Structure-preserving Discretization of the Hessian Complex

Structure-preserving Discretization of the Hessian Complexbased on Spline Spaces

Jeremias Arf and Bernd Simeon ∗

TU Kaiserslautern, Germany

AbstractWe want to propose a new discretization ansatz for the second order Hessian complex

exploiting benefits of isogeometric analysis, namely the possibility of high-order conver-gence and smoothness of test functions. Although our approach is firstly only valid indomains that are obtained by affine linear transformations of a unit cube, we see in theapproach a relatively simple way to obtain inf-sup stable and arbitrary fast convergentmethods for the underlying Hodge-Laplacians. Background for this is the theory of Fi-nite Element Exterior Calculus (FEEC) which guides us to structure-preserving discretesub-complexes.

1 IntroductionThe Hessian complex is a so-called Hilbert complex that pops up in different fields like nu-merical relativity ([13]) or also as underlying complex for a biharmonic problem; see [12].A more famous Hilbert complex in numerical mathematics and physics is the first order deRham complex

R H1(Ω) H(Ω, curl) H(Ω, div) L2(Ω)∇⊂ ∇× ∇· 0

0 ,

which can be used for problems in electromagnetics; see [4]. It has a connection to Maxwell’sequations; compare [2, section 8.6]. An elegant way of discretizing latter Hilbert complexby setting up a finite-dimensional subcomplex can be achieved through the theory of FEECdeveloped mainly by Arnold, Falk and Winther; see e.g. [1]. One of the basic ideas behindFEEC is the usage of test function spaces which are compatible with the complex in thesense that we have projections onto the finite-dimensional spaces that commute with thedifferential operators. In his book [2] Arnold introduces in detail how one can constructFinite Element (FEM) spaces fulfilling the commutation and other properties and in whatway they lead to stable and convergent numerical variational formulations for different kindof equations, e.g. the Hodge-Laplacians. Underlying are the spaces of polynomial differentialforms. Fortunately, the results of FEEC are quite general and the framework is applicablefor every closed Hilbert complex and other types of discrete spaces. For example Buffa et al.

∗E-mail: arf, [email protected],2020 Mathematics Subject Classification. 65N30; 65N12 ,Keywords. Finite Element Exterior Calculus, B-splines, Hodge-Laplacian, Hilbert complexes,structure-preservation.

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presented in the paper [4] a procedure of discretizing for the de Rham complex using splinespaces that satisfy the main aspects of FEEC.Here in this article we want to continue the idea of combining isogeometric analysis (IGA)and FEEC within the scope of numerical methods for the example of the Hessian complex.In other words we adapt the approach in [4] for the case of the Hessian complex and orientourselves very closely towards latter reference.A main reason for our studies is the search for a stable and convergent numerical method

for the Linearized Einstein Bianchi System (LEBS) in numerical relativity. For a derivationof the LEBS we refer the reader to the thesis [13] of Quenneville-Bélair and the referencestherein. Further, the author of [13] uses the concept of FEEC for the numerical computationof solutions to the LEBS, too. Hence we use a similar mind walk since we are also lookingfor structure-preserving discretizations using the results of FEEC. But whereas Quenneville-Bélair uses polynomial de Rham complexes, we exploit isogeometric analysis for the definitionof test function spaces. Because of the possibility to increase the smoothness of splines easilywe are able to discretize the original Hessian complex with its required C1 regularity of thetest functions due to fact that H2-Sobolev spaces are involved. Furthermore, as Quenneville-Bélair pointed out in his thesis, the version of the LEB system as a part of the Hessian complexguarantees automatically some special features of the physics behind the equations. Namely,suitable symmetry and trace properties are fulfilled, or preserved, respectively. Thus one ofthe outcomes of this article is the achievement of a stable high-order convergent method forthe Hessian complex that is feasible for an application in the context of numerical relativity.However, the needed restriction to affine linear parametrizations for the proposed methoddemonstrates the meaningfulness of generalizations. Especially the study of the Hessian com-plex on geometries with curved boundaries is of current interest for the authors.

We also want to mention that the idea of using splines for Hilbert complexes like presentedin [4] should also be applicable in the context of other tensor complexes, e.g. the elasticity-complex; see [12] or [2, Chapter 8].

The paper is structured as follows. In Section 2 we introduce mathematical notation andbasic notions in the context of isogeometric analysis as well as for Hilbert complexes. Af-terwards, we define a discrete Hessian complex using splines. Then we face approximationestimates for quantifying the goodness of the discretization. In Section 5 we introduce twoapplication examples, namely the Hodge-Laplacian and the LEBS. In the last Section 6 wedisplay some numerical tests for checking the convergence statements established before inSection 4.

2 Mathematical preliminaries and notation

2.1 Mathematical notation

In this section we introduce some notation and define several spaces.Given some bounded Lipschitz domain D ⊂ R3 we write for the standard Sobolev spacesH0(D) = L2(D), Hk(D), k ∈ N, where L2(D) stands for the Hilbert space of square-integrable functions endowed with the inner product 〈·, ·〉L2(D). The norms (|·|Hk(D)), ‖·‖Hk(D)denote the classical Sobolev (semi-)norms in Hk(D). In case of vector- or matrix-valued func-tions we can define Sobolev spaces, too, by requiring the component functions to be in suitable

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Sobolev spaces. To distinguish latter case from the scalar-valued one, we use a bold-type no-tation. For example we have for v := (v1, . . . , vd), v ∈ Hk(D) : ⇔ vi ∈ Hk(Ω), ∀i andM :=

(Mij

)3i,j=1, M ∈ H

k(D) : ⇔ Mij ∈ Hk(Ω), ∀i, j and define the norms

‖v‖2Hk(D) :=∑i

‖vi‖2Hk(D) , ‖M‖2Hk(D) :=∑i,j

‖Mij‖2Hk(D) .

Analogously we can proceed in case of the semi-norms. We note that the inner product〈·, ·〉L2(D) introduces straightforwardly an inner product on L2(D). For the definition of thenext spaces and norms we follow partly [12] to introduce further notation. First, let us considervector-valued mappings. Then we set

H(D, curl) := v ∈ L2(D) | ∇ × v ∈ L2(D), ‖v‖2H(D,curl) := ‖v‖2L2(D) + ‖∇ × v‖2L2(D) ,

H(D,div) := v ∈ L2(D) | ∇ · v ∈ L2(D), ‖v‖2H(D,div) := ‖v‖2L2(D) + ‖∇ · v‖2L2(D) .

Above we wrote ∇ for the classical nabla operator and later will write ∇2 for the Hessian. Thedefinitions for H(D, curl), H(D,div), can be generalized to the matrix setting by requiringthat all the rows (as vector-valued mappings) are in the respective spaces. Here, the curl ∇×and divergence ∇· act row-wise, too. Furthermore, we denote the subspace of symmetric andtraceless matrix-valued functions by

L2(D,S) := M ∈ L2(D) | MT = M, L2(D,T) := M ∈ L2(D) | tr(M) = 0,

and set

H(D, curl,S) := H(D, curl) ∩ L2(D,S), H(D,div,T) := H(D,div) ∩ L2(D,T).

Besides we define

H(D, symcurl,T) := T ∈ L2(D,T) | ∇ ×T + (∇×T)T ∈ L2(D),H(D,divdiv, S) := S ∈ L2(D,S) | ∇ · ∇ · S ∈ L2(D),‖M‖2H(D,divdiv) := ‖M‖2L2(D) + ‖∇ · ∇ ·M‖2L2(D) ,

‖M‖2H(D,symcurl) := ‖M‖2L2(D) +∥∥∥∇×M + (∇×M)T

∥∥∥2

L2(D).

Then writingC∞c (D) for the space of smooth compact supported vector-valued, matrix-valuedrespectively, functions, one can introduce some spaces with zero boundary conditions in thesense

Hk0(D) := C∞c (D)Hk(D)

,H(D,divdiv,S) := C∞c (D) ∩ L2(D,S)

H(D,divdiv),

H(D, symcurl,T) := C∞c (D) ∩ L2(D,T)

H(D,symcurl),

where we write XY for the closure of the space X w.r.t. the norm ‖·‖Y .Next we define some abbreviations. If we have arbitrary functions tj , si : D → R, i =

1, . . . , 6, j = 1, . . . , 8 and ml : D → R, l = 1, . . . , 9, we define the corresponding symmetric

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and traceless matrix functions through

SYM(s1, . . . , s6) : D → R3×3, SYM(s1, . . . , s6) =

s1 s2 s3s2 s4 s5s3 s5 s6

,

TR(t1, . . . , t8) : D → R3×3, TR(t1, . . . , t8) =

t1 t2 t3t4 t5 − t1 t6t7 t8 −t5

,

MAT(m1, . . . ,m9) : D → R3×3, MAT(m1, . . . ,m9) =

m1 m2 m3m4 m5 m6m7 m8 m9

.

