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

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 $N = {1, 2, \dots}$ and $N_{0} = N \cup {0}$ ., $p \in N$ , for the approximation of the velocity field, while the pressure is approximated by discontinuous local polynomials of degree $p - 1$ . In their paper, the authors prove that for $p = 1$ and spatial dimension $d = 2, 3$ , 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 $p \geq 4$ 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 $p \geq 4$ 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 $p \geq 5$ and proceed conceptually in the same way. However, it turns out that the macroelements, which are considered in [2], are not suited for odd $p$ . One reason is that the non-conforming basis functions for even $p$ are more local (support on one triangle) compared to odd $p$ (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 $p \geq 4$ . 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 $p$ is introduced in Section 3.1. We identify the critical pressure functions for odd $p \geq 5$ 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 $p \in N$ , let $C R_{p, 0} (T)$ denote the (scalar) Crouzeix-Raviart finite element space with local polynomial order $p$ on regular triangulations $T$ , i.e., without hanging nodes, of a two-dimensional bounded polygonal Lipschitz domain $Ω \subset R^{2}$ obeying zero-boundary conditions in a “Crouzeix-Raviart” sense. Let $P_{p - 1} (T)$ denote the discontinuous finite element space of local polynomial order $p - 1$ on this triangulation. Then

({(C R_{p, 0} (T))}_{p, 0}^{2}, P_{p - 1} (T) / R) is a stable finite element for the Stokes equation

(1.1)

for odd $p \geq 5$ .

In the following we comment on the remaining cases: $p = 1, 2, 3$ and even $p \geq 4$ . In [10], statement (1.1) is proved for $p = 1$ (and for spatial dimensions $d = 2, 3$ ). From [2] we know that assertion (1.1) is true for even $p \geq 4$ . For $p = 2$ , the result we proved already in [13]. For the case $p \geq 5$ and $p$ odd, statement (1.1) follows from [19] (see also [16]) provided the triangulation does not contain critical points. In [9], the case $p = 3$ 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 $p = 3$ 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 $p \geq 3$ 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 $p$ 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 $Ω \subset R^{2}$ denote a bounded polygonal domain with boundary $\partial Ω$ . 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

\begin{matrix} - Δ u & + \nabla p & = f & in Ω, div u & = 0 & in Ω \end{matrix}

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

u=0on ∂Ωand\quad∫Ωp=0.

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 $s \geq 0$ , $1 \leq p \leq \infty$ , $W^{s, p} (Ω)$ denote the classical Sobolev spaces of functions with norm ${∥ \cdot ∥}_{W^{s, p} (Ω)}$ . As usual we write $L^{p} (Ω)$ instead of $W^{0, p} (Ω)$ and $H^{s} (Ω)$ for $W^{s, 2} (Ω)$ . For $s \geq 0$ , we denote by $H_{0}^{s} (Ω)$ the closure with respect to the $H^{s} (Ω)$ norm of the space of infinitely smooth functions with compact support in $Ω$ . Its dual space is denoted by $H^{- s} (Ω)$ .

The scalar product and norm in $L^{2} (Ω)$ are denoted respectively by

\begin{matrix} {(u, v)}_{L^{2} (Ω)} := \int_{Ω} u v & and & {∥ u ∥}_{L^{2} (Ω)} := {(u, u)}_{L^{2} (Ω)}^{1 / 2} & in L^{2} (Ω) . \end{matrix}

Vector-valued and $2 \times 2$ tensor-valued analogues of the function spaces are denoted by bold and blackboard bold letters, e.g., $H^{s} (Ω) = {(H^{s} (Ω))}^{2}$ and $H^{s} = {(H^{s} (Ω))}^{2 \times 2}$ and analogously for other quantities.

The $L^{2} (Ω)$ scalar product and norm for vector valued functions are given by

where $⟨ u, v ⟩$ denotes the Euclidean scalar product in $R^{2}$ . In a similar fashion, we define for $G, H \in L^{2 \times 2} (Ω)$ the scalar product and norm by

(G,H)L2×2(Ω):=∫Ω⟨G,H⟩and% \quad∥G∥L2×2(Ω):=(G,G)1/2L2×2(Ω),

where $⟨ G, H ⟩ = \sum_{i, j = 1}^{2} G_{i, j} H_{i, j}$ . Finally, let $L_{0}^{2} (Ω) := {u \in L^{2} (Ω) : \int_{Ω} u = 0}$ .

