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High order geometric methods with splines: an analysis of discrete Hodge--star operators

Bernard Kapidani, Rafael Vázquez

arXiv:2203.00620v2math.NAcs.NAphysics.comp-ph

TL;DR

Discrete differential-form methods face the challenge of defining discrete Hodge–star operators, with geometric approaches relying on dual meshes that can add complexity. This paper constructs a dual spline complex without explicitly building such a mesh and studies three discrete Hodge–star operators. For elliptic problems, one operator attains the standard Galerkin convergence order, while the other two lose one order; the method is restricted to single-patch domains.

  • Problem

    Defining discrete Hodge–star operators is a central issue, and geometric approaches use primal–dual mesh relationships that can require cumbersome dual-mesh construction.

    Source: §1 · p. 1

  • Method

    The method builds a dual spline complex by changing polynomial degree, without explicitly constructing a dual grid, and links primal and dual complexes using three discrete Hodge–star operators.

    Source: §1 · p. 3, §4.1.1 · p. 10, §4.1.2 · p. 11

  • Results

    The first operator attains the standard Galerkin convergence order for elliptic problems, while the other two reduce the expected order by one because the dual-complex splines have reduced degree.

    Source: §7 · p. 23

  • Takeaways & Limitations

    The spline construction avoids explicit dual-mesh realization and works in arbitrary dimension on tensor-product domains, including curved domains.

    Source: §7 · p. 22

    The method is restricted to a single patch; extending the degree-reduction construction to multiple patches is difficult because cross-patch regularity matches that of standard finite elements.

    Source: §7 · p. 23

Abstract

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A new kind of spline geometric method approach is presented. Its main ingredient is the use of well established spline spaces forming a discrete de Rham complex to construct a primal sequence $\{X^k_h\}^n_{k=0}$, starting from splines of degree $p$, and a dual sequence $\{\tilde{X}^k_h\}_{k=0}^n$, starting from splines of degree $p-1$. By imposing homogeneous boundary conditions to the spaces of the primal sequence, the two sequences can be isomorphically mapped into one another. Within this setup, many familiar second order partial differential equations can be finally accommodated by explicitly constructing appropriate discrete versions of constitutive relations, called Hodge--star operators. Several alternatives based on both global and local projection operators between spline spaces will be proposed. The appeal of the approach with respect to similar published methods is twofold: firstly, it exhibits high order convergence. Secondly, it does not rely on the geometric realization of any (topologically) dual mesh. Several numerical examples in various space dimensions will be employed to validate the central ideas of the proposed approach and compare its features with the standard Galerkin approach in Isogeometric Analysis.

1. Introduction.

Discrete differential-form methods encode topological structure, but geometric Hodge approaches can require cumbersome dual meshes and often have limited convergence or scope. This work introduces a spline-based dual complex without explicitly constructing a dual grid, and analyzes discrete Hodge–star operators for it.

1. Introduction.

  • Geometric Hodge approaches define discrete Hodge–star operators using geometric relationships between primal and dual meshes.Galerkin–Hodge approaches instead weakly satisfy the continuous equation against locally supported polynomial approximations.Source: §1 · pp. 1–2
  • Explicit dual-mesh construction can add algorithmic complexity and unknowns, while barycentric-refinement dual complexes are limited to two dimensions and do not extend to higher approximation orders.Circumcentric dual meshes yield diagonal Hodge–star matrices and fast methods, but error generally decreases at most linearly under refinement.Source: §1 · p. 2
  • Earlier isogeometric discrete differential forms provide spline de Rham complexes and Galerkin discretizations, while another approach uses the primal space as its dual and discretizes the codifferential.The introduction situates the proposed method among spline-based and differential-form methods, including, [44, 19], and.Source: §1 · pp. 2–3
  • The spline method defines a dual complex by changing polynomial degree, avoiding an explicitly constructed dual grid while retaining equal-dimensional pairing spaces.Its exterior derivatives are incidence matrices of a Cartesian grid, and three discrete Hodge–star operators are analyzed.Source: §1 · p. 3
  • The method is proved to achieve high-order convergence for elliptic problems with each of three operators, losing one order relative to a Galerkin method based on the primal complex.Numerical evidence also indicates that it has no spurious solutions for the Maxwell eigenvalue problem.Source: §1 · p. 3

2. Differential forms.

Differential forms provide the objects and operators used to express the paper’s geometric formulation. The section establishes their algebra, de Rham sequences, Hodge–star relation to weighted inner products, and mapping between parametric and physical domains.

2.1. Alternating forms.

  • Alternating k-forms are multilinear maps on k-tuples that change sign under odd permutations; in n dimensions, n-forms are generated by the volume form.Wedge products build higher-degree alternating forms, and repeated coordinate differentials vanish.Source: §2.1 · pp. 3–4

2.2. Differential forms, de Rham diagrams and the Hodge–star operator.

  • Differential k-forms are functions on a domain with alternating k-forms as values, represented as coefficient functions multiplying decomposable forms.The section assumes a contractible domain for its de Rham complexes.Source: §2.2 · pp. 4, 6
  • The wedge product combines differential forms into a form whose degree is the sum of the input degrees, while the exterior derivative raises degree by one and satisfies d∘d=0.For 0-forms, the exterior derivative collects their first derivatives as a 1-form.Source: §2.2 · p. 5
  • A positive, uniformly bounded weight defines the weighted L2 inner product, and the Hodge–star operator’s equivalence with an L2 inner product guides its discrete approximation.The weight can represent a Riemannian metric tensor; when it equals one, the inner product is the standard L2 inner product.Source: §2.2 · pp. 5–6
  • The exterior derivative links successive form spaces in the L2 de Rham complex, with a corresponding complex for forms subject to homogeneous boundary conditions.On a contractible domain, the complexes are exact; non-vanishing boundary conditions add a boundary integral to the integration-by-parts formula.Source: §2.2 · p. 6

2.3. Parametric domain, physical domain, and pullback operators.

  • Pullback operators map differential forms from the physical domain to the parametric domain and commute with the exterior derivative.For 0-forms, pullback composes with the parameterization; for n-forms, it also includes the determinant of its Jacobian.Source: §2.3 · pp. 6–7

2.4. Model problem.

  • The model problems include an elliptic problem for a k-form with homogeneous boundary conditions and an eigenvalue problem, with Maxwell eigenvalues of particular interest.For Maxwell’s eigenvalue problem, k=1 and the material parameters are magnetic reluctivity and electric permittivity.Source: §2.4 · p. 7

3. The B-spline complex of discrete differential forms.

The B-spline complex uses tensor-product splines to form discrete differential-form spaces connected by exterior derivatives. Its construction relies on degree-reduced spline spaces for derivatives and yields a de Rham complex with sparse incidence-matrix derivatives.

3. The B-spline complex of discrete differential forms.

  • The B-spline complex is built from tensor-product B-splines and defines isogeometric discrete differential-form spaces in parametric and physical domains.The physical-domain spaces are obtained using pullback operators.Source: §3 · p. 7, §3.3 · p. 9

3.1. Univariate B-splines.

  • A univariate B-spline space consists of piecewise polynomials of degree p, with continuity at each knot determined by its multiplicity.Each basis function has support on its local knot interval.Source: §3.1 · p. 7
  • When internal knot multiplicities are at most p, differentiating a spline in S^p(Ξ) produces a spline in the degree-reduced space S^(p−1)(Ξ′).The derivative is expressed using the normalized Curry–Schoenberg basis.Source: §3.1 · pp. 7–8

3.2. Multivariate tensor-product B-splines.

  • Multivariate B-splines are tensor products of univariate splines, and the construction uses mixed-degree spaces combining standard B-splines with Curry–Schoenberg splines.The knot vectors define a Cartesian mesh, assumed shape-regular and locally quasi-uniform.Source: §3.2 · p. 8

3.3. The B-spline complex.

  • The spline spaces of differential forms constitute a discrete de Rham complex, including spaces subject to vanishing boundary conditions.The spaces are formed using tensor-product splines and their degree-reduced counterparts.Source: §3.3 · pp. 8–9
  • The exterior derivatives of spline differential forms are represented by sparse incidence matrices with entries 1 and −1, defined on an auxiliary Cartesian Greville mesh.The auxiliary mesh need not be constructed in practice; in three dimensions the derivatives correspond to gradient, curl, and divergence.Source: §3.3 · p. 9

4. The discrete Hodge–dual complex.

The section constructs a dual spline sequence and connects it to the primal sequence to make the final system solvable.