For operators oi : X → Y, i = 1, . . . , 9 we can define analogously the matrix operators, forexample

MAT(o1, . . . , o9) : X3×3 → Y 3×3, MAT(o1, . . . , o9)(m1, . . . ,m9) =

o1(m1) o2(m2) o3(m3)o4(m4) o5(m5) o6(m6)o7(m7) o8(m8) o9(m9)

.

Further, for a matrix M = (Mij) we use an upper index M j to denote the j-th column and alower indexMi for the i-th row. Then, for matrix-valued mapping, we write for the deviatoricgradient dev∇ and the symmetric curl operator sym∇×.From functional analysis we know the Hilbert space adjoint for a densely-defined linear

operator T : W 1 →W 2, W 1,W 2 Hilbert spaces. It is the linear mapping T ∗ : W 2 →W 1 forwhich 〈Tx, y〉W 2 = 〈x, T ∗y〉W 1 , ∀x ∈ D(T ), y ∈ D(T ∗), where the angle brackets stand forthe inner product and D(L) denotes the domain of some operator L.After stating some basic notation we proceed with the consideration of Hilbert complexes,B-splines and spaces involving splines.

2.2 Hilbert complexes

The following definitions and explanations are based on the references [2, 12].A Hilbert complex is a chain of Hilbert spaces (W k, 〈·, ·〉Wk) together with closed and denselydefined linear operators dk : W k → W k+1, where one requires the range R(dk) of dk to bea subset of the nullspace N (dk+1) of dk+1. Hence dk+1 dk = 0. For our purposes we aremainly interested in the so-called domain complex (V k, dk), where the Hilbert spaces W k arereplaced by the dense domains V k of the operators dk, i.e. we have a sequence

dk−2. . . V k−1 V k V k+1 . . . .dk−1 dk dk+1

Using the graph inner product with induced graph norm ‖v‖2V k := ‖v‖2Wk +∥∥∥dkv∥∥∥2

Wk+1,

we obtain with (V k, 〈·, ·〉V k) again Hilbert spaces and thus the domain complex is indeed aHilbert complex. We call a Hilbert complex closed if the ranges R(dk) are closed in V k+1

and we denote the domain complex exact, if R(dk) = N (dk+1). Another important notion isthe dual complex which is built up by means of the adjoint operators d∗k := (dk−1)∗. Moreprecisely, the dual complex of the domain complex has the form

d∗k−1. . . V ∗k−1 V ∗k V ∗k+1 . . . ,d∗k d∗k+1 d∗k+2

where the V ∗k indicate the domains of the adjoint operators.

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Now we tend to the Hessian complex on which we focus in this article. It is the domaincomplex

Definition 1. (Hessian complex)

H2(Ω) H(Ω, curl,S) H(Ω, div,T) L2(Ω)∇2 ∇× ∇· 0ιP1 0,P1(Ω)

derived from the Hilbert complex

L2(Ω) L2(Ω,S) L2(Ω,T) L2(Ω)∇2 ∇× ∇· 0ιP1 0 .P1(Ω)

Above the ιP1 stands for the inclusion map, P1(Ω) denotes the linear polynomial space andΩ ⊂ R3. Further we make here the assumption:

Assumption 1. Ω is a bounded and simply connected Lipschitz domain with connected bound-ary.

And the dual complex has the form

L2(Ω)H(Ω,divdiv,S)

H(Ω, symcurl,T) H1

0(Ω)(∇·)2 sym∇× −dev∇ 0πP1 0 ,P1(Ω)

where the circles should indicate that the domains of the dual operators are subspaces ofL2(Ω) with suitable zero boundary conditions. Further, we wrote πP1 for the L2-orthogonalprojection onto the linear polynomial space.One can show that the mentioned Hessian complex is a closed and exact complex. For a

proof of the exactness we refer to Theorem 3.3 in [6]. And since the exactness also impliesthat the Hessian sequence is closed ( cf. Theorem 3.8 or section 4.1 in [2] ) we will be able toset up a discrete Hessian complex based on spline spaces to approximate several PDEs. Tohave the mathematical notation available for defining such a finite-dimensional version of thecomplex we face some very basic definitions from the field of isogeometric analysis in the nextsection.

2.3 Spline spaces

Here, we state a short overview of B-spline functions, spaces respectively, and some basicresults in the univariate as well as in the multivariate case.Following [5, 3] for a brief exposition, we call an increasing sequence of real numbers Ξ :=ξ1 ≤ ξ2 ≤ · · · ≤ ξn+p+1 for some p ∈ N knot vector, where we assume 0 = ξ1 = ξ2 = · · · =ξp+1, ξn+1 = ξn+2 = · · · = ξn+p+1 = 1, and call such knot vectors p-open. Furthermore,the multiplicity of the j-th knot is denoted by m(ξj). Then the univariate B-spline functionsBj,p(·) of degree p corresponding to a given knot vector Ξ are defined recursively by theCox-DeBoor formula :

Bj,0(ζ) :=

1, if ζ ∈ [ξj , ξj+1)0, else,

and if p ∈ N≥1 we set

Bj,p(ζ) := ζ − ξjξj+p − ξj

Bj,p−1(ζ) + ξj+p+1 − ζξj+p+1 − ξj+1

Bj+1,p−1(ζ),

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where one puts 0/0 = 0 to obtain well-definedness. The knot vector Ξ without knot repeti-tions is denoted by ψ1, . . . , ψm.The multivariate extension of the last spline definition is achieved by a tensor product con-struction. In other words, we set for a given knot vector Ξ := Ξ1 × · · · × Ξd, where theΞl = ξl1, . . . , ξlnl+pl+1, l = 1, . . . , d are pl-open, and a given degree vector p := (p1, . . . , pd)for the multivariate case

Bi,p(ζ) :=d∏l=1

Bil,pl(ζl), ∀ i ∈ I, ζ := (ζ1, . . . , ζd),

with d as the underlying dimension of the parametric domain Ω = (0, 1)d and I the multi-index set I := (i1, . . . , id) | 1 ≤ il ≤ nl, l = 1, . . . , d.B-splines fulfill several properties and for our purposes the most important ones are:

• If for all internal knots the multiplicity satisfies 1 ≤ m(ξj) ≤ m ≤ p, then the B-splinebasis functions Bi,p(ξ) are globally Cp−m-continuous. Therefore we define in this casethe regularity integer r := p−m. Obviously, by the product structure, we get splines Bi,pwhich are Crl-smooth w.r.t. the l-th coordinate direction if the internal multiplicitiesfulfill 1 ≤ m(ξlj) ≤ ml ≤ pl, rl := pl − rl, ∀l in the multivariate case. We write in thefollowing r := (r1, . . . , rd), for the regularity vector to indicate the smoothness. In caseof ri < 0 we have discontinuous splines w.r.t. the i-th coordinate direction.

• The B-splines Bi,p | i ∈ I are linearly independent.

• For univariate splines Bi,p, p ≥ 1 we have

∂ζBi,p(ζ) = p

ξi+p − ξiBi,p−1(ζ)− p

ξi+p+1 − ξi+1Bi+1,p−1(ζ), (1)

with B1,p−1(ζ) := Bn+1,p−1(ζ) := 0.

• The support of the spline Bi,p is a subset of the interval [ξi, ξi+p+1]. Moreover, the knotsψj define a subdivision of the interval (0, 1) and for each element I = (ψj , ψj+1) we findan i with (ψj , ψj+1) = (ξi, ξi+1) and write I := (ξi−p, ξi+p+1) for the so-called supportextension.

The space spanned by all univariate splines Bi,p corresponding to given knot vector and degreep and global regularity r is denoted by

Srp := spanBi,p | i = 1, . . . , n.

For the multivariate case we just define the spline space as the product space

Sr1,...,rdp1,...,pd

:= Sr1p1 ⊗ · · · ⊗ S

rdpd

= spanBi,p | i ∈ I

of proper univariate spline spaces.To define discrete spaces based on splines we require a parametrization mapping F : Ω :=(0, 1)d → Rd which parametrizes the computational domain. In fact we will assume in thesubsequent parts that F is an affine linear map and hence smooth. The knots stored in the knotvector Ξ, corresponding to the underlying splines, determine a mesh in the parametric domain

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Ω, namely M := Kj := (ψ1j1 , ψ

1j1+1)× · · · × (ψdjd , ψ

djd+1) | j = (j1, . . . , jd), with 1 ≤ ji < mi,

and with Ψ = ψ11, . . . , ψ

1m1 × · · · × ψ

d1 , . . . , ψ

dmd as the knot vector Ξ without knot

repetitions. The image of this mesh under the mapping F, i.e. M := F(K) | K ∈ M,gives us a mesh structure in the physical domain. By inserting knots without changing theparametrization we can refine the mesh, which is the concept of h-refinement; see [10, 5, 3].For a meshM we define the global mesh size h := maxhK | K ∈ M, where for K ∈ M wedenote with hK := diam(K) the element size.

Assumption 2. (Regular mesh)There exists a constant cu independent from the mesh size such that hK ≤ h ≤ cu hK for allmesh elements K ∈M.