We introduce the bilinear form $a : H^{1} (Ω) \times H^{1} (Ω) \to R$ by

a (u, v) := {(\nabla u, \nabla v)}_{L^{2 \times 2} (Ω)},

where $\nabla u$ and $\nabla v$ denote the derivatives (Jacobi matrices) of $u$ and $v$ . The variational form of the Stokes problem is given by: For given $f \in H^{- 1} (Ω),$

(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 $T = {K_{i} : 1 \leq i \leq n}$ denote a regular triangulation of $Ω$ consisting of closed triangles $K_{i}$ which have the property that the intersection of two different triangles $K_{i}$ , $K_{j}$ is either empty, a common edge, or a common point. We also assume $Ω = dom T$ , where

dom T := int (⋃ K \in T K)

(2.2)

and $int (M)$ denotes the interior of a set $M \subset R^{2}$ . An important measure for the quality of a finite element triangulation is the shape-regularity constant, which we define by

γ_{T} := max K \in T \frac{h_{K}}{ρ_{K}}

(2.3)

with the local mesh width $h_{K} := diam K$ and $ρ_{K}$ denoting the diameter of the largest inscribed ball in $K$ .

The set of edges in $T$ are denoted by $E$ , while the subset of boundary edges is $E_{\partial Ω} := {E \in E : E \subset \partial Ω}$ ; the subset of inner edges is given by $E_{Ω} := E ∖ E_{\partial Ω}$ . The set of triangle vertices in $T$ is denoted by $V$ , while the subset of inner vertices is $V_{Ω} := {V \in V : V \notin \partial Ω}$ and $V_{\partial Ω} := V ∖ V_{Ω}$ . For $E \in E$ , we define the edge patch by

TE:={K∈T:E⊂K}and\quadωE:=⋃K∈TEK.

For $z \in V$ , the nodal patch is defined by

(2.4)

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

T^{'} := ⋃ z \in V_{Ω} T_{z} .

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

d_{T} := ∣ ∣ T ∖ T^{'} ∣ ∣

(2.5)

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

For $m \in N$ , we employ the usual multiindex notation for $\boldmathμ=(μi)mi=1∈Nm0$ and points $x = {(x_{i})}_{i}^{m} \in R^{m}$

∣∣\boldmathμ∣∣:=μ1+…+μm,x\boldmathμ:=m∏j=1xμjj.

For $k \in N_{0}$ and $m \in N$ , we define the index sets

Im≤k:={\boldmathμ∈Nm0∣∣∣\boldmathμ∣∣≤k}and\quadIm=k:={\boldmathμ∈Nm0∣∣∣% \boldmathμ∣∣=k}

Let $P_{m, k}$ denote the space of $m$ -variate polynomials of maximal degree $k$ , consisting of functions of the form

∑\boldmathμ∈Im≤ka\boldmathμx\boldmathμ

for real coefficients $a\boldmathμ$ . Formally, we set $P_{m, - 1} := {0}$ . To indicate the domain of definition we write sometimes $P_{k} (D)$ for $D \subset R^{m}$ and skip the index $m$ since it is then clear from the argument $D$ .

For $s \geq 0$ and a regular triangulation $T$ for the domain $Ω$ , let

H^{s} (T) := {u \in L^{2} (Ω) ∣ \forall K \in T : {u |}_{K} \in H^{1} (K)} .

