Critical Functions and Inf-Sup Stability of Crouzeix-Raviart Elements

In this paper, we prove that Crouzeix-Raviart finite elements of polynomial order p≥5, p odd, are inf-sup stable for the Stokes problem on triangulations. For p≥4, p even, the stability was proved by Á. Baran and G. Stoyan in 2007 by using the macroelement technique, a dimension formula, the concept of critical points in a triangulation and a representation of the corresponding critical functions. Baran and Stoyan proved that these critical functions belong to the range of the divergence operator applied to Crouzeix-Raviart velocity functions and the macroelement technique implies the inf-sup stability. The generalization of this theory to cover odd polynomial orders p≥5 is involved; one reason is that the macroelement classes, which have been used for even p, are unsuitable for odd p. In this paper, we introduce a new and simple representation of non-conforming Crouzeix-Raviart basis functions of odd degree. We employ only one type of macroelement and derive representations of all possible critical functions. Finally, we show that they are in the range of the divergence operator applied to Crouzeix-Raviart velocities from which the stability of the discretization follows.

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1 Introduction

In the seminal paper [10] in 1973, Crouzeix and Raviart developed a non-conforming family of finite elements with the goal to obtain a stable discretization of the Stokes equation with relatively few unknowns. The family was indexed by the (local polynomial) order, say111 and ., , for the approximation of the velocity field, while the pressure is approximated by discontinuous local polynomials of degree . In their paper, the authors prove that for and spatial dimension , their non-conforming finite element leads to a stable discretization of the Stokes equation. Since then the development of pairs of finite elements for the velocity and for the pressure of the Stokes problem was the topic of vivid research in numerical analysis. Surprisingly, the problem is still not fully settled with some open issues for problems in two spatial dimension, while much less is known for the discretization of three-dimensional Stokes problems.

An important theoretical approach to prove the stability of Crouzeix-Raviart elements for the Stokes problem is based on the macroelement technique which goes back to [20], [21]. If the divergence operator maps a localized velocity space on the macroelements onto the local pressure space (modulo the constant function), the stability of the discretization follows. It has been shown in [19] that this surjectivity holds already for continuous velocities for if the mesh does not contain critical points (cf. Def. 3.10). If the mesh contains critical points an established idea to prove inf-sup stability is as follows: first, one identifies the critical functions whose span has zero intersection with the range of the local divergence operator applied to continuous velocities. Then, one proves that the critical functions lie in the span of the divergence operator applied to Crouzeix-Raviart velocities and stability of the discretization follows.

The case for even was proved along these lines in [2]. In that paper, seven types of non-overlapping macroelements are considered consisting of two and three triangles, the critical functions are identified, and it is shown that these belong to the range of the local divergence operator applied to the localized Crouzeix-Raviart velocities so that inf-sup stability follows for this case. In our paper, we focus on the case of odd and proceed conceptually in the same way. However, it turns out that the macroelements, which are considered in [2], are not suited for odd . One reason is that the non-conforming basis functions for even are more local (support on one triangle) compared to odd (support on two adjacent triangles). Instead, we consider here nodal patches (element stars) for the interior vertices of the triangulation as the only type of overlapping macroelements. We identify the critical functions in §3.2, which are related to critical points (cf. Def. 3.10) in the nodal patches. We show that these critical functions form a basis of the complement in the pressure space of the range of the divergence operator applied to the localized continuous velocities; the proof employs the “dimension formula” in [19] and hence is restricted to . Then we show that the critical functions belong to the range of the divergence operator applied to Crouzeix-Raviart velocities and this implies the stability of the discretization.

The main achievements in this paper are as follows. A new and simple representation of Crouzeix-Raviart basis functions for odd is introduced in Section 3.1. We identify the critical pressure functions for odd and continuous localized velocities in Definition 3.12 and show that they form a basis for the complement of the range of the divergence operator applied to localized continuous velocities. Finally, we prove that the critical functions belong to the range of the divergence operator applied to localized Crouzeix-Raviart velocities.

This leads to the main conclusion: for , let denote the (scalar) Crouzeix-Raviart finite element space with local polynomial order on regular triangulations , i.e., without hanging nodes, of a two-dimensional bounded polygonal Lipschitz domain obeying zero-boundary conditions in a “Crouzeix-Raviart” sense. Let denote the discontinuous finite element space of local polynomial order on this triangulation. Then

(1.1)

for odd .

In the following we comment on the remaining cases: and even . In [10], statement (1.1) is proved for (and for spatial dimensions ). From [2] we know that assertion (1.1) is true for even . For , the result we proved already in [13]. For the case and odd, statement (1.1) follows from [19] (see also [16]) provided the triangulation does not contain critical points. In [9], the case is considered and the claim (1.1) is proved if the triangulation does not contain critical points and an additional technical condition for the nodal points is satisfied. The case of statement (1.1) for any regular triangulation is proved in [4] which is the companion to this paper. The proof in [4] circumvents the concept of critical points and functions and reduces the problem to a purely algebraic problem in that to determe the nullspace for a coefficient matrix of a linear system on the local nodal patches. The proof in [4] applies also to odd but does not give insight on the mechanism how the critical functions are eliminated by the non-conforming Crouzeix-Raviart functions.

The paper is structured as follows. In Section 2 we formulate the Stokes equation in variational form and introduce the functional analytic setting in §2.1. The discretization is based on non-conforming Galerkin finite element methods on regular triangulations. The finite element spaces, in particular the Crouzeix-Raviart finite elements of polynomial order are introduced in Section 2.2. The inf-sup condition and the main theorem (Thm. 2.2) are stated at the end of this section.

The final section §3 of the paper is devoted to the proof of the main theorem. In the appendices §A, §B, §C we provide some technical properties on derivatives of barycentric coordinates, determinants of tridiagonal matrices, and closed form integrals of some products of Jacobi polynomials.

2 Setting

2.1 The Continuous Stokes Problem

Let denote a bounded polygonal domain with boundary . Our goal is to find a family of pairs of finite element spaces for the stable numerical solution of the Stokes equation. On the continuous level, the strong form of the Stokes equation is given by

with boundary conditions for the velocity and a normalization condition for the pressure

To state the classical existence and uniqueness result we formulate this equation in a variational form and first introduce the relevant function spaces. Throughout the paper we restrict to vector spaces over the field of real numbers.

For , , denote the classical Sobolev spaces of functions with norm . As usual we write instead of and for . For , we denote by the closure with respect to the norm of the space of infinitely smooth functions with compact support in . Its dual space is denoted by .

The scalar product and norm in are denoted respectively by

Vector-valued and tensor-valued analogues of the function spaces are denoted by bold and blackboard bold letters, e.g., and and analogously for other quantities.

The scalar product and norm for vector valued functions are given by

where denotes the Euclidean scalar product in . In a similar fashion, we define for the scalar product and norm by

where . Finally, let .

We introduce the bilinear form by

where and denote the derivatives (Jacobi matrices) of and . The variational form of the Stokes problem is given by: For given

(2.1)

It is well-known (see, e.g., [14]) that (2.1) is well posed.

2.2 Numerical Discretization of the Stokes Problem

In the following we introduce a discretization for problem (2.1). Let denote a regular triangulation of consisting of closed triangles which have the property that the intersection of two different triangles , is either empty, a common edge, or a common point. We also assume , where

(2.2)

and denotes the interior of a set . An important measure for the quality of a finite element triangulation is the shape-regularity constant, which we define by

(2.3)

with the local mesh width and denoting the diameter of the largest inscribed ball in .

The set of edges in are denoted by , while the subset of boundary edges is ; the subset of inner edges is given by . The set of triangle vertices in is denoted by , while the subset of inner vertices is and . For , we define the edge patch by

For , the nodal patch is defined by

(2.4)

We will need an additional mesh parameter. For a regular triangulation of , let

Then, the constant222By we denote the cardinality of a discrete set (cf. Notation 3.1).

(2.5)

denotes the number of triangles in the triangulation which are not connected to an inner point.

For , we employ the usual multiindex notation for and points

For and , we define the index sets

Let denote the space of -variate polynomials of maximal degree , consisting of functions of the form

for real coefficients . Formally, we set . To indicate the domain of definition we write sometimes for and skip the index since it is then clear from the argument .

For and a regular triangulation for the domain , let

We introduce the following finite element spaces

(2.6)

Furthermore, let

The vector-valued versions are denoted by and . Finally, we define the Crouzeix-Raviart space by

(2.7a)
(2.7b)

Here, denotes the jump of across an edge and is the space of polynomials of maximal degree with respect to the local variable in .

We have collected all ingredients for defining the Crouzeix-Raviart discretization for the Stokes equation. For , let the discrete velocity space and pressure space be defined by

Then, the discretization is given by: find such that

(2.8)
Definition 2.1

Let denote a regular triangulation for . A pair is inf-sup stable if there exists a constant such that

(2.9)

We are now in the position to formulate our main theorem.

Theorem 2.2

Let be a bounded polygonal Lipschitz domain and let denote a regular triangulation of , which contains at least one inner node. Then, the inf-sup condition (2.9) holds for a constant , which depends on the shape regularity of the mesh, the constant in (2.5), and the polynomial degree .

We emphasize that the original definition in [10] allows for slightly more general finite element spaces, more precisely, the spaces can be enriched by locally supported functions. From this point of view, the definition (2.7) describes the minimal Crouzeix-Raviart space. The possibility for enrichment has been used frequently in the literature to prove inf-sup stability for the arising finite element spaces (see, e.g., [10], [15], [18]). In contrast, we will prove the stability for the minimal Crouzeix-Raviart family.

3 Proof of Theorem 2.2

In [10], Theorem 2.2 is proved for (and for spatial dimensions ). From [2] we know that the theorem is true for even . In [13], the result is proved for . In this section, we will prove the result for odd and refer for the proof of the case to [4].

3.1 Barycentric Coordinates and Basis Functions for the Velocity

In this section, we introduce basis functions for the finite element spaces in Section 2.2. We begin with introducing some general notation.

Notation 3.1

For vectors , , we write for the matrix with column vectors . For we set . Let  be the -th canonical unit vector in

Vertices in a triangle are always numbered counterclockwise. In a triangle with vertices , , the angle at is called or alternatively where are pairwise different. If a triangle is numbered by an index (e.g., ), the angle at is called or alternatively . For quantities in a triangle as, e.g., angles , , we use the cyclic numbering convention and .

For a -dimensional measurable set we write for its measure; for a discrete set, say , we denote by its cardinality.

In the proofs, we consider frequently nodal patches for inner vertices . The number denotes the number of triangles in . Various quantities in this patch such as, e.g., the triangles in , have an index which runs from to . Here, we use the cyclic numbering convention and and apply this analogously for other quantities in the nodal patch.

Let the closed reference triangle be the triangle with vertices , , . The nodal points on the reference element of order are given by

For a triangle , we denote by an affine bijection. The mapped nodal points of order on are given by

The nodal points of order on are defined by

We introduce the well-known Lagrange basis for the space , which is indexed by the nodal points and characterized by

(3.1)

where is the Kronecker delta. A basis for the space is given by , .

Next, we define a basis for the Crouzeix-Raviart space. Let and . The Jacobi polynomial is a polynomial of degree such that

for all polynomials of degree less than , and (cf. [11, Table 18.6.1])

(3.2)

Here the shifted factorial is defined by for and . Note that are the Legendre polynomials (see [11, 18.7.9]).

Let denote a triangle with vertices , , and let be the barycentric coordinate for the node defined by

(3.3)

If the numbering of the vertices in is fixed, we write short for and for :

(3.4)

For the barycentric coordinate on the reference element for the vertex we write , .

Definition 3.2

Let be even and . Then, the non-conforming triangle bubble is given by

For odd and , the non-conforming edge bubble is given by

where denotes the vertex in which is opposite to the edge .

Different representations of the functions , exist in the literature, see [23], [1], [5, for ], [7] while the formula for has been introduced in [2].

Theorem 3.3

A basis for the space is given

  • for even by

  • for odd by

Proof. For even , this follows from [23, Rem. 3] in combination with [7, Thm. 22].

For odd , we first observe that

and for and , the restriction is the Legendre polynomial (lifted to the edge ) with endpoint values at and at . Hence, the assertion follows from [7, Thm. 22] 

Corollary 3.4

A basis for the space is given

  1. for even by

    (3.5)
  2. for odd by

    (3.6)

Here, for any nodal point

, the linearly independent vectors

can be chosen arbitrarily. The same holds for any triangle for the vectors in (3.5) and for any for the vectors in (3.6).

Remark 3.5

The original definition by [10] is implicit and given for regular simplicial finite element meshes in , . For their practical implementation, a basis is needed and Corollary 3.4 provides a simple definition. A basis for Crouzeix-Raviart finite elements in has been introduced in [12] for and a general construction is given in [8].

3.2 The Pressure Kernel

For the investigation of the discrete inf-sup condition we employ the macroelement technique in the form described in [21] (see also [20]).

Let us first assume that every triangle has a vertex . As a consequence, the sets , , with nodal patches form a macroelement partitioning of in the sense of [21]. We define the spaces

(3.7)
Remark 3.6

The definition of the Crouzeix-Raviart spaces (2.7b) implies and, in turn,

(3.8)

Let denote the function with constant value . Then, an integration by parts implies so that .

The following Theorem is a direct consequence of [21, Thm. 2.1].

Theorem 3.7

Let be a regular finite element triangulation of a bounded polygonal domain as in §2.2 with shape regularity constant and at least one inner vertex. Let . If

(3.9)

then the discrete inf-sup condition (2.9) is satisfied with a constant depending only on , , and (cf. (2.3), (2.5)).