After the introduction of elementary notions we face now the spline-based discretization forthe Hessian complex.

3 Structure-preserving discretization for the Hessian complexHere we consider a Lipschitz domain Ω ⊂ R3 as computational domain for the Hessian complexdefined in Section 2.2 (Def. 1). As already mentioned we assume the parametrization mappingto be affine linear, i.e. F : (0, 1)3 → Ω , ζ 7→ A · ζ + b, A ∈ R3×3 invertible, b ∈ R3.Aim of this section is the establishment of discrete spaces for the Hessian complex which

fulfill the structure-preserving properties of the FEEC framework; see [2, Section 5.2.2].To construct an appropriate discretization we follow the approach in [4] for the case of thede Rham complex. In fact, most of the proofs and results are very similar to the ones in thelatter reference, but are stated for reasons of completeness and since there arise differences insome points. We have to be careful mainly due to the fact that we consider a second-ordercomplex in contrast to the first order sequence in [4].First, we define three mappings which connect the spaces in the reference cube, i.e. in theparametric domain Ω, with the spaces in the physical domain Ω. More precisely, we have thediagram

H2(Ω) H(Ω, curl, S) H(Ω,div,T) L2(Ω)∇2 ∇× ∇· 0ιP1 0P1(Ω)

H2(Ω) H(Ω, curl, S) H(Ω,div,T) L2(Ω)∇2 ∇× ∇· 0ιP1 0 ,P1(Ω)

Y1 Y1 Y2 Y3 Y4 id

whereY1(φ) := φ F,Y2(S) := JT (S F) J,Y3(T) := det(J)JT (T F)J−T ,Y4(v) := det(J)JT (v F),

(2)

with ∇ denoting the nabla operator w.r.t. to the parametric coordinates and (Jij) := J := DFdenoting the Jacobian of the parametrization mapping. The above transformations Yi arecompatible with the Hessian complex as follows.

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Lemma 1. The latter diagram commutes, i.e.

∇2 Y1 = Y2 ∇2, (∇×) Y2 = Y3 (∇×), (∇·) Y3 = Y4 (∇·) .

Proof. We show the three equations separately.

1. Let Fk be the k-th component function of F. By the chain rule of the hessian operator(see [15, Corollary 1]) we obtain directly

∇2(Y1(φ)) = ∇2(φ F) = JT((∇2φ) F

)J +

∑k

(∇2Fk)∂kφ

= JT((∇2φ) F

)J = Y2(∇2φ).

2. Here we assume that S is a symmetric 3×3 matrix-valued function. Then for the secondequation in the assertion we first note the relation1

∇ ×[(a1, a2, a3) ·C(ζ)

]=∑i

[∇ai × Ci + ai ∇ × Ci

],

where (a1, a2, a3) ∈ R3 is some row vector, C(ζ) a matrix-valued function and Ci thei-th row of the matrix C(ζ). The last equation can be simplified to

∇ ×[(a1, a2, a3) ·C(ζ)

]=∑i

[ai ∇ × Ci

]. (3)

Using (3) and setting C := (S F)J we can write

∇ ×[JT (S F)J

]= ∇ ×

[J11 J21 J31J12 J22 J32J13 J23 J33

C]

=

∇ × (J1 ·C)∇ × (J2 ·C)∇ × (J3 ·C)

=

∑i Ji1 ∇ × Ci∑i Ji2 ∇ × Ci∑i Ji3 ∇ × Ci

.Then the next application of the chain rule for the covariant Piola transfomation ∇ ×(JT (v F)) = det(J)J−1(∇× v) F (see e.g. 2.15 and 2.17 in [9]) yields

∇ × CTi = ∇ × ((S F)i · J)T = ∇ × (JT · (S F)i)= det(J)J−1((∇× Si) F).

This and the symmetry of S makes it possible to write

∇ ×[JT (S F)J

]=

∑i Ji1 det(J)((∇× Si) F)J−T∑i Ji2 det(J)((∇× Si) F)J−T∑i Ji3 det(J)((∇× Si) F)J−T

= det(J)JT ((∇× S) F) J−T

= Y3(∇× S).1we use the well-known generalized product rule ∇×

(a v)

= ∇a × v + a∇× v.

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3. For the last part of the proof we note that for a vector-valued function (a1, a2, a3)T anda matrix-valued function C it is

∇ ·((a1, a2, a3) ·C(ζ)

)=∑i

Ci · (∇ai) + ai (∇ · Ci).

This means with the definition C := det(J) (M F)J−T , where M is a matrix-valuedfunction, we obtain

∇ ·(Y3(M)

)= ∇ ·

(det(J)JT (M F)J−T

)=

∑i Ji1 ∇ · Ci∑i Ji2 ∇ · Ci∑i Ji3 ∇ · Ci

.In view of the divergence preserving Piola transformation ∇ ·

(det(J)J−1(v F)

)=

det(J) (∇ · v) F (compare 2.15 and 2.18 in [9]), it is

∇ · Ci = ∇ ·(det(J)(M F)J−T )i = ∇ ·

(det(J)J−1 (MT F)i)

= det(J)(∇ ·Mi) F.

And hence one gets

∇ ·(Y3(M)

)= det(J)JT

((∇ ·M) F

)= Y4(∇ ·M).

An important feature of the mappings is the symmetry and trace preservation. We havethe next simple lemma.

Lemma 2. The mapping Y2 is symmetry-preserving and Y3 preserves the zero trace.

Proof. Let S be a symmetric matrix ,then

Y2(S)T =(JT (S F)J

)T = JT (S F)TJ = JT (S F)J = Y2(S).

Furthermore, since similar matrices have the same traces, the mapping Y3 maps tracelessmatrices to traceless matrices. More precisely, two matrices A, B are called similar, if thereexists an invertible matrix C with A = CBC−1. Then it holds tr(A) = tr(B). Consequentlywe have for T ∈ H(Ω,div,T) the equality chain

tr(Y3(T)) = tr(det(J)JT (T F)J−T ) = det(J) tr(JT (T F)J−T ) = 0.

Before exploiting the above transformations, we come back to splines and introduce auxil-iary spline function spaces on the parametric domain. They will be used later to introducediscrete spaces on the actual domain Ω.

Definition 2. (Discrete spaces in the parametric domain)As already defined we denote with Sr1,r2,r3

p1,p2,p3 the scalar-valued spline space with global regularityCri w.r.t. the i-th coordinate. Clearly, this parametric spline space depends on the underlyingmesh structure, which we assume to have the global mesh size h. Here we require the regularity

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to be greater or equal to one, i.e. ri ≥ 1. Then we can define the parametric test functionspaces

V 0h := P1(Ω) ⊂ Sr1,r2,r3

p1,p2,p3 ,

V 1h := Sr1,r2,r3

p1,p2,p3 ⊂ H2(Ω),

V2h := SYM(s1, . . . , s6) | (s1, . . . , s6) ∈(Sr1−2,r2,r3p1−2,p2,p3 × S

r1−1,r2−1,r3p1−1,p2−1,p3 × S

r1−1,r2,r3−1p1−1,p2,p3−1 × S

r1,r2−2,r3p1,p2−2,p3 × S

r1,r2−1,r3−1p1,p2−1,p3−1 × S

r1,r2,r3−2p1,p2,p3−2),

V3h := TR(t1, . . . , t8) | (t1, . . . , t8) ∈

(Sr1−1,r2−1,r3−1p1−1,p2−1,p3−1 × S

r1−2,r2,r3−1p1−2,p2,p3−1 × S

r1−2,r2−1,r3p1−2,p2−1,p3 × S

r1,r2−2,r3−1p1,p2−2,p3−1

× Sr1−1,r2−1,r3−1p1−1,p2−1,p3−1 × S

r1−1,r2−2,r3p1−1,p2−2,p3 × S

r1,r2−1,r3−2p1,p2−1,p3−2 × S

r1−1,r2,r3−2p1−1,p2,p3−2),

V4h := Sr1−2,r2−1,r3−1

p1−2,p2−1,p3−1 × Sr1−1,r2−2,r3−1p1−1,p2−2,p3−1 × S

r1−1,r2−1,r3−2p1−1,p2−1,p3−2.

Remark 1. Due to the product structure of the splines it is easy to see that the differentialoperators ∇2, ∇× and ∇·match with our parametric spline spaces in the sense that ∇2(V 1

h ) ⊂V

2h, ∇×(V 2

h) ⊂ V 3h and further ∇·(V 3

h) ⊂ V 4h. This is an easy consequence of (1). Moreover,

looking at the regularity property of splines in Section 2.3, we have indeed V 1h ∈ H2(Ω), V 2

h ⊂H(Ω, curl,S), V 3

h ⊂ H(Ω, div,T). Besides, we want to remark that the space P1(Ω) is asubspace of every spline space if we have pi ≥ 1. Hence we can ignore the discretization stepfor the first space in the Hessian complex chain; see Def. 1.In the next step we define the discrete spaces in the physical domain by means of the

mappings Yi; see (2).