We introduce the following finite element spaces

\begin{matrix} P_{k} (T) := {q \in L^{2} (Ω) ∣ \forall K \in T : {q |}_{K} \in P_{k} (K)}, and (cf. (???)) & P_{k} (T) / R := {q \in P_{k} (T) : \int_{dom T} q = 0} . \end{matrix}

(2.6)

Furthermore, let

\begin{matrix} S_{k} (T) := {v \in C^{0} (Ω) ∣ \forall K \in T : {v |}_{K} \in P_{k} (K)}, and & S_{k, 0} (T) := S_{k} (T) \cap H_{0}^{1} (dom T) . \end{matrix}

The vector-valued versions are denoted by $S_{k} (T) := S_{k} {(T)}^{2}$ and $S_{k, 0} (T) := S_{k, 0} {(T)}^{2}$ . Finally, we define the Crouzeix-Raviart space by

	$C R_{k} (T)$	$:= {v \in P_{k} (T) ∣ \forall q \in P_{k - 1} (E) \forall E \in E_{Ω} \int_{E} {[v]}_{E} q = 0},$		(2.7a)
	$C R_{k, 0} (T)$	$:= {v \in C R_{k} (T) ∣ \forall q \in P_{k - 1} (E) \forall E \in E_{\partial Ω} \int_{E} v q = 0} .$		(2.7b)

Here, ${[v]}_{E}$ denotes the jump of $v \in P_{k} (T)$ across an edge $E \in E_{Ω}$ and $P_{k - 1} (E)$ is the space of polynomials of maximal degree $k - 1$ with respect to the local variable in $E$ .

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

Then, the discretization is given by: find $(u_{CR}, p_{disc}) \in {C R}_{p, 0} (T) \times M_{p - 1} (T)$ such that

⎧ ⎨ ⎩ \begin{matrix} a (u_{CR}, v) - {(p_{disc}, div v)}_{L^{2} (Ω)} & = {(f, v)}_{L^{2} (Ω)} & \forall v \in {C R}_{p, 0} (T), {(div u_{CR}, q)}_{L^{2} (Ω)} & = 0 & \forall q \in M_{p - 1} (T) . \end{matrix}

(2.8)

Definition 2.1

Let $T$ denote a regular triangulation for $Ω$ . A pair ${C R}_{p, 0} (T) \times M_{p - 1} (T)$ is inf-sup stable if there exists a constant $c_{T, p}$ such that

inf p \in M_{p - 1} (T) ∖ {0} sup v \in S_{p, 0}^{CR} (T) ∖ {0} \frac{{(p, div v)}_{L^{2} (Ω)}}{{∥ v ∥}_{H^{1} (Ω)} {∥ p ∥}_{L^{2} (Ω)}} \geq c_{T, p} > 0.

(2.9)

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

Theorem 2.2

Let $Ω \subset R^{2}$ be a bounded polygonal Lipschitz domain and let $T$ denote a regular triangulation of $Ω$ , which contains at least one inner node. Then, the inf-sup condition (2.9) holds for a constant $c_{T, p}$ , which depends on the shape regularity of the mesh, the constant $d_{T}$ in (2.5), and the polynomial degree $p$ .

We emphasize that the original definition in [10] allows for slightly more general finite element spaces, more precisely, the spaces $C R_{p} (T)$ 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 $p = 1$ (and for spatial dimensions $d = 2, 3$ ). From [2] we know that the theorem is true for even $p \geq 4$ . In [13], the result is proved for $p = 2$ . In this section, we will prove the result for odd $p \geq 5$ and refer for the proof of the case $p = 3$ 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 $a_{i} \in R^{n}$ , $1 \leq i \leq m$ , we write $[a_{1} ∣ a_{2} ∣ \dots ∣ a_{m}]$ for the $n \times m$ matrix with column vectors $a_{i}$ . For $v = (\frac{v_{1}}{v_{2}}) \in R^{2}$ we set $v^{⊥} = {(v_{2}, - v_{1})}_{2}^{T}$ . Let $e_{k, i} \in R^{k}$ be the $i$ -th canonical unit vector in $R^{k} .$

Vertices in a triangle are always numbered counterclockwise. In a triangle $K$ with vertices $A_{1}$ , $A_{2}$ , $A_{3}$ the angle at $A_{i}$ is called $α_{i}$ or alternatively $α_{j, k}$ where $i, j, k \in {1, 2, 3}$ are pairwise different. If a triangle is numbered by an index (e.g., $K_{ℓ}$ ), the angle at $A_{ℓ, i}$ is called $α_{ℓ, i}$ or alternatively $α_{ℓ, j, k}$ . For quantities in a triangle $K$ as, e.g., angles $α_{j}$ , $1 \leq j \leq 3$ , we use the cyclic numbering convention $α_{3 + 1} := α_{1}$ and $α_{1 - 1} := α_{3}$ .

