7 · p. 23
• The present construction is limited to planar disk-like domains with a single polar point; double-pole and sphere-like geometries are identified as extensions.The paper also identifies generalized spline spaces as a possible extension.Source: §7 · p. 23

A Extraction operators defining Bk,int

• For k = 0, 1, 2, each intermediate space is a subspace of Bk,per defined using an extraction operator Ek,int.Source: A Extraction operators defining Bk,int · p. 23
• The notation assumes the polar map of (10) and uses Ia for an identity matrix of size a and 0a×b for a zero matrix of size a × b.Source: A Extraction operators defining Bk,int · p. 23
• The intermediate-space dimensions are n0,int = mθ,per(mρ − 1) + 1, n1,int = 2mθ,per(mρ − 1), and n2,int = mθ,per(mρ − 1).Source: A Extraction operators defining Bk,int · p. 23

A.2 Definition of E1,int

• E1,int is defined as a matrix of size n1,int × mθ,per(2mρ − 1).Source: §A.2 · p. 23

• E2,int is the identity matrix of size n2,int × n2,int.Source: §A.3 · p. 23

B Extraction operators defining Pk

• Pk is represented as a subspace of Bk,int, with extraction matrices Ek defined for k = 0, 1, 2.Source: B Extraction operators defining Pk · p. 23
• The dimensions are n0 = n0,int − mθ,per + 2, n1 = n1,int − 2mθ,per + 2, and n2 = n2,int − mθ,per.Source: B Extraction operators defining Pk · p. 23

B.1 Definition of E0

• E0 is defined as a block-diagonal matrix.Source: §B.1 · p. 24

B.2 Definition of E1

• E1 is defined as a block-diagonal matrix.Source: §B.2 · p. 24

B.3 Definition of E2

• E2 is defined as a matrix.Source: §B.3 · p. 24

C Definition of level-ℓk-form polar basis functions

• At levels ℓ > 0, control points are recursively updated, and the extraction-matrix construction is repeated with those points.The level-0 control points are assumed given.Source: C Definition of level-ℓk-form polar… · p. 24