Definition 3. (Discrete spaces in the physical domain)The test spaces in the physical domain are defined by

V 0h := p | Y1(q) ∈ P1(Ω),V 1h := φh | Y1(φh) ∈ V 1

h ,

V 2h := Sh | Y2(Sh) ∈ V 2

h,

V 3h := Th | Y3(Th) ∈ V 3

h,

V 4h := Mh | Y4(Mh) ∈ V 4

h.

In particular due to the commutativity property of the mappings Yi in Lemma 1 we obtainthe discrete commuting diagram in Fig. 1 .

V 1h V 2

h V 3h V 4

h

∇2 ∇× ∇· 0ιP1 0P1(Ω)

V1h V

2h V

3h V

4h

∇2 ∇× ∇· 0ιP1 0P1(Ω)

Y1 Y1 Y2 Y3 Y4 id

Figure 1: The transformations Yi are used to relate the test spaces in parametric and physical domain.

10

For the subsequent considerations we use the abbreviations

V 1 := H2(Ω), V 1 := H2(Ω),

V 2 := H(Ω, curl,S), V2 := H(Ω, curl,S),

V 3 := H(Ω, div,T), V3 := H(Ω, div,T),

V 4 := L2(Ω), V4 := L2(Ω).

(4)

Besides we write V 0 := P1(Ω) and V 0 := P1(Ω).Aim for the next part is to show the existence of bounded projection operators Π1

h : V 1 →V 1h , Πi

h : V i → V ih, i > 1, which satisfy suitable approximation estimates. As the approach

in [4], we first look at the parametric case, i.e. we define projections of the form

Π1h : V 1 → V 1

h , Πih : V i

→ Vi

h.

Underlying for the projection construction are projections onto spaces of univariate splines.Hence the approach exploits the product structure of our spline spaces. We define threedifferent projections, where two of them are the same as in the paper [4, Section 3.1.2]. Theseprojection operators are the basis for the projection definitions in the multivariate case.Definition 4. For p1 − 1 ≥ r1 ≥ −1, p2 − 1 ≥ r2 ≥ 0 and p3 − 1 ≥ r3 ≥ 1 we set

Πp1 : L2((0, 1))→ Sr1p1 , v 7→

∑i

(λp1i v)Bi,p1 , analogous to (3.6) in [4],

Πc,1p2−1 : L2((0, 1))→ Sr2−1

p2−1, v 7→d

dζΠp2

∫ ζ

0v(τ) dτ, analogous to (3.9) in [4],

Πc,2p3−2 : L2((0, 1))→ Sr3−2

p3−2, v 7→d

dζΠc,1p3−1

∫ ζ

0v(τ) dτ = d2

dζ2 Πp3

∫ ζ

0

∫ τ

0v(r) dr dτ.

Above λpi denote the canonical dual basis functionals corresponding to the spline basis func-tions, i.e. λpi (Bj,p) = δij. For more information we refer to [14, Section 4.6].

Useful for our purposes are the subsequent properties.Lemma 3. The above projections of the univariate case satisfy the following properties, wherewe assume r ≥ −1, p ≥ 0 for the first two lines and p − 1 ≥ r ≥ 0 otherwise and letI := (ψi, ψi+1) denote an arbitrary sub-interval induced by the discretization.

Πps = s ∀s ∈ Srp , (5)|Πpv|Hl(I) ≤ C |v|Hl(I) ∀v ∈ H l((0, 1)), 0 ≤ l ≤ p+ 1,

Πc,1p−1s = s ∀s ∈ Sr−1

p−1 , (6)Πc,1p−1∂ζv = ∂ζΠpv ∀v ∈ H1((0, 1)), (7)

|Πc,1p−1v|Hl(I) ≤ C |v|Hl(I) ∀v ∈ H l((0, 1)), 0 ≤ l ≤ p, (8)

for some constant C independent of mesh refinement.And if it is r ≥ 1, p ≥ 2 we further have

Πc,2p−2s = s ∀s ∈ Sr−2

p−2 , (9)Πc,2p−2∂

2ζ v = ∂2

ζ Πpv = ∂ζΠc,1p−1∂ζv ∀v ∈ H2((0, 1)), (10)

|Πc,2p−2v|Hl(I) ≤ C |v|Hl(I) ∀v ∈ H l((0, 1)), 0 ≤ l ≤ p− 1. (11)

11

Proof. The first five statements correspond to the properties (3.7), (3.8) and (3.10)-(3.12) in[4] if we keep the regular mesh assumption in mind. Hence we only check the points (9)-(11).In the following we use (3.4) of [4], which gives immediately that ∂2

ζSrp = Sr−2

p−2 . Firstly, withproperty (6) we have

Πc,2p−2s = d

dζΠc,1p−1

∫ ζ

0s(τ) dτ︸︷︷︸∈Sr−1

p−1

= d

dζ

∫ ζ

0s(τ) dτ = s.

In view of (5) and (7) one gets

Πc,2p−2∂

2ζ v = d2

dζ2 Πp

∫ ζ

0

∫ τ

0∂2rv(r) dr dτ = d2

dζ2 Πp(v(ζ) + a · ζ + b

)= d2

dζ2 Πp(v(ζ)

),

Πc,2p−2∂

2ζ v = d2

dζ2 Πp

∫ ζ

0∂rv(r) + a dr = d2

dζ2 Πp

∫ ζ

0∂rv(r) dr = d

dζΠc,1p−1∂ζv.

In the last two lines we wrote a, b ∈ R for suitable constants of integration. Hence (10) follows.The last point can be proven in a similar fashion like equation 3.12 in [4] using the fact thatif v ∈ H l((0, 1)) then w(ζ) :=

∫ ζ0 vdτ ∈ H l+1((0, 1)) and using estimate (8). It is

|Πc,2p−2v|Hl(I) ≤ |Π

c,1p−1w|Hl+1(I) ≤ C |w|Hl+1(I) ≤ C |v|Hl(I).

For the multivariate case we define projections as the tensor product ⊗ of the univariatecase operators, e.g. Πp1,p2,p3 : L2((0, 1)3)→ Sr1,r2,r3

p1,p2,p3 , v 7→ (Πp1 ⊗ Πp2 ⊗ Πp3)v. Here Πpi actsin some sense only in the i-th coordinate. For more information concerning the interpretationof tensor product projections we refer to Section 4.1. in [4].Next we can define the projections for the parametric Hessian complex, namely we set for

pi ≥ 2 and ri ≥ 1.Definition 5. (Projections in the parametric domain)

Π1h := Πp1 ⊗ Πp2 ⊗ Πp3 ,

Π2h := SYM

((Πc,2

p1−2 ⊗ Πp2 ⊗ Πp3

)×(Πc,1

p1−1 ⊗ Πc,1p2−1 ⊗ Πp3

)×(Πc,1

p1−1 ⊗ Πp2 ⊗ Πc,1p3−1

)×(Πp1 ⊗ Πc,2

p2−2 ⊗ Πp3

)×(Πp1 ⊗ Πc,1

p2−1 ⊗ Πc,1p3−1

)×(Πp1 ⊗ Πp2 ⊗ Πc,2

p3−2)),

Π3h := MAT

((Πc,1

p1−1 ⊗ Πc,1p2−1 ⊗ Πc,1

p3−1)×(Πc,2

p1−2 ⊗ Πp2 ⊗ Πc,1p3−1

)×(Πc,2

p1−2 ⊗ Πc,1p2−1 ⊗ Πp3

)×(Πp1 ⊗ Πc,2

p2−2 ⊗ Πc,1p3−1

)×(Πc,1

p1−1 ⊗ Πc,1p2−1 ⊗ Πc,1

p3−1)×(Πc,1

p1−1 ⊗ Πc,2p2−2 ⊗ Πp3

)×(Πp1 ⊗ Πc,1

p2−1 ⊗ Πc,2p3−2

)×(Πc,1

p1−1 ⊗ Πp2 ⊗ Πc,2p3−2

)×(Πc,1

p1−1 ⊗ Πc,1p2−1 ⊗ Πc,1

p3−1)),

Π4h :=

((Πc,2

p1−2 ⊗ Πc,1p2−1 ⊗ Πc,1

p3−1)×(Πc,1

p1−1 ⊗ Πc,2p2−2 ⊗ Πc,1

p3−1)×(Πc,1

p1−1 ⊗ Πc,1p2−1 ⊗ Πc,2

p3−2)).

The latter projection operators preserve the structure of the Hessian complex since theycommute with the respective differential operators. We have the subsequent lemma.Lemma 4. It holds

∇2(Π1hφ)

= Π2h

(∇2φ

), ∀ φ ∈ V 1, (12)

∇ ×(Π2hS)

= Π3h

(∇ × S

), ∀ S ∈ V

2,

∇ ·(Π3hT)

= Π4h

(∇ · T

), ∀ T ∈ V

3.