For a $d$ -dimensional measurable set $D,$ we write $| D |$ for its measure; for a discrete set, say $J$ , we denote by $| J |$ its cardinality.

In the proofs, we consider frequently nodal patches $T_{z}$ for inner vertices $z \in V_{Ω}$ . The number $m$ denotes the number of triangles in $T_{z}$ . Various quantities in this patch such as, e.g., the triangles in $T_{z}$ , have an index which runs from $1$ to $m$ . Here, we use the cyclic numbering convention $K_{m + 1} := K_{1}$ and $K_{1 - 1} := K_{m}$ and apply this analogously for other quantities in the nodal patch.

Let the closed reference triangle $ˆ K$ be the triangle with vertices ${^A}_{1} := (0, 0)$ , ${^A}_{2} := (1, 0)$ , ${^A}_{3} := (0, 1)$ . The nodal points on the reference element of order $k \in N_{0}$ are given by

ˆNk:=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩{1k%\boldmath$μ$∣\boldmathμ∈I2≤k}k≥1,{(13,13)}k=0.

For a triangle $K \subset R^{2}$ , we denote by $χ_{K} : ˆ K \to K$ an affine bijection. The mapped nodal points of order $k \in N_{0}$ on $K$ are given by

N_{k} (K) := {χ_{K} (z) : z \in {ˆ N}_{k}} .

The nodal points of order $k$ on $T$ are defined by

Nk(T):=⋃K∈TNk(K)and\quadNk∂Ω(T):=Nk(T)∩∂Ω,Nk,Ω(T):=Nk(T)∩Ω.

We introduce the well-known Lagrange basis for the space $S_{k} (T)$ , which is indexed by the nodal points $z \in N_{k} (T)$ and characterized by

Bk,z∈Sk(T)and\quad∀z′∈Nk(T)Bk,z(z′)=δz,z′,

(3.1)

where $δ_{z, z^{'}}$ is the Kronecker delta. A basis for the space $S_{k, 0} (T)$ is given by $B_{k, z}$ , $z \in N_{k, Ω} (T)$ .

Next, we define a basis for the Crouzeix-Raviart space. Let $α, β > - 1$ and $n \in N_{0}$ . The Jacobi polynomial $P_{n}^{(α, β)}$ is a polynomial of degree $n$ such that

\int_{- 1}^{1} P_{n}^{(α, β)} (x) q (x) {(1 - x)}^{α} {(1 + x)}^{β} d x = 0

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

P_{n}^{(α, β)} (1) = \frac{{(α + 1)}_{n}}{n!}, P_{n}^{(α, β)} (- 1) = {(- 1)}^{n} \frac{{(β + 1)}_{n}}{n!} .

(3.2)

Here the shifted factorial is defined by ${(a)}_{n} := a (a + 1) \dots (a + n - 1)$ for $n > 0$ and ${(a)}_{0} := 1$ . Note that $P_{k}^{(0, 0)}$ are the Legendre polynomials (see [11, 18.7.9]).

Let $K$ denote a triangle with vertices $A_{i}$ , $1 \leq i \leq 3$ , and let $λ_{K, A_{i}} \in P_{1} (K)$ be the barycentric coordinate for the node $A_{i}$ defined by

λ_{K, A_{i}} (A_{j}) = δ_{i, j} 1 \leq i, j \leq 3.

(3.3)

If the numbering of the vertices in $K$ is fixed, we write $λ_{K, i}$ short for $λ_{K, A_{i}}$ and for $\boldmathμ∈N30$ :

\boldmathλ\boldmathμK=λμ1K,1λμ2K,2λμ3K,3.

(3.4)

For the barycentric coordinate on the reference element $ˆ K$ for the vertex ${^A}_{j}$ we write ${ˆ λ}_{j}$ , $j = 1, 2, 3$ .

Definition 3.2

Let $p \in N$ be even and $K \in T$ . Then, the non-conforming triangle bubble is given by

B_{p, K}^{CR} := ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{1}{2} (- 1 + 3 \sum i = 1 P_{p}^{(0, 0)} (1 - 2 λ_{K, i})) & on K, 0 & on Ω ∖ K . \end{matrix}

For $p$ odd and $E \in E$ , the non-conforming edge bubble is given by

B_{p, E}^{CR} := ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} P_{p}^{(0, 0)} (1 - 2 λ_{K, A_{K, E}}) & on K for K \in T_{E}, 0 & on Ω ∖ ω_{E}, \end{matrix}

where $A_{K, E}$ denotes the vertex in $K$ which is opposite to the edge $E$ .

Different representations of the functions $B_{p, E}^{CR}$ , $B_{p, K}^{CR}$ exist in the literature, see [23], [1], [5, for $p = 4, 6.$ ], [7] while the formula for $B_{p, K}^{CR}$ has been introduced in [2].

Theorem 3.3

A basis for the space $C R_{p, 0} (T)$ is given

for even $p$ by

${B_{p, z}, z \in N_{p, Ω} (T)} \cup {B_{p, K}^{CR}, K \in T},$
for odd $p$ by

${B_{p, z}, z \in N_{p, Ω} (T) ∖ V_{Ω}} \cup {B_{p, E}^{* C R}, E \in E_{Ω}} .$

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

For odd $p$ , we first observe that

{B_{p, E}^{CR} ∣ ∣}_{E} = 1 so that {B_{p, E}^{CR} ∣ ∣}_{ω_{E}} \in C^{0} (ω_{E})

and for $K \in T_{E}$ and $E^{'} \subset \partial K ∖ E$ , the restriction ${B_{p, E}^{CR} ∣ ∣}_{E^{'}}$ is the Legendre polynomial (lifted to the edge $E^{'}$ ) with endpoint values $1$ at $\partial E$ and $- 1$ at $A_{K, E}$ . Hence, the assertion follows from [7, Thm. 22].

Corollary 3.4

A basis for the space $S_{p, 0}^{CR} (T)$ is given

for even $p$ by

(3.5)
for odd $p$ by

(3.6)

Here, for any nodal point $z$

, the linearly independent vectors

v_{z}, w_{z} \in R^{2}

can be chosen arbitrarily. The same holds for any triangle

K

for the vectors

v_{K}, w_{K} \in R^{2}

in (3.5) and for any

E \in E_{Ω}

for the vectors

v_{E}, w_{E} \in R^{2}

in (3.6).

Remark 3.5

The original definition by [10] is implicit and given for regular simplicial finite element meshes in $R^{d}$ , $d = 2, 3$ . For their practical implementation, a basis is needed and Corollary 3.4 provides a simple definition. A basis for Crouzeix-Raviart finite elements in $R^{3}$ has been introduced in [12] for $p = 2$ 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 $K \in T$ has a vertex $z \in Ω$ . As a consequence, the sets $T_{z}$ , $z \in V_{Ω}$ , with nodal patches $ω_{z}$ form a macroelement partitioning of $Ω$ in the sense of [21]. We define the spaces

	$N_{p, z}^{CR}$			(3.7)
	$N_{p, z}$

Remark 3.6

The definition of the Crouzeix-Raviart spaces (2.7b) implies $S_{p, 0} (T_{z}) \subset C R_{p, 0} (T_{z})$ and, in turn,

N_{p, z}^{CR} \subset N_{p, z} .

(3.8)

Let $1_{z} : ω_{z} \to R$ denote the function with constant value $1$ . Then, an integration by parts implies $1_{z} \in N_{p, z}^{CR}$ so that $dim N_{p, z}^{CR} \geq 1$ .

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

Theorem 3.7

Let $T$ be a regular finite element triangulation of a bounded polygonal domain $Ω \subset R^{2}$ as in §2.2 with shape regularity constant $γ_{T}$ and at least one inner vertex. Let $p \in N$ . If

(3.9)

then the discrete inf-sup condition (2.9) is satisfied with a constant $c_{T, p}$ depending only on $p$ , $γ_{T}$ , and $d_{T}$ (cf. (2.3), (2.5)).