4. The discrete Hodge–dual complex.

  • The section’s goal is to construct a dual spline sequence and connect the two sequences so the resulting system can be solved.The construction follows the primal spline de Rham complex introduced earlier.Source: §4 · p. 10

4.1. The dual spline complex.

The dual spline complex is built by lowering spline degree and exploiting primal-space continuity, with homogeneous boundary conditions enabling isomorphic primal–dual spaces. Tensorization extends the construction, and pairing matrices encode discrete integration by parts.

4.1.1. The one-dimensional dual complex.

  • In one dimension, the dual 0-form space uses the spline space of the primal 1-forms, and homogeneous boundary conditions make the relevant primal and dual spaces isomorphic.Their proxy fields are identical, while constructing an explicit isomorphism as a discrete Hodge-star operator is not immediate.Source: §4.1.1 · p. 10

4.1.2. The tensor-product dual complex.

  • Tensorization extends the construction to higher dimensions, defining the dual sequence with splines one degree lower on the same knot vector.The exterior derivative is well defined across the dual complex and represented by incidence matrices on an auxiliary control mesh.Source: §4.1.2 · pp. 10–11

4.1. The dual spline complex.

  • High spline continuity permits a dual complex without a dual mesh, using a degree change on the same knot-vector mesh, provided the primal starting space is C1 continuous.The construction is stated to work for any degree, regularity, and non-uniform knot vectors under that condition.Source: §4.1 · p. 10, §4.1.2 · p. 11

4.1.3. Pairing matrices.

  • Pairing matrices represent the discrete integration-by-parts relation and are invertible because the primal–dual pairing is stable.These pairing properties do not yet involve the metric.Source: §4.1.3 · p. 12

4.2. Discrete Hodge–star operators.

The paper defines three discrete Hodge-star choices that preserve the spline discretization’s high-order nature. Two are global; the third uses local projections to obtain a sparser operator.

4.2. Discrete Hodge–star operators.

  • Three discrete Hodge-star operators are proposed to preserve the high-order nature of the spline-based discretization.The first two choices are global, while the third is based on local projection operations.Source: §4.2 · p. 12

4.2.1. Global discrete Hodge operators.

  • The first two operators are global constructions induced by continuous Hodge-star properties and adapted from existing discrete Hodge operators.Their main drawback is their global nature.Source: §4.2 · p. 12, §4.2.1 · p. 12

4.2.2. A local discrete Hodge operator.

  • The local option composes the continuous Hodge star with a spline quasi-interpolant whose approximation properties are equivalent to those of the L2 projection.It uses local L2 projections rather than point interpolation, which is meaningful only for 0-forms.Source: §4.2.2 · p. 13
  • For the one-dimensional example, local integration uses central knot intervals and produces at most p + 2 nonzero entries per matrix row, with endpoint choices adjusted for reduced supports.The figure marks the central integration element and distinguishes contributing basis functions in the projection matrix from those in the starting space.Source: §4.2.2 · p. 14, Figure 1 · p. 14

4.3. The discrete method.

The discrete method seeks a primal-space solution using discrete Hodge-star operators. Its matrix system is square despite the non-Galerkin approximation, and the right-hand side can be projected consistently.

4.3. The discrete method.

  • The discrete problem seeks a solution in the primal space using one of the three proposed discrete Hodge-star constructions.The method also gives an analogous discrete eigenvalue problem.Source: §4.3 · pp. 14–15
  • In matrix form, the resulting system is square even though the approximation is not performed in Galerkin fashion.The square system follows from the dimensions of the dual complex spaces.Source: §4.3 · p. 14
  • The right-hand side must be a discrete form in the dual space, and an L2 projection into that space is an alternative that gives a consistent right-hand side.Source: §4.3 · p. 15

5. Error analysis of the elliptic problem.

The error analysis finds that the first discrete Hodge-star choice is equivalent to standard Galerkin under a consistent right-hand side, while the other two have reduced convergence order. For the latter pair, coercivity is proved for sufficiently fine meshes, although numerical tests suggest uniform coercivity.

