Error analysis for a CrouzeixRaviart approximation of the p-Dirichlet problem Alex Kaltenbach1

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Error analysis for a Crouzeix–Raviart approximation

of the p-Dirichlet problem

Alex Kaltenbach∗1

Department of Applied Mathematics, University of Freiburg, Ernst–Zermelo–Straße 1, 79104 Freiburg

im Breisgau, Germany

October 21, 2022

Abstract

In the present paper, we examine a Crouzeix–Raviart approximation for non-linear partial

diﬀerential equations having a (

p, δ

)-structure for some

p∈

,∞

) and

δ≥

0. We establish

priori

error estimates, which are optimal for all

p∈

,∞

) and

δ≥

0, medius error estimates,

i.e., best-approximation results, and a primal-dual a posteriori error estimate, which is both

reliable and eﬃcient. The theoretical ﬁndings are supported by numerical experiments.

Keywords:

-Dirichlet problem; Crouzeix–Raviart element; a priori error analysis; medius error

analysis

;

a posteriori error analysis.

AMS MSC (2020): 49M29; 65N15; 65N50.

1. Introduction

We examine the numerical approximation of a non-linear system of p-Dirichlet type, i.e.,

−div A(∇u) = fin Ω ,

u= 0 on ΓD,

A(∇u)·n= 0 on ΓN,

(1.1)

using the Crouzeix–Raviart element, cf. [

]. More precisely, for a given right-hand side

f∈Lp′(Ω)

p′:

p−1

p∈

,∞

), we seek

u∈W1,p

D(Ω) :={v∈W1,p(Ω) |tr v= 0 in Lp(ΓD)}

solving

(1.1)

Here,

Ω⊆Rd

d∈N

, is a bounded Lipschitz domain, whose topological boundary

∂

Ω is disjointly

divided into a Dirichlet part Γ

and a Neumann part Γ

, and the non-linear

operator A:Rd→Rd

has a (

p, δ

)-structure for some

p∈

,∞

) and

δ≥

0. The relevant example, falling into this class,

for every a∈Rd, is deﬁned by

A(a):= (δ+|a|)p−2a . (1.2)

Problems of type

(1.1)

arise in various mathematical models describing physical processes, e.g.,

in plasticity, bimaterial problems in elastic-plastic mechanics, non-Newtonian ﬂuid mechanics,

blood rheology, and glaciology, cf. [

]. Most of these models admit equivalent formulations

as convex minimization problems, e.g., for the non-linear system

(1.1)

, if the non-linear operator

A:Rd→Rd

possesses a potential, i.e., there is a strictly convex function

φ:R≥0→R≥0

such that

(

φ◦|·|

)(

) =

(

) for all

a∈Rd

, e.g.,

(1.2)

, then each solution

u∈W1,p

D(Ω)

(1.1)

is unique

minimizer of the energy functional I:W1,p

D(Ω) →R≥0, for every v∈W1,p

D(Ω) deﬁned by

I(v):=ˆΩ

φ(|∇v|) dx−ˆΩ

f v dx , (1.3)

and vice-versa, leading to a primal and a dual formulation of

(1.1)

, as well as to convex duality

relations.

∗Email: alex.kaltenbach@mathematik.uni-freiburg.de

arXiv:2210.12116v4 [math.NA] 7 Jun 2024

A. Kaltenbach 2

1.1 Related contributions

The ﬁnite element approximation of

(1.1)

has been intensively analyzed by numerous authors:

The ﬁrst contributions addressing a priori error estimation as well as a posteriori estimation,

measured in the conventional

W1,p

(Ω)-semi-norm, can be found in [

]. Sharper (optimal)

a priori error estimates for the conforming Lagrange ﬁnite element method applied to

(1.1)

measured in the so-called quasi-norm or natural distance, resp., were established in [

Furthermore, residual a posteriori error estimates for the conforming Lagrange ﬁnite element

method and the non-conforming Crouzeix–Raviart ﬁnite element method applied to

(1.1)

, each

measured in the quasi-norm or natural distance, resp., were established in

[38,39,18,23,11,10]

In addition, there exist optimal a priori error estimates for Discontinuous Galerkin (DG) methods,

cf. [

]. In [

], if

p≥

2 and

= 0 in

(1.2)

, a priori and a posteriori error estimates for the

Crouzeix–Raviart ﬁnite element method applied to

(1.1)

, measured in the

quasi-norm

, were

derived

However, in [

], the optimality of the a priori error estimates and the eﬃciency of the a posteriori

error estimates remain unclear. In [

], if

= 2 and

= 0 in

(1.2)

, by means of a so-called medius

error analysis, i.e., a best-approximation result, for the Crouzeix–Raviart ