On the unisolvence for the quasi-polynomial spaces of differential forms

03/31/2020
by   Shuonan Wu, et al.
Penn State University
Peking University
0

We consider quasi-polynomial spaces of differential forms defined as weighted (with a positive weight) spaces of differential forms with polynomial coefficients. We show that the unisolvent set of functionals for such spaces on a simplex in any spatial dimension is the same as the set of such functionals used for the polynomial spaces. The analysis in the quasi-polynomial spaces, however, is not standard and requires a novel approach. We are able to prove our results without the use of Stokes' Theorem, which is the standard tool in showing the unisolvence of functionals in polynomial spaces of differential forms. These new results provide tools for studying exponentially-fitted discretizations stable for general convection-diffusion problems in Hilbert differential complexes.

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1. Introduction

The Finite Element (FE) Exterior Calculus (EC) [2, 1] is a powerful technique that combines tools from differential geometry and finite element analysis in constructing discretizations which inherit the natural structure of the underlying physical models. In our work, we consider general convection-diffusion equations on Hilbert complexes, such as the ones involving , , and in 3D, describing diffusion (by Hodge Laplacian) and corresponding transport driven by different velocity fields.

The design of stable discretizations for convection-diffusion problems, even in the scalar case, is a challenging task as these are singularly perturbed differential equations with small, and even vanishing, diffusion (see, e.g. [25] and the references therein for discussion on such topics). There is a vast amount of literature on various techniques designed to take care of the numerical instabilities associated with this type of equations. We refer the reader to recent and classical works on the subject focused on some of these techniques: mixed FE methods [11, 8, 19, 7, 10]; discontinuous Galerkin methods [12, 15, 17, 20]; discontinuous Petrov-Galerkin methods [5, 14, 13].

Our results show unisolvence for the quasi-polynomial (weighted) spaces used in simplex-averaged finite element (SAFE) discretization [27] for convection-diffusion equations in Hilbert complexes. Such exponentially fitted finite element schemes have been used with success for scalar convection-diffusion equations, i.e., in our terminology for convection-diffusion problems in . A rough explanation of the ideas behind SAFE discretizations could be as follows: (1) define a variable representing the flux, as in mixed methods, and use a variable change to symmetrize the equation; (2) discretize the differential operator using discretization for the flux and the primal variable; (3) eliminate the flux (locally) and change the variables to obtain a discretization of the original problem. Such a path for the derivation of discrete problems is seen in the pioneering work on discretizing drift-diffusion models in 1D [26] and later in FE and finite volume schemes in higher spatial dimensions [3, 28, 21]. Recently, a more general FEEC approach has brought mechanisms that can utilize higher degree polynomials and can work in any spatial dimension. In addition to the SAFE discretizations [27], the FEEC approach was an important tool in designing exponentially fitted space-time discretizations in [4].

In this work we consider one of the key ingredients needed in steps (2) and (3) above, namely, determining a set of unisolvent functionals for the numerical flux. A typical situation in the discretizations discussed above is the following: Given a polynomial vector space

, on an -dimensional simplex , we discretize the flux using a quasi-polynomial space of differential forms,

where denotes the space of -forms in with coefficients from . For example, for a convection-diffusion equation in

in 3D the degrees of freedom (unisolvence functionals) which uniquely determine an element

, are the moments of

on edges of , faces of and itself. The classical works [6, 18, 1, 2] usually use the Stokes’ Theorem, when verifying the unisolvence of such functionals for , and the arguments involve differentiation of . For quasi-polynomial spaces, with a non-constant weight , such differentiation results in terms that have both derivatives of and derivatives of . The standard arguments are, therefore, not applicable except in some special cases, such as the lowest order first kind of Nédélec-Raviart-Thomas elements. Our analysis here circumvents the use of Stokes’ Theorem, and we are able to show that the unisolvence functionals for are also unisolvence functionals for for differential forms of all orders , in any spatial dimension , and all polynomial spaces of the first and second kind (Nédélec–Raviart–Thomas [24, 23, 22], Nédélec–Brezzi–Douglas–Marini [23, 22, 9] spaces).

The landscape of the paper can be mapped as follows: Preliminaries and FEEC notation is introduced in Section 2. The unisolvence sets of functionals for the first kind (Nédélec–Raviart–Thomas) and second kind (Nédélec–Brezzi–Douglas–Marini) are discussed in Section 3 and Section 4, respectively. Examples for constructing discretizations of the flux are then given in Section 5, and numerical tests are shown in Section 6.