12

Proof. The proof is analogous to the one for Lemma 4.3 in [4] and a consequence of thecommutation properties (7) and (10) and the tensor product construction of the splines spacesand projections. For reasons of explanations we show the assertion for two different entries ofthe matrix (12). The rest follows with very similar ideas. Let φ be a smooth function withcompact support in Ω.For example we have(

∇2(Π1hφ))

11= ∂2

ζ1

(Π1hφ)

= ∂2ζ1

(Πp1 ⊗ Πp2 ⊗ Πp3

)φ = ∂2

ζ1(Πp1(Πp2(Πp3 φ))) =

= (Πc,2p1−2(Πp2(Πp3∂

2ζ1 φ))) =

(Πc,2p1−2 ⊗ Πp2 ⊗ Πp3

)∂2ζ1 φ =

(Π2h∇2(φ))

11.

On the other hand we get e.g.(∇2(Π1

hφ))

12= ∂ζ1∂ζ2

(Π1hφ)

= ∂ζ1∂ζ2(Πp1(Πp2(Πp3 φ))) = ∂ζ1(Πp1(Πc,1p2−1(Πp3∂ζ2 φ)))

= (Πc,1p1−1(Πc,1

p2−1(Πp3∂ζ1∂ζ2 φ))) =(Πc,1p1−1 ⊗ Πc,1

p2−1 ⊗ Πp3

)∂ζ1∂ζ2 φ

=(Π2h∇2(φ))

12.

Doing similar computations for the other components together with the boundedness of theprojections in the univariate case, Lemma 6 respectively, and a density argument leads to(12).The proof of the rest of the statement uses analogous steps and is not shown here.

Now the projections in the physical domain are defined through the mappings Yi. We set

Π1h := Y−1

1 Π1h Y1,

Π2h := Y−1

2 Π2h Y2,

Π3h := Y−1

3 Π3h Y3,

Π4h := Y−1

4 Π4h Y4.

One notes the invertibility of the Yi and further, because of the fact P1(Ω) ⊂ Sr1,r2,r3p1,p2,p3 , we can

set Π0h := id. Due to the definition of the latter projections combined with the commutativity

properties of the parametric projections as well as of the mappings Yi, the next diagramcommutes.

V 1 V 2 V 3 V 4∇2 ∇× ∇· 0ιP1 0P1(Ω)

V 1h V 2

h V 3h V 4

h

∇2 ∇× ∇· 0ιP1 0V 0h

id Π1h Π2

h Π3h Π4

h id

Figure 2: The diagram commutes and thus the discretization is compatible with the complex.

The commutativity of latter diagram makes it possible to proof later approximation esti-mates for different model problems. Prior to that we look at approximation properties of thediscrete spaces from above. This is subject of the following section.

13

4 Approximation estimatesIn the previous section we showed how one can use splines to construct in some sense structure-preserving discrete spaces for the Hessian complex, i.e. the diagram in Fig. 2 commutes.Hence it is reasonable to face the approximation behavior of the discrete spaces from abovein order to set up convergence estimates for different PDE problems later. As in [4] we dothis by relating the approximation behavior in the parametric domain to the spline basedspaces on the physical domain. Underlying for this approach are the next three lemmaswhich correspond to the Lemma 5.2 , Lemma 4.2 and Lemma 5.1 in [4], where we considerthe case without boundary conditions. Here and in the following we assume for reasons ofsimplification p1 = p2 = p3 = p ≥ 2 and r1 = r2 = r3 = r ≥ 1. Further, we introduce thebroken-Sobolev spaces Hl by

Hl(Ω) := v ∈ L2(Ω) | v|K ∈ H l(K), ∀K ∈ M,

where the corresponding semi-norm is defined via

|v|2Hl(Q):=

∑K∈M

K∩Q6=∅

|v|2Hl(K),

with obvious component-wise generalization to vector- or matrix-valued mappings. In lattercase we use again a bold-type notation, i.e. Hl.

Lemma 5. (Regularity preservation)Due to the fact that F is smooth we have for some constant C, independent of mesh-refinementand of the mesh element K = F(K), K ∈ M ,

1C‖φ‖Hl(K) ≤ ‖Y1(φ)‖Hl(K) ≤ C ‖φ‖Hl(K) , ∀φ ∈ H l(K),

1C‖S‖Hl(K) ≤ ‖Y2(S)‖Hl(K) ≤ C ‖S‖Hl(K) , ∀S ∈ Hl(K),

1C‖T‖Hl(K) ≤ ‖Y3(T)‖Hl(K) ≤ C ‖T‖Hl(K) , ∀T ∈ Hl(K),

1C‖v‖Hl(K) ≤ ‖Y4(v)‖Hl(K) ≤ C ‖v‖Hl(K) , ∀v ∈ Hl(K).

Proof. Follows directly by the smoothness of F.

Lemma 6. (Stability of the projections)The projections Π1

h, . . . , Π4h are continuous in the sense∥∥∥Π1hφ∥∥∥L2(K)

≤ C∥∥∥φ∥∥∥

L2(K), ∀φ ∈ L2(Ω),∥∥∥Π2

hS∥∥∥

L2(K)≤ C

∥∥∥S∥∥∥L2(K)

, ∀S ∈ L2(Ω),∥∥∥Π3hT∥∥∥

L2(K)≤ C

∥∥∥T ∥∥∥L2(K)

, ∀T ∈ L2(Ω),∥∥∥Π4hv∥∥∥

L2(K)≤ C ‖v‖L2(K) , ∀v ∈ L2(Ω),

14

for a suitable constant C = C(F). Above K stands for the extended support of the meshelement K. This means if K = (ψ1

i1 , ψ1i1+1) × · · · × (ψ3

i3 , ψ3i3+1), we find indices jl with

K = (ξ1j1 , ξ

1j1+1)× · · · × (ξ3

j3 , ξ3j3+1) . Then we set

K := (ξ1j1−p1 , ξ

1j1+p1+1)× · · · × (ξ3

j3−p3 , ξ3j3+p3+1).

Proof. The statement follows by the regular mesh assumption, the product structure of theprojections onto the multivariate spline spaces and Lemma 3.

Lemma 7. (Approximation property in the parametric domain)Let p > r ≥ 1. It holds

|φ− Π1hφ|Hl(K) ≤ C hs−l |φ|Hs(K) , 0 ≤ l ≤ s ≤ p+ 1, ∀φ ∈ V 1 ∩ Y1(Hs(Ω)),

|S − Π2hS|Hl(K) ≤ C hs−l |S|Hs(K) , 0 ≤ l ≤ s ≤ p− 1, ∀S ∈ V 2

∩ Y2(Hs(Ω)),

|T − Π3hT |Hl(K) ≤ C hs−l |T |Hs(K) , 0 ≤ l ≤ s ≤ p− 1, ∀T ∈ V 3

∩ Y3(Hs(Ω)),

|v − Π4hv|Hl(K) ≤ C hs−l |v|Hs(K) , 0 ≤ l ≤ s ≤ p− 1, ∀v ∈ V 4

∩ Y4(Hs(Ω)).

Proof. The proof is completely analogous to the one of Lemma 5.1 in [4]. We only show theassertion for the third inequality as an example since all the other inequalities follow withsimilar arguments. Let now ∈ T ∈ V 3

∩ Y3(Hs(Ω)). Then, due to the smoothness of F,we have T ∈ V 3

∩ Hs(Ω). Further let 0 ≤ l ≤ s ≤ p − 1. In view of Lemma 3.1 in [3]we find splines s1 ∈ Sr−1,r−1,r−1

p−1,p−1,p−1, s2 ∈ Sr−2,r,r−1p−2,p,p−1, s3 ∈ Sr−2,r−1,r

p−2,p−1,p, s4 ∈ Sr,r−2,r−1p,p−2,p−1, s5 ∈

Sr−1,r−1,r−1p−1,p−1,p−1, s6 ∈ Sr−1,r−2,r

p−1,p−2,p, s7 ∈ Sr,r−1,r−2p,p−1,p−2, s8 ∈ Sr−1,r,r−2

p−1,p,p−2, s9 ∈ Sr−1,r−1,r−1p−1,p−1,p−1 with

|si − T (i)|Hl(K) ≤ C hs−l|T (i)|Hs(K), ∀i, T =: MAT(T (1), . . . , T (9)). (13)

Here we used the regular mesh assumption. This implies directly|M − T |Hl(K) ≤ C hs−l|T |Hs(K), where M := MAT(s1, . . . , s9). Triangle inequality and thespline preserving property of the projections lead to

|T − Π3hT |Hl(K) ≤ |T − M |Hl(K) + |Π3

h

(M − T

)|Hl(K). (14)

The last term on the right-hand side can be estimated by means of the stability result for theprojections (see Lemma 6) and a standard inverse estimate for polynomials. More precisely,

|Π3h

(M − T

)|Hl(K) ≤ C h

−l∥∥∥Π3

h

(M − T

)∥∥∥L2(K)

≤ C h−l∥∥∥M − T

∥∥∥L2(K)

.

The combination of the last estimate with the two lines (13) and (14) leads to the thirdinequality in the assertion. All the other estimates are obtained similarly.

Finally we can state the approximation properties of the projections Πih in the physical

domain.

