High order biorthogonal functions for the discrete de Rham complex on simplices*

1 Introduction

• Basis choice is an important hp-FEM ingredient because it directly influences the condition number of the involved matrices.The introduction also cites condition number and sparsity as properties affected by basis choices in particular settings.Source: §1 · p. 2
• The paper modifies existing basis constructions so the basis of the second sequence contains that of the first, allowing spaces to be switched by enabling or disabling degrees of freedom.The authors state that the construction applies to the other spaces of the de Rham complex in two and three dimensions.Source: §1 · p. 4
• The authors choose linear-combination factors depending on the polynomial indices i, j, and k to obtain the nested bases.They contrast this with, where the factors are chosen independently of those indices.Source: §1 · p. 4
• The paper extends the construction of biorthogonal functions from H(curl) to H(div), with functions biorthogonal to the BDM interior basis.The construction principle is similar to [26].Source: §1 · pp. 4–5
• The paper introduces new basis functions for second-kind Nédélec and BDM spaces, separating first-kind Nédélec functions from H1 gradients and RT functions from solenoidal fields, respectively.Source: §1 · p. 5

2.1 Jacobi polynomials

• Jacobi polynomials are introduced for parameters α, β > −1 with weight ω(x) = (1 − x)^α(1 + x)^β.The paper gives particular attention to β ∈ {0, 1}.Source: §2.1 · pp. 5–6
• The section records relations among Jacobi, integrated Jacobi, and Legendre polynomials for use later in the paper.It notes that integrated Legendre and Jacobi polynomials can also be written as Jacobi polynomials.Source: §2.1 · pp. 5–7
• Integral identities are justified using polynomial orthogonality and symmetry, with separate cases depending on the indices.The text identifies the case i > k and the case where k − i is odd.Source: §2.1 · p. 7

3. In the remaining case, Lemma 2.1 gives

• Lemma 2.1's remaining case is handled using Jacobi orthogonality and integration by parts.The proof then applies the results for α = 1 and α = 2.Source: §3 · p. 8

2.2 Basis functions

• The element-based H(curl) functions on the reference triangle are developed from H1 basis functions by taking partial derivatives.The tetrahedral construction is described as similar.Source: §2.2 · pp. 8–9
• For the case using weights α_i = 2i, the mass and stiffness matrices have an optimal sparsity pattern with respect to the number of nonzero entries.Source: §2.2 · p. 9
• Redefining type II to allow disabling the highest polynomial degree yields the Nédélec basis of first kind.The passage identifies this as the modification made in the following section.Source: §2.2 · p. 9
• The H(div) construction takes curls of H(curl) basis types II, III, and IV, then completes them with Raviart–Thomas functions; removing the highest degree yields the BDM space.The passages describe the tetrahedral construction and the degree reduction for the BDM space.Source: §2.2 · p. 10

3 New H(curl) basis functions on a triangle and a tetrahedron

• The modified gradient construction keeps the basis hierarchical, but its interior functions do not split into homogeneous and non-homogeneous parts as required for the first kind.That mismatch motivates modifying the functions to obtain Nédélec conformity.Source: §3 · pp. 10–11

3.1 Triangle

• The modified triangle functions belong to the first Nédélec space Rp−1 when their indices satisfy i ≥ 2 and j ≥ 1.The construction addresses that the earlier triangle functions were not Nédélec-conforming.Source: §3 · p. 10, §3.1 · p. 11

3.2 Tetrahedron

• For the tetrahedron, a polynomial combination is a first-kind Nédélec function when c1i + c2j + c3k = 0.The condition follows from the relation between the radial product and the polynomial combination.Source: §3.2 · p. 14
• The constructed tetrahedral interior functions form a basis of the second-kind Nédélec space; using the type-I functions only through degree p−1 yields the first-kind functions.The theorem states the result for indices with i + j + k ≤ p.Source: §3.2 · p. 15

4 New H(div) basis functions on a tetrahedron

• The proposed Raviart–Thomas functions use coefficients i, j, and k for i ≥ 2, with a separate construction for i = 1.The theorem describes the resulting polynomial form and bounds the remainder's degree for the two cases.Source: §4 · pp. 16–17
• The system matrix is sparse, with O(p^3) nonzero entries for choices (22), but its constant exceeds that of optimal choice (27); Jacobi weights 2i − 1 and 2i + 2j − 2 are applied.Source: §4 · pp. 20–21
• Full biorthogonality requires modifying the Jacobi weights, and the next section develops closed-form biorthogonal functions for the constructed basis.The passage compares the resulting system-matrix sparsity with the optimal choice.Source: §4 · pp. 20–21

5 Biorthogonal functions

• The auxiliary basis functions form an upper triangular system, so their biorthogonal functions are obtained from a lower triangular system.The third dual function can be chosen directly because only the corresponding auxiliary function has a third component.Source: §5 · p. 23
• Theorem 5.3 establishes biorthogonality between the auxiliary basis functions and their duals, with scaling constants specified for types I, II, and III.The scaling is c_ijkm = 16(2i−1)(2i+2j−1)(2i+2j+2k−1).Source: §5 · p. 28
• Low-order dual functions are constructed to be orthogonal to the auxiliary basis functions, with explicit pairings reported for types I and IV.The construction uses Legendre-polynomial orthogonality and the Duffy transformation.Source: §5 · pp. 29–30
• The dual functions for the original interior basis are defined using adjusted Jacobi weights and are biorthogonal to the basis functions defined by (28).The stated Jacobi-weight choice is minimal.Source: §5 · pp. 30–32

• The indicated exponents in (29) would imply a reduction of the weight-function exponents in (37).Source: §5 · p. 32
• That reduction would introduce poles, making the biorthogonal functions rational; to the authors’ knowledge, no biorthogonal polynomial is available in closed form in this case.Source: §5 · p. 32
• Using the sparsity-optimal choice with (44) leads to a very sparse system.Source: §5 · p. 32