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Higher-order finite element de Rham complexes on sparse grids

Yihanqi Hu, Yuwen Li

arXiv:2609.27768v1math.NA

TL;DR

Higher-order conforming sparse-grid discretizations for differential forms remain limited, despite the need to preserve de Rham structure in vector-field problems. This paper constructs compatible Alpert-based hierarchies and proves commuting, exact, and stable sparse-grid properties; source experiments show predicted convergence, while corner singularities limit the benefit of higher degree for a square-annulus eigenproblem.

  • Problem

    Higher-order conforming sparse-grid methods that preserve de Rham structure are limited, while componentwise discretizations of H(curl) and H(div) can produce spurious solutions.

    Source: §1 · pp. 1–2

  • Method

    The paper tensorizes compatible one-dimensional continuous and discontinuous polynomial spaces linked by differentiation, using Alpert multiwavelets and integrated counterparts.

    Source: §1 · p. 2, §7 · p. 26

  • Results

    Source experiments exhibit the predicted convergence order, whereas reentrant-corner singularities substantially reduce the benefit of increasing polynomial degree for the square-annulus Maxwell eigenproblem.

    Source: §7 · p. 26

  • Takeaways & Limitations

    On the unit cube, an explicit sparse-grid-preserving homotopy proves exactness and yields potentials stable in grid level and polynomial degree, together with an H(d)-bounded commuting projection.

    Source: §7 · p. 26

    Reentrant-corner singularities limit Maxwell eigenspace regularity, and sparse approximation depends on mixed regularity, reducing the benefit of higher degree for the square-annulus problem.

    Source: §6.2 · p. 26

Abstract

from arXiv · show

We construct, for the first time, a family of higher-order finite element differential forms on tensor-product sparse grids. The construction starts from compatible one-dimensional spaces of continuous piecewise polynomials and discontinuous piecewise polynomials of one degree lower, linked by differentiation in each coordinate direction. Their hierarchical decompositions combine Alpert multiwavelets with their integrated counterparts. We establish commuting canonical interpolation operators and corresponding approximation error bounds under mixed Sobolev regularity. For the sparse-grid de Rham complex of arbitrary polynomial degree on the unit cube in arbitrary dimension, we also prove exactness, polynomial-degree-robust stable discrete potentials, and an H(d)-bounded commuting projection by developing a novel stable homotopy operator. Numerical experiments for curl-curl source problems on a cube and a Maxwell eigenproblem on a square-annulus illustrate the effectiveness of the proposed higher-order sparse-grid method.

1. Introduction

Tensor-product finite elements face a dimensionality barrier, while sparse-grid discretizations for differential forms remain limited at higher polynomial orders. The paper builds a higher-order sparse-grid de Rham framework with commuting interpolation, approximation estimates, exactness, stable potentials, and bounded commuting projections.

Motivation

  • Sparse grids reduce degrees of freedom to O(h^-1_n |log h_n|^(d-1)) while retaining full-grid approximation power up to logarithmic factors under sufficient mixed Sobolev regularity.Full tensor-product spaces instead have O(h_n^-d) degrees of freedom.Source: §1 · p. 1
  • Sparse-grid discretization of H(curl) and H(div) remains underdeveloped, and naive componentwise approaches can produce spurious solutions in curl-curl and eigenvalue problems.Existing de Rham sparse-grid constructions are restricted to lowest polynomial order because higher-order bases do not retain the hat–Haar derivative relation.Source: §1 · pp. 1–2

Construction

  • The construction pairs Alpert multiwavelets with integrated continuous detail functions whose derivatives are exactly the original Alpert functions.Tensorizing these compatible hierarchies produces conforming sparse-grid differential forms.Source: §1 · p. 2

Framework

  • The paper uses finite element exterior calculus because differential forms and traces extend naturally to the intended high-dimensional domains, and bounded commuting projections support stability.The projection is constructed using a stable homotopy operator and a tensor trick.Source: §1 · pp. 2–3

Contributions

  • Compatible canonical interpolants and hierarchical decompositions yield commuting sparse-grid interpolation and approximation error bounds under mixed Sobolev regularity.The continuous 1D hierarchy is new and differs from the hierarchy in.Source: §1 · p. 3
  • On the unit cube, a sparse-grid-preserving tensor-product homotopy proves exactness and yields polynomial-degree-robust stable potentials and an H(d)-bounded commuting projection.The construction applies to arbitrary dimension and form degree.Source: §1 · p. 3

2. Finite element differential forms and hierarchies

The section defines cubical differential-form spaces and develops compatible one-dimensional hierarchies for their sparse-grid construction. Canonical interpolation, orthogonal detail decompositions, and computed Alpert and integrated-Alpert bases provide the building blocks.