2. Preliminaries

In this section, we present some preliminary results which will be used in the following sections. We begin by a simple result, frequently used in the analysis.

Lemma 2.1.

Let be an open domain and be a Riemann integrable function. If (or ), then implies that almost everywhere.

We denote the spaces of polynomials in variables of degree at most and of homogeneous polynomial functions of degree by and , respectively. We will abbreviate them to and at times. Next, following [1], we present some basic notation commonly used in FEEC when working with polynomial differential forms.

2.1. Simplices and barycentric coordinates

Let denote the set of increasing maps , for . is the complementary map of with . For any , denote by . Similarly, let denote the set of increasing maps for . The map complementary to , denoted by , satisfies such that . Here, represents the range of in ascending order, which is also denoted by if there is no ambiguity. In addition, denotes the cardinality of .

Let be an -simplex with the vertices . For each , the set is a subsimplex of dimension . For , is the -dimensional subsimplex of opposite to the -subsimplex . The set of subsimplices of dimension of is denoted by , and the set of all subsimplices of is denoted by .

We denote by the barycentric coordinates satisfying . Clearly, form a basis of and satisfy . For a sub-simplex with , there is an isomorphism between of polynomial functions on and the space . That is, each can be expressed as

for a unique . The extension, denoted by , is defined by extending the right-hand side for . It is readily seen that the extension is an injective mapping from to .

Since the vectors form a basis for , the dual basis functions form a basis for . For any face , the restrictions of to the tangent space of at any point of give a basis for .

The algebraic -forms , , form a basis for . Hence, a differential -form can be uniquely written in the form

By definition of wedge product, we have , which implies that

(2.1)

where denotes the volume form in .

2.2. Whitney forms

For any and , an associated differential -form (called Whitney form) is given by

(2.2)

where the inverted hat represents a suppressed argument. As shown below, the Whitney form gives an explicit formulation of the basis of .

Theorem 2.2 (Theorem 4.1 in [1]).

The Whitney -forms corresponding to form a basis for .

3. Unisolvence for quasi-polynomial spaces of first kind

In this section we consider the first type quasi-polynomial

(3.1)

Here, is a general positive weight on .

3.1. Geometrical decomposition of the first kind polynomial spaces

We now introduce the geometrical decomposition of and . The degrees of freedom of as given in [1, Section 4.6] are

(3.2)

The proof (see, e.g. [1]) that these functionals form a unisolvent set uses induction argument and the Stokes’ Theorem, and the arguments do not carry over to quasi-polynomial spaces. An attempt to prove the result for quasi-polynomial spaces (3.1), however, reveals that the characterization of the trace free part of the plays a crucial role in showing the unisolvence. Such a characterization is given in the theorem below.

Theorem 3.1 (Theorem 4.16 in [1]).

For , , the map

(3.3)

where the , defines an isomorphism of onto . Here, .

Next, we use this Theorem to show that the unisolvence functionals for the polynomial space also work for the quasi-polynomials.

3.2. Polynomials of first kind with vanishing traces

The main result in this section is the following lemma.

Lemma 3.2.

Let . Suppose that

(3.4)

Then .

Postponing the proof of this lemma for later, we note that its implications show the desired unisolvence results. Indeed, by induction argument (from low dimension sub-simplices to high dimension sub-simplices), it can be easily shown that the functionals given in (3.2) also give degrees of freedom for , and we have the following theorem.

Theorem 3.3.

Let , . Suppose that satisfies

Then .

Proof.

For any , the trace of on vanishes, as it is a -form on a manifold of dimension . Noting that , applying in Lemma 3.2, we have . Next, for any ,

which implies that by applying Lemma 3.2 again. The proof is completed by an induction argument. ∎

Before we proceed the proof of Lemma 3.2 for general case, we first give some examples to fix the ideas as abstractions can often be difficult to grasp. Noting that cases for -forms in which or are trivial.

Proof of Lemma 3.2 for 1-forms in .

In this case, there are two maps in , namely

In light of (3.3), can be uniquely written as

where . We choose a special test form defined as

We note here that the sign of is in accordance with the isomorphism defined in (3.3). Using that , , and collecting the coefficients of then shows that

On the other hand the polynomial function under the integral is non-negative as seen below,

Finally, using that the weight is positive, i.e., together with Lemma 2.1, shows that and therefore proves Lemma 3.2 for and . ∎

Proof of Lemma 3.2 for -forms and any spatial dimension .