The discrete Stokes equation (2.8) has a unique solution $(u_{CR}, p_{disc}) \in S_{p, 0}^{CR} (T) \times M_{p - 1} (T)$ .

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 $p = 1, 2$ and $p \geq 4$ , $p$ even. Also note that we may restrict to triangulations with the property that any $K \in T$ 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 $T$ denote a regular triangulation of a domain $Ω$ . Let $K^{'}$ denote a further triangle (not contained in $T$ ) which is attached to $T$ by an edge $E$ , more precisely:

$K^{'} \cap Ω = \emptyset$ and there exists some $K \in T$ and $E \in E$ such that $K^{'} \cap K = E$ ,
Let $T^{'} := T \cup {K}$ and $Ω^{'} := dom T^{'}$ (cf. (2.2)) is a bounded, polygonal Lipschitz domain in $R^{2}$ .

If the pair ${C R}_{p, 0} (T) \times M_{p - 1} (T)$ is inf-sup stable, then is inf-sup stable with a constant $c_{T^{'}, p}$ which depends on $c_{T, p}$ , the shape regularity of ${K^{'}}$ and the polynomial degree $p$ .

A proof for $p = 3$ is given in [9, Lem. 6.2]. Inspection of the proof shows that it applies also to general $p$ 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

a)p≥3 is odd andb)T is a regular triangulation as in \lx@sectionsign??? s.t. every K∈T has one % inner vertex.

(3.10)

In the following we investigate the condition (3.9) and start with the definition of critical points of a nodal patch $T_{z}$ for $z \in V_{Ω}$ (see [19]).

Definition 3.10

Let $T$ denote a triangulation as in §2.2. For $z \in V_{Ω}$ , we denote by $T_{z}$ , $ω_{z}$ the nodal patch as in (2.4) and let

A point $z^{'} \in V_{z}$ is a critical point for $T_{z}$ if there exist two straight infinite lines $L_{1}$ , $L_{2}$ in $R^{2}$ such that all edges $E \in E_{z}$ having $z^{'}$ as an endpoint satisfy $E \subset L_{1} \cup L_{2}$ . The set of all critical points in $T_{z}$ is $C_{z}$ and its cardinality denoted by $σ_{z} := | C_{z} |$ .

Remark 3.11

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

$T_{z}$ consists of four triangles and $z$ is the intersections of the two diagonals in $ω_{z}$ . Then $z$ is a critical point; see Fig. 1. In this case, it holds $σ_{z} = 1$ .
$z^{'} \in V_{z} ∖ {z}$ . Let $E$ denote the edge with endpoints $z$ and $z^{'}$ and let $K, K^{'} \in T_{z}$ be the adjacent triangles. Then, $z^{'}$ is a critical point if the sum of the two angles at $z^{'}$ in the triangles $K, K^{'} \in T_{z}$ equals $π$ ; see Fig. 2. In this case, it holds $z \notin C_{z}$ .

From [19] we know that for $p \geq 4$ it holds

dim N_{p, z} = 1 + σ_{z} .

(3.11)

Remark 3.6 implies $1 \leq dim N_{p, z}^{CR} \leq dim N_{p, z}$ . In this section, we define functions $q_{p - 1, z^{'}}$ for the critical points $z^{'} \in C_{z}$ such that the $(1 + σ_{z})$ functions $1_{z}$ and $q_{p - 1, z^{'}}$ , $z^{'} \in C_{z}$ are linearly independent and belong to $N_{p, z}$ .

Afterwards, we will prove the implication:

⎛ ⎝ η = β_{0} 1_{z} + \sum z^{'} \in C_{z} β_{z^{'}} q_{p - 1, z^{'}} % satisfies η \in N_{p, z}^{CR} ⎞ ⎠ ⟹ η \in span {1_{z}} .