5. Error analysis of the elliptic problem.

  • The analysis applies standard variational techniques and treats equal starting degree in every parametric direction for simplicity.Source: §5 · p. 15
  • With no right-hand-side consistency error, the first Hodge-star operator yields solutions equivalent to standard Galerkin and total error behaving like the interpolation error, despite different matrices.Source: §5 · p. 16
  • The second and third Hodge-star choices have reduced approximation order because they project onto spline spaces one degree lower than the primal sequence.Their analysis assumes no consistency error in the right-hand side.Source: §5 · pp. 16–17
  • For the second and third operators, coercivity is proved only when the mesh is sufficiently fine, while numerical tests suggest the bilinear form is uniformly coercive.Source: §5 · p. 18

6. Numerical results.

Numerical tests compare three discrete Hodge operators on elliptic problems and Maxwell eigenvalue problems. The global operator matches standard Galerkin convergence rates, while the projection-based alternatives show different convergence and sparsity trade-offs.

6. Numerical results.

  • The elliptic tests include a mapped unit-square domain with a=0.3, while the eigenvalue tests use a one-eighth ring and a deformed unit cube.The ring and deformed cube are used to examine spectral fidelity.Source: §6.1 · p. 19, Figure 2 · p. 19, §6.2 · p. 21

6.1. Elliptic boundary value problem.

  • For the mapped-domain elliptic problem, the first global operator converges as fast as standard Galerkin IGA, while the second and local operators usually lose one order.The standard Galerkin approach is algebraically derivable from the first approach, though the two are not computationally equivalent.Source: §6.1 · p. 19, Figure 3 · p. 20
  • With a smooth material-parameter perturbation, the three operators maintain the previous convergence rates without adding quadrature points.The test uses the same Gauss-Legendre quadrature rules as the preceding test.Source: §6.1 · p. 20, Figure 4 · p. 21
  • Under h-refinement, condition-number growth matches standard IGA asymptotically up to a constant factor, while the local projector yields sparser matrices by an h-independent factor.The sparsity comparison is for degree p=3 and maximum continuity.Source: §6.1 · p. 20, Figure 5 · p. 21

6.2. Second order eigenvalue problem.

  • The eigenvalue study targets both absence of spurious eigenfunctions and optimal 2p convergence in eigenvalue errors as measures of structure preservation and high-order accuracy.The tests use the Maxwell eigenvalue problem with k=1 and material parameters γ=β=1.Source: §6.2 · p. 21
  • The Galerkin-equivalent operator retains optimal eigenvalue convergence; the second loses orders at h^2(p-1), and the local operator loses more.Spectral correctness is numerically corroborated for the local operator on the one-eighth ring and deformed cube, although its projector does not commute with the sequence's differential operators.Source: Figure 6 · p. 22, §6.2 · p. 22

7. Conclusions and future work.

The proposed spline geometric method uses a de Rham complex and three discrete Hodge–star operators, without explicitly constructing a dual mesh. For an elliptic problem, the first operator matches standard Galerkin IGA's convergence order, while the other two lose one order.

7. Conclusions and future work.

  • The method uses spline spaces forming a de Rham complex and three discrete Hodge–star operators; it avoids explicit dual meshes and works in arbitrary dimensions.The first two operators are global, while the third is local and has a reduced sparsity pattern; exterior derivatives use Cartesian-mesh incidence matrices, including for curved domains.Source: §7 · p. 22
  • The figures compare projector choices using second-eigenvalue error on the eighth ring in 2D and twentieth-eigenvalue error on the unit cube in 3D.The eighth-ring case uses starting degree p=3 with C2 continuity; the unit-cube cases use p=4 and p=5 with maximum continuity.Source: Figure 7 · p. 23, Figure 8 · p. 23
  • For the elliptic problem, the first operator is equivalent to standard Galerkin IGA with the same convergence order; the other two lose one order.The reduced order for the latter operators is attributed to the lower degree of splines in the dual complex.Source: §7 · p. 23
  • The method is currently restricted to single-patch domains; multipatch and multimaterial cases need modifications because degree reduction alone cannot construct the dual complex.The authors expect the necessary modifications to be the same for multipatch and multimaterial domains.Source: §7 · pp. 23–24
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