15

Theorem 1. (Approximation in the physical domain)There exist constants C not depending on the mesh size, refinement respectively, such that∥∥∥φ−Π1

hφ∥∥∥Hl(Ω)

≤ C hs−l ‖φ‖Hs(Ω) , 0 ≤ l ≤ s ≤ p+ 1, ∀φ ∈ V 1 ∩Hs(Ω),∥∥∥S−Π2hS∥∥∥

Hl(Ω)≤ C hs−l ‖S‖Hs(Ω) , 0 ≤ l ≤ s ≤ p− 1, ∀S ∈ V 2 ∩Hs(Ω),∥∥∥T−Π3

hT∥∥∥

Hl(Ω)≤ C hs−l ‖T‖Hs(Ω) , 0 ≤ l ≤ s ≤ p− 1, ∀T ∈ V 3 ∩Hs(Ω),∥∥∥v −Π4

hv∥∥∥

Hl(Ω)≤ C hs−l ‖v‖Hs(Ω) , 0 ≤ l ≤ s ≤ p− 1, ∀v ∈ V 4 ∩Hs(Ω).

Proof. The proof is nearly the same as the one for Theorem 5.3. in [4] . For reasons ofcompleteness we show the assertion for the third estimate. Let K ∈ M be a mesh elementand T ∈ V 3 ∩Hs(Ω). Set T := Y3(T). Lemma 5 yields for some constant C the inequality∥∥∥T−Π3

hT∥∥∥

Hl(K)≤ C

∥∥∥T − Π3hT∥∥∥

Hl(K). (15)

Above we wrote K = F(K) and note that for T ∈ Hs(Ω) it is T ∈ Hs(Ω). Thus Lemma 7gives ∥∥∥T − Π3

hT∥∥∥

Hl(K)≤ C

l∑i=0|T − Π3

hT |Hi(K) ≤ C hs−l|T |Hl(K).

The repeated application of Lemma 5 , together with (15), leads to the third inequality of theassertion. All the other approximation estimates can be proven in a similar fashion.

Exploiting the commutativity rules for the differential operators and projections we obtaina modified approximation estimate which is useful for example in the subsequent section.Corollary 1. Let 1 ≤ s ≤ p− 1. Then, assuming sufficient regularity, it is∥∥∥φ−Π1

hφ∥∥∥H2(Ω)

≤ C hs ‖φ‖Hs+2(Ω) ,∥∥∥S−Π2hS∥∥∥

H(Ω,curl)≤ C hs ‖S‖Hs(Ω,curl) ,∥∥∥T−Π3

hT∥∥∥

H(Ω,div)≤ C hs ‖T‖Hs(Ω,div) ,∥∥∥v −Π4

hv∥∥∥

L2(Ω)≤ C hs ‖v‖Hs(Ω) .

We use the notation ‖S‖2Hs(Ω,curl) := ‖S‖2Hs(Ω) +‖∇ × S‖2Hs(Ω) and ‖T‖2Hs(Ω,div) := ‖T‖2Hs(Ω) +

‖∇ ·T‖2Hs(Ω).Proof. The estimates are a consequence of Theorem 1 . We show the assertion only for thesecond line. The rest can be derived applying an analogous procedure.∥∥∥S−Π2

hS∥∥∥2

H(Ω,curl)=∥∥∥S−Π2

hS∥∥∥2

L2(Ω)+∥∥∥∇× S−∇×Π2

hS∥∥∥2

L2(Ω)

=∥∥∥S−Π2

hS∥∥∥2

L2(Ω)+∥∥∥∇× S−Π3

h(∇× S)∥∥∥2

L2(Ω)

≤ C(h2s ‖S‖2Hs(Ω) + h2s ‖∇ × S‖2Hs(Ω)

)≤ C h2s ‖S‖2Hs(Ω,curl) .

After we focused on standard approximation inequalities for the introduced discrete spaces,we move on with examples for which such a FEEC based discretization is helpful.

16

5 ApplicationsIn this part we show how one can apply the structure-preserving discretization to set up nu-merical schemes, following the results of FEEC (see Chapter 5 in [2]), for different differentialequations. Exploiting FEEC we will obtain relatively easy stability and convergence state-ments.We look at the case of the Hodge-Laplacians corresponding to the above studied Hessiancomplex. Thus let us first shortly define the class of the Hodge-Laplace equation. For moreinformation on that we refer to [2] and [1].

5.1 Hodge-Laplace equation

For a given closed Hilbert-complex (W k, dk) with a domain complex (V k, dk) we call theproblem (

d∗k+1dk + dk−1d∗k)u = f (16)

(abstract) Hodge-Laplacian of level k, where f ∈ W k is some given mapping. Clearly, theoperator Lk := d∗k+1d

k + dk−1d∗k on the left-hand side has the domain

D(Lk) = v ∈ V k ∩ V ∗k | dkv ∈ V ∗k+1, d∗kv ∈ V k−1

and one requires u ∈ D(Lk). Furthermore we assume from now on, in view of the Hessiancomplex from above, that the domain complex is exact. Then, Theorem 4.8 in [2] gives usthe well-posedness of the problem (16). For the computation of approximate solutions forthe Hodge-Laplacian of level k it is useful to introduce an equivalent (see Theorem 4.7 in [2])mixed weak formulation. It reads

Find σ ∈ V k−1, u ∈ V k with〈σ, τ〉 − 〈u, dk−1τ〉 = 0, ∀τ ∈ V k−1, (17)

〈dk−1σ, v〉+ 〈dku, dkv〉 = 〈f, v〉, ∀v ∈ V k .

The connection between the original formulation and the mixed weak form is given by d∗ku = σ.Due to readability we neglected the indices W k for inner product brackets.The basic idea for a numerical method of the latter problem is to use test functions fromfinite-dimensional subspaces V k

h ⊂ V k instead of taking the whole spaces V k into account.As in the case of spline spaces the index h indicates refinement, dimension increase of thesubspaces respectively, meaning for h → 0 the dimension of the subspaces grow to infinity.The work from Arnold guides us to stable and convergent numerical methods, by requiringthree key properties for the subspaces. In view of Theorems 4.8, 4.9 and 5.5 as well as chapter5 of [2] we summarize the mentioned properties and results in the next Theorem.

Theorem 2. (Structure-preserving discretization of the Hodge-Laplacian)Let (W k, dk) be a closed Hilbert complex with exact domain complex (V k, dk). Further let thenext three properties be fulfilled:

• "Approximation property:" For h→ 0 we have infv∈V i

h

‖w − v‖V i → 0, for all w ∈ V i.

17

• "Subcomplex property": The subspaces form a subcomplex in the sense dk−1V k−1h ⊂

V kh , dkV k

h ⊂ Vk+1h .

• "Bounded cochain projections": There existW i-bounded projections Πih : V i → V i

h, whichcommute with the operators di. In other words the next diagram commutes.

V k−1

V k−1h

V k

V kh

V k+1

V k+1h .

Πk−1h Πk

h Πk+1h

dk−1

dk−1

dk

dk

Then the discrete formulation of (17), i.e.

Find σh ∈ V k−1h , uh ∈ V k

h with〈σh, τh〉 − 〈uh, dk−1τh〉 = 0, ∀τh ∈ V k−1

h ,

〈dk−1σh, vh〉+ 〈dkuh, dkvh〉 = 〈f, vh〉, ∀vh ∈ V kh ,

(18)

has a unique solution and satisfies an inf-sup stability criterion, namely

inf06=(σh,uh)∈Xk

h

sup06=(τh,vh)∈Xk

h

B(σh, uh; τh, vh)‖(σh, uh)‖Xk ‖(τh, vh)‖Xk

≥ γ > 0,

with B(σh, uh; τh, vh) := 〈σh, τh〉 − 〈uh, dk−1τh〉+ 〈dk−1σh, vh〉+ 〈dkuh, dkvh〉,Xkh = V k−1

h × V kh and ‖(τh, vh)‖Xk := ‖τh‖V k−1 + ‖vh‖V k .

Further, the discrete solution (σh, uh) converges to the exact (σ, u) one and it holds

‖σ − σh‖V k−1 + ‖u− uh‖V k ≤ C(

infτ∈V k−1

h

‖τ − σ‖V k−1 + infv∈V k

h

‖u− v‖V k

),

where C is a constant independent of h.Now we use Theorem 2 for the Hodge-Laplacians with underlying Hessian complex and the

above derived discretization of it. Namely, it is easy to see that the three important propertiesstated in latter theorem are satisfied by the discretization ansatz in Section 3. Therefore weapply again the notation of Sections 3 and 5, especially use (4). Besides, below the mappingswithout index h stand for the exact solutions of the respective mixed weak form; compare(17).