(3.12)

Hence, in all cases in (3.10), where the dimension formula (3.11) holds (e.g. for $p \geq 4$ ) 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., $C_{z} \neq \emptyset$ , and we will construct basis functions for $N_{p, z}$ explicitly. We fix a (non-unique) sign function $σ : T_{z} \to {- 1, 1}$ by the condition:

if\ K,K′∈Tz share an edge, then σK=−σK′.

Definition 3.12

Let $z^{'} \in C_{z}$ be a critical point for $T_{z}$ . The critical function $q_{p - 1, z^{'}} \in P_{p - 1} (T_{z})$ for $z^{'}$ is given by

q_{p - 1, z^{'}} := ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \frac{σ_{K}}{| K |} P_{p - 1}^{(0, 2)} (1 - 2 λ_{K, z^{'}}) & on K \in T_{z} with z^{'} \in K, 0 & otherwise. \end{matrix}

Lemma 3.13

Let (3.10) be satisfied. The functions $1_{z}$ and $q_{p - 1, z^{'}}$ , $z^{'} \in C_{z}$ , are linearly independent and belong to $N_{p, z}$ . If the dimension formula (3.11) holds, they form a basis of $N_{p, z}$ .

Proof. From Remark 3.6 it follows that $1_{z} \in N_{p, z}$ . Next, we prove that $q_{p - 1, z^{'}} \in N_{p, z}$ . Let $K \in T_{z}$ with $z^{'} \in K$ . Let $A_{i}$ , $1 \leq i \leq 3$ , denote the vertices of $K$ with the convention $A_{1} = z$ .

Recall the notation for barycentric coordinates in (3.3), (3.4) and that $e_{3, ℓ}$ , $ℓ = 1, 2, 3$ , denotes the $ℓ$ -th canonical unit vector in $R^{3}$ . Let $v_{1}$ , $v_{2} \in R^{2}$ denote two linearly independent vectors – the precise choice will be fixed later.

The restriction of the space $S_{p, 0} (T_{z})$ to $K$ is spanned by (cf. (3.4))

\boldmathλ\boldmathμKvj%forj=1,2 and \boldmathμ=e3,1+\boldmathαwith \boldmathα∈I3=p−1%.

(3.13)

Let $w \in {v_{1}, v_{2}}$ and let $\partial_{w}$ denote the Gâteaux derivative of a function in the direction $w$ . Then

∫Kqp−1,z′div(\boldmathλ\boldmathμKw)=∫Kqp−1,z′∂w\boldmathλ% \boldmathμK=3∑s=1μs∂wλK,s∫Kqp−1,z′\boldmathλ\boldmathμ−e3,sK.

There exists $ℓ \in {1, 2, 3}$ such that $z^{'} = A_{ℓ}$ . We employ [2, Prop. 4.1] which tells us (as a consequence of [17, (3.10)]) that for $j \in {1, 2, 3} ∖ {ℓ}$ it holds

\int_{K} w λ_{j} P_{p - 1}^{(0, 2)} (1 - 2 λ_{K, ℓ}) = 0 \forall w \in P_{p - 2} (K) .

(3.14)

For $j \in {1, 2, 3} ∖ {ℓ}$ with $μ_{j} - δ_{j, s} \geq 1$ , this implies

I(ℓ,s)K:=μs∫Kqp−1,Aℓ% \boldmathλ\boldmathμ−e3,sK=0.

(3.15)

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

$(ℓ, s)$	$=(1,1) and \boldmathμ=(p,0,0),$
$(ℓ, s)$	$=(2,1) and \boldmathμ=(1,p−1,0),$
$(ℓ, s)$	$=(3,1) and \boldmathμ=(1,0,p−1),$	(3.16)
$(ℓ, s)$	$=(1,2) and \boldmathμ=(p−1,1,0),$
$(ℓ, s)$	$=(1,3) and \boldmathμ=(p−1,0,1)$

with corresponding integrals

All other integrals in (3.15) vanish.

A standard affine transformation to the reference element $ˆ K$ shows that

\frac{1}{p} I_{K}^{(1, 1)} = I_{K}^{(2, 1)} = I_{K}^{(3, 1)} = I_{K}^{(1, 2)} = I_{K}^{(1, 3)} = σ_{K} c_{p},

(3.17)

where

c_{p} := 2 \int_{ˆ K} P_{p - 1}^{(0, 2)} (1 - 2 {ˆ λ}_{2}) {ˆ λ}_{2}^{p - 1} .