Consider the general case . In a similar way, there are maps in , denoted by , where . Hence, can be uniquely written as

where . Taking a special test -form as

which gives

and with

(3.5)

If we now denote , , we have

or

(3.6)

Since , for in the interior of , this shows Lemma 3.2 for and any spatial dimension . ∎

Proof of Lemma 3.2 for -forms in any spatial dimension .

Consider the case in which . Again, there are maps in , whose complements we denote by with . Next, we rewrite as

where . Choose a special test function

Then, we have , where and with

We then have

(3.7)

Therefore, we see that Lemma 3.2 holds for any with . ∎

3.3. Summary (spacial cases of Lemma 3.2)

Let us summarize what we have shown so far: Lemma 3.2 holds for any -form for . As a consequence, for spatial dimensions , we have proved Lemma 3.2 in all the possible cases.

To generalize the ideas for other values of and , we proceed as in the special cases considered above. For a given , we find a special test form so that does not change sign on . This gives us a “mass” matrix which corresponds to the Whitney form bases in and .

We now follow this plan and generalize the unisolvence result to discrete differential forms of arbitrary order in any spatial dimension and any quasi-polynomial Hilbert complex of first kind.

3.4. Calculating the “mass” matrix

Let us fix the spatial dimension and recall that

We now define as the number of inversions of the array corresponding to . For instance, when such that , then associates with the array and hence . It is easy to show that for any .

We first give the following result relating the maps , and the Whitney forms.

Lemma 3.4.

For any ,

(3.8)
Proof.

We use the definition of the Whitney form (2.2), to obtain that

(3.9)

Note that for any ,

The result follows by summing up the identities above. ∎

Lemma 3.5.

For and , it holds that

(3.10)
Proof.

Since , , we easily see that . Moreover, if , we deduce from equation (3.9).

For the case in which , we see that . Notice that and , then

This completes the proof. ∎

Corollary 3.6.

For any , it holds that

(3.11)
Proof.

We verify the statement case by case:

  • : obvious.

  • : both left hand side (LHS) and right hand side (RHS) are zero.

  • : Recall that , by (3.10),

This completes the proof. ∎

3.5. Proof of Lemma 3.2

Let us consider a differential form . By Theorem 3.1, we can write as

where . We take a special test form in (3.4) as

By Lemma 3.4 and Lemma 3.5, we have

(3.12)

where is symmetric (by Corollary 3.6) and

(3.13)

Recall that the exterior algebra has as a basis for , where is the orthogonal basis of . Further, the inner product on is defined as (c.f. [1, pp. 11])

(3.14)

We have the following result which generalizes (3.6) and (3.7) for arbitrary and .

Lemma 3.7.

Let be given in (3.13). Then,

(3.15)

where

Proof.

We write

Next, we verify (3.15) for each component:

  • , the coefficient of is

  • : the coefficient of is obviously zero.

  • : Recalling that , we find the coefficient of as

    Here, in the last step, we have used

    where represent the common indices, which contribute equally in counting both and .

Combining the three cases above and using (3.13), we obtain (3.15). ∎

The main result in this section is shown next.

Proof of Lemma 3.2.

From (3.12) and (3.15), it is obvious that the coefficient function of for does not change sign in , as . By Lemma 2.1 and equation (3.15), we have

In the interior of , we have , , which implies that . ∎

4. Unisolvence for the quasi-polynomial spaces of second kind

We now prove the unisolvence of the degrees of freedom for quasi-polynomials derived from polynomial Hilbert complexes of second kind which are defined as

(4.1)

4.1. Geometrical decomposition of the polynomial spaces of second kind

We now consider the geometrical decomposition of the polynomial spaces associated with the Hilbert complexes of second kind: and . We point out that there is a little (if any) analogy in the proofs for these spaces.

Following [1, Section 4.5], the degrees of freedom for are

(4.2)

Further, the characterization of the trace free part of the is stated below.

Theorem 4.1 (Theorem 4.22 in [1]).

For , , the map

(4.3)

where the , defines an isomorphism of onto .

4.2. Polynomials of second kind with vanishing trace

We now state the main result of this section showing unisolvence of the functionals used for degrees of freedom (4.2).

Lemma 4.2.

Let . Suppose that

(4.4)