Hodge-Laplacian: k = 1

We have the equation:(ιP1πP1 +∇ · ∇ · ∇2)φ = f, with φ ∈ H2(Ω),∇2φ ∈

H(Ω, divdiv,S), f ∈ L2(Ω),

and a corresponding discrete mixed weak formulation

Find ph ∈ P1(Ω), φh ∈ V 1h ⊂ H2(Ω) with〈ph, qh〉 − 〈φh, qh〉 = 0, ∀qh ∈ P1(Ω),

〈ph, τh〉+ 〈∇2φh,∇2τh〉 = 〈f, τh〉, ∀τh ∈ V 1h .

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Theorem 2 gives the stability and convergence of the method and an error estimate. If weassume 1 ≤ s ≤ p− 1 and φ ∈ Hs+2(Ω), then Corollary 1 yields

‖p− ph‖L2(Ω) + ‖φ− φh‖V 1 ≤ C(

infqh∈V 0

h

‖p− qh‖L2(Ω) + infτh∈V 1

h

‖φ− τh‖H2(Ω))

≤ C hs ‖φ‖Hs+2(Ω) .

For the last inequality we used the fact that P1(Ω) ⊂ V 0h and note ‖φ‖2V 1 := ‖φ‖2L2(Ω) +∥∥∇2φ

∥∥2L2(Ω).


The level k = 2 Hodge-Laplacian can be written as(sym∇×∇×+∇2 ∇ · ∇ ·

)S = f, S ∈ H(Ω, curl,S), f ∈ L2(Ω,S),

where one searches for S ∈ H(Ω, curl, S) ∩H(Ω,divdiv, S) with ∇ × S ∈

H(Ω, symcurl,T),

∇ · ∇ · S ∈ H2(Ω). Thus the discrete mixed weak formulation in view of (18) is:Find φh ∈ V 1

h ⊂ H2(Ω), Sh ∈ V 2h ⊂ H(Ω, curl, S) with〈φh, τh〉 − 〈Sh,∇2τh〉 = 0, ∀τh ∈ V 1

h ,

〈∇2φh,Mh〉+ 〈∇ × Sh,∇×Mh〉 = 〈f,Mh〉, ∀Mh ∈ V 2h.

Using again Theorem 2 and Corollary 1 we obtain stability and convergence with estimate‖φ− φh‖V 1 + ‖S− Sh‖H(Ω,curl) ≤ C

(inf

τh∈V 1h

‖τh − φ‖H2(Ω) + infMh∈V 2

h

‖S−Mh‖H(Ω,curl))

≤ C hs(‖φ‖Hs+2(Ω) + ‖S‖Hs(Ω,curl)

).

Above we assumed 1 ≤ s ≤ p− 1 and φ ∈ Hs+2(Ω) as well as S,∇× S ∈ Hs(Ω).


In this case equation (16) has the form(− dev∇∇ ·+∇× (sym∇×)

)T = f, f ∈ L2(Ω,T).

And the strong form seeks T ∈ H(Ω,div,T) ∩H(Ω, symcurl,T) such that sym∇ × T ∈

H(Ω, curl,S), ∇·T ∈ H10(Ω). The finite-dimensional problem based on the mixed weak form

reads:Find Sh ∈ V 2

h ⊂ H(Ω, curl,S), Th ∈ V 3h ⊂ H(Ω, div,T) with

〈Sh,Mh〉 − 〈Th,∇×Mh〉 = 0, ∀Mh ∈ V 2h,

〈∇ × Sh,Nh〉+ 〈∇ ·Th,∇ ·Nh〉 = 〈f,Nh〉, ∀Nh ∈ V 3h.

Again if we require 1 ≤ s ≤ p− 1 and S,∇× S ∈ Hs(Ω) and T,∇ ·T ∈ Hs(Ω), then we getwith Corollary 1:

‖S− Sh‖H(Ω,curl) + ‖T−Th‖H(Ω,div)

≤ C(

infMh∈V 2

h

‖Mh − S‖H(Ω,curl) + infNh∈V 3

h

‖T−Nh‖H(Ω,div))

≤ C hs(‖S‖Hs(Ω,curl) + ‖T‖Hs(Ω,div)

),

where T and S are the exact solution of the mixed weak form of the Hodge-Laplacian.

19


In view of d4 = 0 we can write down the Hodge-Laplacian of level 4 as

−(∇ · dev∇

)v = f, v ∈ L2(Ω), f ∈ L2(Ω),

and we look for v ∈ H10(Ω) with dev∇v ∈ H(Ω,div,T). Now using the template (18) we

obtain a finite-dimensional weak form:

Find Th ∈ V 3h ∈ H(Ω, div,T), vh ∈ V 4

h ⊂ L2(Ω) with〈Th,Nh〉 − 〈vh,∇ ·Nh〉 = 0, ∀Nh ∈ V 3

h,

〈∇ ·Th,wh〉 = 〈f,wh〉, ∀wh ∈ V 4h.

Applying Theorems 2 and Corollary 1 yields, with 1 ≤ s ≤ p − 1 and enough regularity, theinequality

‖T−Th‖H(Ω,div) + ‖v− vh‖L2(Ω) ≤ C(

infNh∈V 3

h

‖Nh −T‖H(Ω,div) + infwh∈V 4

h

‖v−wh‖L2(Ω))

≤ C hs(‖T‖Hs(Ω,div) + ‖v‖Hs(Ω)

),

where T and v are the solution of the corresponding variational form (17).

Another application for our structure-preserving discretization would be the LinearizedEinstein-Bianchi (LEBS) system that is utilized to compute solutions to the linearized EinsteinField Equations in numerical relativity. For the next explanations we follow [13].

5.2 Linear Einstein-Bianchi system

Using a suitable auxiliary variable and requiring consistent initial data the LEBS can bewritten as

∂tσ = ∇ · ∇ ·E,∂tE = −∇2σ − sym∇×B,∂tB = ∇×E,

with

σ ∈ C0([0, T ];H2(Ω)) ∩ C1([0, T ];L2(Ω)),

E ∈ C0([0, T ];H(Ω, curl,S) ∩H(Ω,divdiv, S)) ∩ C1([0, T ];L2(Ω)),

B ∈ C0([0, T ];H(Ω, symcurl,T)) ∩ C1([0, T ];L2(Ω)).

Above 0 < T <∞ stands for some final time.The spatial part comprises a Hodge-Laplacian and actually the upper equation belongs

to the class of Hodge wave equations, the hyperbolic time-dependent version of the Hodge-Laplacian; we refer to Chapter 8 in [2]. In his thesis Quenneville-Bélair [13] proofs an ap-proximation estimate for a weak discrete formulation. More precisely, due to Theorem 4.14 in

20

[13], if one has a structure-preserving spatial discretization in the sense of Theorem 2 of theHessian complex (like the one we introduced in this article), then the solution of the problem

Find σh ∈ C1([0, T ];V 1h ) ⊂ C1([0, T ];H2(Ω)),

Eh ∈ C1([0, T ];V 2h) ⊂ C1([0, T ];H(Ω, curl,S)),

Bh ∈ C1([0, T ];V 3h) ⊂ C1([0, T ];H(Ω, div,T)), where

〈∂tσh, τh〉 − 〈Eh,∇2τh〉 = 0, ∀τh ∈ V 1h ,

〈∂tEh,Mh〉+ 〈∇2σh,Mh〉+ 〈Bh,∇×Mh〉 = 0, ∀Mh ∈ V 2h,

〈∂tBh,Nh〉 − 〈∇ ×Eh,Nh〉 = 0, ∀Nh ∈ V 3h.

converges to the exact solution of the LEBS. And since there exists a unique exact solution,we have a well-defined problem; compare [13, Theorem 4.6., Theorem 4.14.]. One can proofthat with proper initial data (σ0,E0,B0) ∈ V 1 × (V2 ∩V∗2)×V∗3 the difference between theexact and the approximate solution satisfies

supt∈[0,T ]

(‖σ − σh‖L2(Ω) + ‖E−Eh‖L2(Ω) + ‖B−Bh‖L2(Ω)

)≤ C

(supt∈[0,T ]

( ∥∥∥σ −Π1hσ∥∥∥L2(Ω)

+∥∥∥E−Π2

hE∥∥∥

L2(Ω)+∥∥∥B−Π3

hB∥∥∥

L2(Ω)

) )

+ C

∫ T

0

∥∥∥σ −Π1hσ∥∥∥V 1

+∥∥∥E−Π2

hE∥∥∥V 2 +

∥∥∥B−Π3hB∥∥∥V 3 dt

∈ O(hs),

assuming 1 ≤ s ≤ p− 1 and enough regularity of the exact solution 2. In the second last linethe dots denote the time derivative.We want to remark that in [13] also other formulations and methods are established and

considered. Main reason for this is to simplify the discretization procedure, since the H2-regularity is quite hard to achieve in the context of classical FEM.

We proceed now with the consideration of some numerical test problems to check thetheoretical findings from above.

6 Numerical examplesIn this part we want to display some numerical examples which consist of the computation ofapproximate solutions to the Hodge-Laplacians to verify the convergence statements. Basisof the calculations is the GeoPDEs package ([8, 16]) together with the MATLAB software;see [11]. The appearing linear systems are mainly solved by means of the mldivide functionof MATLAB. Nevertheless for some very large systems we utilized the minres-solver, wherea large number of iterations (40000) and small residuals, compared to the computed actual

2We have to assume that the exact solution variables are elements of C1([0, T ]; V 1), C1([0, T ]; Vi) respec-tively.