It remains to prove the second equality in

∫Kqp−1,z′div(\boldmathλ\boldmathμKw)=3∑s=1I(ℓ,s)K∂wλK,s=0

for $(ℓ, s) \in {(1, r), (r, 1) : r \in {1, 2, 3}}$ (cf. (3.16)). By using (3.17) we obtain

∫Kqp−1,Aℓdiv(\boldmathλ\boldmathμKw)=σKcp⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩p∂wλK,1ℓ=1∧% \boldmathμ=(p,0,0),∂wλK,1ℓ=2∧\boldmathμ=(1,p−1,0),∂wλK,1ℓ=3∧\boldmathμ=(1,0,p−1),∂wλK,2ℓ=1∧\boldmathμ=(p−1,1,0),∂wλK,3ℓ=1∧\boldmathμ=(p−1,0,1),0otherwise.

(3.18)

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

The basis functions in $S_{p, 0} (T_{z})$ can be grouped into three different types.

Basis functions whose support is one triangle $K \in T_{z}$ . Let $A_{i}$ , $1 \leq i \leq 3$ , denote the vertices of $K$ . Then, these basis functions are given by

$B\boldmathμ,jK:=\boldmathλ% \boldmathμKvj,for \boldmathμ∈I3=p and min{μi:1≤i≤3}≥1.$

From (3.18) it follows that

$∫ωzqp−1,z′div(B\boldmathμ,jK)=0.$
Basis functions whose support is an edge patch. Let $E \in E_{z}$ with $z \in E$ and denote by $K$ and $K^{'}$ the triangles in $T_{z}$ which share $E$ . We denote the vertices in $K$ and $K^{'}$ by $A_{ℓ}$ and $A^{'}$

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

Authors

On the Inf-Sup Stabillity of Crouzeix-Raviart Stokes Elements in 3D

A CutFEM divergence–free discretization for the Stokes problem

Fortin Operator for the Taylor-Hood Element

On the Sobolev and L^p-Stability of the L^2-projection

On a seventh order convergent weakly L-stable Newton Cotes formula with application on Burger's equation

Critical Points for Two-view Triangulation

Isogeometric Mortar Coupling for Electromagnetic Problems

1 Introduction

2 Setting

2.1 The Continuous Stokes Problem

2.2 Numerical Discretization of the Stokes Problem

Definition 2.1

Theorem 2.2

3 Proof of Theorem 2.2

3.1 Barycentric Coordinates and Basis Functions for the Velocity

Notation 3.1

Definition 3.2

Theorem 3.3

Corollary 3.4

Remark 3.5

3.2 The Pressure Kernel

Remark 3.6

Theorem 3.7

Remark 3.8

Lemma 3.9

Definition 3.10

Remark 3.11

Definition 3.12

Lemma 3.13

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

Authors

Related Research

On the Inf-Sup Stabillity of Crouzeix-Raviart Stokes Elements in 3D

A CutFEM divergence–free discretization for the Stokes problem

Fortin Operator for the Taylor-Hood Element

On the Sobolev and L^p-Stability of the L^2-projection

On a seventh order convergent weakly L-stable Newton Cotes formula with application on Burger's equation

Critical Points for Two-view Triangulation

Isogeometric Mortar Coupling for Electromagnetic Problems

This week in AI

1 Introduction

2 Setting

2.1 The Continuous Stokes Problem

2.2 Numerical Discretization of the Stokes Problem

Definition 2.1

Theorem 2.2

3 Proof of Theorem 2.2

3.1 Barycentric Coordinates and Basis Functions for the Velocity

Notation 3.1

Definition 3.2

Theorem 3.3

Corollary 3.4

Remark 3.5

3.2 The Pressure Kernel

Remark 3.6

Theorem 3.7

Remark 3.8

Lemma 3.9

Definition 3.10

Remark 3.11

Definition 3.12

Lemma 3.13