The discrete Stokes equation (2.8) has a unique solution .

Remark 3.8

If the assumptions of Theorem 3.7

are satisfied, various types of error estimates follow from well-established theory. Since this is not the major theme of our paper, we briefly summarize two approaches: The error

can be estimated by means of the second Strang lemma (see [3], [6, Thm. 4.2.2]) via the sum of a quasi-optimal term and a consistency error. The latter one converges with optimal rates to zero by assuming sufficiently high regularity of the continuous solution (see [10, Thm. 3], [5, Thm. 2.2]).

There are also methods to establish quasi-optimality of non-conforming Crouzeix-Raviart discretizations. We mention the paper [24], where a mapping from the non-conforming space to a conforming one is introduced and employed in the discretization. For this method, quasi-optimal error estimates can be proved.

Assumption (3.9) is proved for and , even. Also note that we may restrict to triangulations with the property that any has an inner vertex since the result for triangulations, which contain at least one inner vertex, is implied by the following lemma.

Lemma 3.9

Let denote a regular triangulation of a domain . Let denote a further triangle (not contained in ) which is attached to by an edge , more precisely:

  1. and there exists some and such that ,

  2. Let and (cf. (2.2)) is a bounded, polygonal Lipschitz domain in .

If the pair is inf-sup stable, then is inf-sup stable with a constant which depends on , the shape regularity of and the polynomial degree .

A proof for is given in [9, Lem. 6.2]. Inspection of the proof shows that it applies also to general provided the assumptions of the lemma are satisfied. We omit the repetition of the arguments here.

Taking into account Lemma 3.9 and the known cases from the quoted literature, we assume for the following

(3.10)

In the following we investigate the condition (3.9) and start with the definition of critical points of a nodal patch for (see [19]).

Definition 3.10

Let denote a triangulation as in §2.2. For , we denote by , the nodal patch as in (2.4) and let

A point is a critical point for if there exist two straight infinite lines , in such that all edges having as an endpoint satisfy . The set of all critical points in is and its cardinality denoted by .

Remark 3.11

Geometric configurations where critical points occur are well studied in the literature (see, e.g., [19]). For nodal patches , any critical point belongs to one of the following cases:

  1. consists of four triangles and is the intersections of the two diagonals in . Then is a critical point; see Fig. 1. In this case, it holds .

  2. . Let denote the edge with endpoints and and let be the adjacent triangles. Then, is a critical point if the sum of the two angles at in the triangles equals ; see Fig. 2. In this case, it holds .

From [19] we know that for it holds

(3.11)

Remark 3.6 implies . In this section, we define functions for the critical points such that the functions and , are linearly independent and belong to .

Afterwards, we will prove the implication:

(3.12)

Hence, in all cases in (3.10), where the dimension formula (3.11) holds (e.g. for ) the condition (3.9) and, in turn, the assumptions of Theorem 3.7 are satisfied.

To prove (3.12) it is sufficient to consider nodal patches with critical points, i.e., , and we will construct basis functions for explicitly. We fix a (non-unique) sign function by the condition:

Definition 3.12

Let be a critical point for . The critical function for is given by

Lemma 3.13

Let (3.10) be satisfied. The functions and , , are linearly independent and belong to . If the dimension formula (3.11) holds, they form a basis of .

Proof. From Remark 3.6 it follows that . Next, we prove that . Let with . Let , , denote the vertices of with the convention .

Recall the notation for barycentric coordinates in (3.3), (3.4) and that , , denotes the -th canonical unit vector in . Let , denote two linearly independent vectors – the precise choice will be fixed later.

The restriction of the space to is spanned by (cf. (3.4))

(3.13)

Let and let denote the Gâteaux derivative of a function in the direction . Then

There exists such that . We employ [2, Prop. 4.1] which tells us (as a consequence of [17, (3.10)]) that for it holds

(3.14)

For with , this implies

(3.15)

Taking into account the orthogonality relations (3.14) and that the factor in (3.15) vanishes in many cases, it remains to consider the cases

(3.16)

with corresponding integrals

All other integrals in (3.15) vanish.

A standard affine transformation to the reference element shows that

(3.17)

where

It remains to prove the second equality in

for (cf. (3.16)). By using (3.17) we obtain

(3.18)

Evaluation of the right-hand side in (3.18).

The basis functions in can be grouped into three different types.

  1. Basis functions whose support is one triangle . Let , , denote the vertices of . Then, these basis functions are given by

    From (3.18) it follows that

  2. Basis functions whose support is an edge patch. Let with and denote by and the triangles in which share . We denote the vertices in and by and