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errors, ensure meaningful results.As computational domain we consider a deformed cube obtained by the parametrization map

F : (0, 1)3 → R3 , ζ 7→ A · ζ, A :=

1 0.5 0.50 1 0.5

0.5 0 1

.For our experiments we divide the parametric reference cube into equal smaller cubes withedge lengths h to generate regular meshes in the physical domain. In Fig. 3 the mesh withsize h = 1/4 is displayed.Now we go through the cases of k = 1, . . . , 4 and compute numerical solutions for the

Hodge-Laplacian by means of the discrete mixed weak formulation (18). In all test exampleswe use right-hand sides such that the exact solutions are smooth. Hence looking at Theorem2 and Corollary 1, we expect for the different graph norms ‖·‖V k , ‖·‖V k a convergence orderone lower than the chosen degree p for the space V 1

h . The polynomial degree p in V 1h is chosen

to be equal w.r.t. each parametric coordinate, i.e. pi = p, and the splines have the regularitiesri = r := p− 1, ∀i. Latter is done in order to save computational costs.


The right-hand side f is adapted in such a way that φF = sin(π ζ1)4 sin(π ζ2)4 sin(π ζ3)4 isthe exact solution. Obviously, we get φ ∈ D(L1). If we compute the errors between numericaland exact solution for polynomial degrees p = 2, 3, 4 we obtain the results displayed in Fig. 4.The convergence behavior in the mentioned figure Fig. 4 confirms the theoretical assertion.

Figure 3: Physical domain we use for all testexamples.

Figure 4: The errors ‖φ− φh‖H2 for thelevel k = 1 Hodge-Laplacian.


Here we set the source term f s.t. the exact solution has the form

S = MAT(v, v, . . . , v), v := (sin(π ζ1)2 sin(π ζ2)2 sin(π ζ3)2) F−1(x, y, z).

One notes the small deviation between the predicted convergence order 1 for p = 2 and theplot in Fig. 5, where the errors in the norm ‖·‖H(Ω,curl) are plotted. Nevertheless, there is nocontradiction to Theorem 2. And in Fig. 6 we also see the errors ‖d∗2S− φh‖H2 which decaysteadily in good accordance with the theory.

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Figure 5: H(Ω, curl)-error for the level k = 2. Figure 6: The errors ‖d∗2S− φh‖H2 (k = 2).


As above we use a manufactured exact solution to study the convergence behavior.We have as exact solution

T =

v v v0 v 00 0 −2 v

.In view of Theorem 2 and Section 5.1, we compute the errors ‖T−Th‖H(Ω,div) and‖S− Sh‖H(Ω,curl), with S = sym∇×T. One can see the results in Fig. 7 and Fig. 8.

Figure 7: H(Ω,div)-error for level k = 3. Figure 8: The errors ‖d∗3T− Sh‖H(curl).


For our last level in the context of the Hodge-Laplacian, we use as test case the exact solutionv = (v, v, v). The errors ‖v− vh‖L2(Ω) and ‖T−Th‖H(Ω,div) between exact and numericalsolution are shown in the figures Fig. 9 and Fig.10. One notes the relation T = −dev∇v.In another small example we want to point out the superiority of the structure-preserving

ansatz in contrast to straight-forward IGA discretizations. For this purpose we consideragain the Hodge-Laplacian test cases for k = 2 and k = 3 from above. But now we use foreach occurring component function the standard IGA test spaces V 1

h and ignore the otherconstructed finite-dimensional spaces. The results for the errors and different mesh sizes are

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Figure 9: L2-error for the level k = 4. Figure 10: The H(Ω,div)-errors for k = 4.

shown in the figures Fig. 11-14. One sees the non-stable decay behavior for the errors andwe interpret it as a clue for possible instability problems if one devotes oneself with classicaltest function spaces. In particular we can not guarantee that the error decays match withthe theoretically predicted rates. For example in Fig. 14, we observe partly an error increasedespite mesh refinement.

Figure 11: H2-error for k = 2 and a non-structure-preserving discretization.

Figure 12: H(Ω, curl)-error for k = 2 and anon-structure-preserving discretization.

So we can conclude for this section that the test examples confirm the statements in The-orem 2, the structure-preserving property of our discretization respectively.

7 Discussion and further problemsWe introduced spline spaces which are suitable for a discretization of the Hessian complex,since they mimic the complex structure. In other words, they can be used to built up asub-complex. Then the theory of FEEC shows us how to write down mixed-weak formula-tions of the Hodge-Laplacian problem that guarantee stability and convergence. Further, thediscretization can also be used in other fields like numerical relativity. And here we want toremark, that the symmetry and trace properties as parts of the original Hessian complex arepreserved with our method exactly. But, there are also questions and new problems arising.On the one hand, we could only show the meaningfulness of the transformation mappings

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Figure 13: H(Ω, curl)-error for k = 3 and anon-structure-preserving discretization.

Figure 14: H(Ω,div)-error for k = 3 and anon-structure-preserving discretization.

Yi in case of affine linear parametrizations of the physical domain. Clearly, it is a reason-able thought to check for possible generalizations to exploit the benefit of IGA, namely theexact representation of curved boundary domains. The authors have already addressed theissue of curved boundary geometries and although one can show the existence of generalizedstructure-preserving transformations for non-trivial geometries we have to study the problemin more detail to obtain practicable outcomes. On the other hand, as a task for further studiesit would be natural to consider other second order complexes in a similar fashion, e.g. the di-vdiv-complex. Furthermore, there exists also a Hessian complex involving Dirichlet boundaryconditions. Hence the construction of proper spline spaces satisfying the boundary conditionsis also an interesting problem.

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References[1] D. Arnold, R. Falk, and R. Winther, Finite Element Exterior Calculus: From

Hodge Theory to Numerical Stability, Bulletin of the American Mathematical Society, 47(2010), pp. 281–354.

[2] D. N. Arnold, CBMS-NSF Regional Conference Series in Applied Mathematics, 93. Fi-nite Element Exterior Calculus, SIAM (Society for Industrial and Applied Mathematics),(2018).

[3] Y. Bazilevs, L. Veiga, J. Cottrell, T. Hughes, and G. Sangalli, IsogeometricAnalysis: Approximation, Stability and Error Estimates for h-Refined Meshes, Mathe-matical Models and Methods in Applied Sciences, 16 (2006), pp. 1031–1090.

[4] A. Buffa, J. Rivas, G. Sangalli, and R. Vázquez, Isogeometric Discrete Differen-tial Forms in Three Dimensions, SIAM J. Numer. Anal., 49 (2011), pp. 818–844.

[5] A. Buffa and G. Sangalli, IsoGeometric Analysis: A New Paradigm in the Numer-ical Approximation of PDEs, Lecture Notes in Mathematics 2161, CIME FoundationSubseries, Springer, Cham, Switzerland, 2016.

[6] L. Chen and X. Huang, Discrete Hessian complexes in three dimensions,arXiv:2012.10914, (2020).

[7] L. B. da Veiga, D. Cho, and G. Sangalli, Anisotropic nurbs approximation inisogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 209-212 (2012), pp. 1 – 11.

[8] C. de Falco, A. Reali, and R. Vázquez, GeoPDEs: A research tool for isogeometricanalysis of PDEs, Advances in Engineering Software, 42 (2011), pp. 1020–1034.

[9] R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, 11(2002), pp. 237 – 339.

[10] T. Hughes, J. Cottrell, and Y. Bazilevs, Isogeometric Analysis: CAD, Finite El-ements, NURBS, Exact Geometry and Mesh Refinement, Computer Methods in AppliedMechanics and Engineering, 194 (2005), pp. 4135–4195.

[11] MATLAB, Version 9.6 (R2019a), The MathWorks Inc., Natick, Massachusetts,USA,2019.

[12] D. Pauly and W. Zulehner, On Closed and Exact Grad-grad- and div-Div-Complexes,Corresponding Compact Embeddings for Tensor Rotations, and a Related DecompositionResult for Biharmonic Problems in 3D, arXiv: Analysis of PDEs, (2016).

[13] V. Quenneville-Bélair, A New Approach to Finite Element Simulations of GeneralRelativity, Ph.D. thesis, Department of Mathematics, University of Minnesota, 2015.

[14] L. Schumaker, Spline Functions: Basic Theory, Cambridge Mathematical Library,Cambridge University Press, 3 ed., 2007.

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[15] M. Skorski, Chain Rules for Hessian and Higher Derivatives Made Easy by TensorCalculus, arXiv:1911.13292, (2019).

[16] R. Vázquez, A new design for the implementation of isogeometric analysis in Octaveand Matlab: GeoPDEs 3.0, Computers and Mathematics with Applications, (2016). Toappear.

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Structure-preserving Discretization of the Hessian Complex