2.1. Cubical finite element forms

  • Cubical form spaces use degree k in form directions and degree k+1 in transverse directions, with face assembly enforcing the trace continuity required for HΛl conformity.Their tensor-product degrees of freedom are standard moments on faces and cell interiors.Source: §2.1 · pp. 4–5

2.2. Compatible one-dimensional hierarchy

  • In the one-dimensional de Rham complex, Sk+1_m is a 0-form space, and Qk_m dx is a 1-form space.Source: §2.2 · p. 5

2.2.1. Canonical interpolation and commutation

  • Endpoint values and interior moments define the canonical interpolants, whose derivative relation commutes and supplies the basis for later commuting relations.The discontinuous-space interpolant is the L2 projection.Source: §2.2.1 · p. 6

2.2.2. Alpert and integrated-Alpert hierarchies

  • The discontinuous detail spaces admit L2-orthogonal hierarchical decompositions, while integrated continuous partners provide the corresponding continuous hierarchy.The figure shows discontinuous functions and their integrated continuous partners.Source: §2.2.2 · pp. 6–8, Figure 1 · p. 8
  • The local hierarchy is built on parent cells by extending functions from child intervals by zero, giving the detail construction a cellwise structure.The local detail space has dimension k + 1.Source: §2.2.2 · p. 6

2.2.3. Computation of hierarchical basis

  • Reference Alpert detail functions are constructed as an orthonormal basis of a null space, then mapped to cells and paired with integrated partners.The reference null space is computed once for each polynomial degree.Source: §2.2.3 · pp. 8–9

3. Tensor-product interpolation and sparse truncation

The section builds tensor-product sparse-grid spaces of differential forms from directional detail spaces and defines interpolants that commute with the exterior derivative. Using the same sparse index set across form degrees makes the spaces a discrete de Rham subcomplex.

3.1. Tensor-product detail spaces.

  • The exterior derivative of the interpolated form agrees with interpolation of its derivative for sufficiently regular forms.The identity follows by applying the one-dimensional commuting relation in each coordinate and summing over form components.Source: §3.1 · p. 10
  • The tensor-product detail construction yields a basis for the scalar coefficient spaces and supports canonical interpolation of differential-form components.The canonical interpolant is associated with the geometric degrees of freedom.Source: §3.1 · pp. 9–11

3.2. Sparse spaces and commuting interpolation.

  • Using one common downward-closed index set in every form degree makes the sparse spaces a discrete de Rham subcomplex with commuting interpolants.Differentiation maps paired continuous and discontinuous factors at the same level, so it does not change the sparse index.Source: §3.2 · pp. 11–12

4. Sparse-grid approximation

The section derives sparse-grid approximation bounds for differential forms under mixed Sobolev regularity. The resulting rates depend on polynomial degree and logarithmic factors, with a more favorable degrees-of-freedom exponent than the stated full-grid rate.

4. Sparse-grid approximation.

  • The approximation analysis is stated for functions with mixed Sobolev regularity, whose mixed norm controls the derivatives arising in coordinatewise detail estimates.The text notes that this norm is slightly stronger than necessary.Source: §4 · p. 12, §4.1 · pp. 13–14

4.1. One-dimensional and tensor detail estimates.

  • Under the mixed Sobolev regularity considered, the full hierarchical expansion converges in L2.The stated mixed norm is stronger than necessary but controls the mixed derivatives used in the successive one-dimensional detail estimates.Source: §4 · p. 12, §4.2 · p. 14

4.2. Summation of the sparse tail.

  • Summing the sparse tail yields an error bound with a constant independent of grid level n.The section obtains the bound by summing component estimates and using the commuting property for the graph error.Source: §4.2 · pp. 15–16
  • Higher-order sparse-grid methods for differential forms converge approximately as O(N_dof^{-(k+1)}) modulo logarithmic factors, versus O(N^{-(k+1)/d}) for full-grid tensor-product finite elements, where N_dof is the number of DoFs.For mesh size h_n = 2^{-n} and a sufficiently regular l-form ω, the H(d) error is governed by O(h_n^{k+1}|log h_n|^l).Source: §4.2 · p. 16

5. Exactness and discrete stable potentials

A tensor-product homotopy provides the key tool for proving exactness and constructing stable potentials in sparse-grid finite element spaces. These results also yield a bounded commuting projection between the continuous and sparse discrete complexes.

5.1. A tensor-product homotopy.

  • Tensorizing the one-dimensional construction gives the homotopy identity dH + Hd = id − P on HΛ forms.The construction uses the tensor trick and extends from smooth forms by boundedness.Source: §5.1 · pp. 17, 19

5.2. Exactness and stable potentials.

  • The sparse-grid complex is exact, and the stable-potential constant 1/π is uniform in polynomial degree k.The homotopy preserves sparse-grid spaces, enabling the exactness and potential results.Source: §5.2 · pp. 19–21

5.3. Bounded cochain projection.

  • An H(d)-bounded commuting projection connects the continuous and sparse discrete complexes, yielding quasi-optimal convergence for sparse-grid mixed Hodge–Laplacian source problems.The stated applications include mixed Poisson and magnetostatics.Source: §5.3 · pp. 21–22

6. Numerical experiments

The experiments test predicted convergence on two- and three-dimensional curl-curl source problems and examine polynomial-degree effects on a square-annulus Maxwell eigenproblem.

6.1. Two- and three-dimensional source problems.

  • The source problems impose homogeneous tangential boundary conditions, and their manufactured fields are divergence free and satisfy the stated curl-curl identities.The variational problem uses the curl-curl bilinear form with an additional L2 term.Source: §6.1 · p. 23
  • Sparse-grid slopes are close to k + 1 and rise by about one per degree, while measured full-grid rates agree with (k + 1)/d.Figures 2 and 3 show the H(curl) convergence histories for the two source problems.Source: §6.1 · pp. 23–24, Figure 2 · p. 24, Figure 3 · p. 24

6.2. Maxwell eigenproblem on a square with a hole.

  • For the square-annulus test, eight rectangles are joined by tangential trace moments, and a full-grid computation with k = 3 and n = 7 supplies reference eigenvalues.The reported comparison concerns the first ten positive eigenvalues.Source: §6.2 · p. 25, Table 1 · p. 25
  • On the square annulus, fitted sparse-grid slopes are approximately 0.55–0.58 versus 0.64–0.68 for full grids; increasing k mainly lowers the error constant, not the slope.The authors relate the weak order dependence to limited eigenfunction regularity near reentrant corners and the mixed-regularity requirement of sparse approximation.Source: §6.2 · pp. 25–26, Figure 4 · p. 26

7. Concluding remarks

The paper constructs higher-order sparse-grid differential forms and proves key structural and stability properties on the unit cube. Source experiments exhibit predicted convergence, while corner singularities constrain degree benefits for the square-annulus eigenproblem.

7. Concluding remarks

  • Compatible Alpert and integrated-Alpert hierarchies yield conforming sparse-grid de Rham subcomplexes, commuting interpolation, and mixed-Sobolev approximation estimates.On the unit cube, a tensor-product homotopy proves exactness and gives degree- and level-robust stable potentials, hence an H(d)-bounded commuting projection.Source: §7 · p. 26
  • Source experiments exhibit predicted convergence, whereas reentrant-corner singularities limit eigenfunction regularity and reduce the square-annulus problem’s benefit from higher polynomial degree.Mapped geometries and adaptive sparse grids under low regularity are identified as future directions.Source: §7 · p. 26

Declaration of AI use

  • The authors report using ChatGPT to polish writing, generate code, and conduct computational experiments, and say they reviewed and edited its output.Source: Declaration of AI use · p. 27
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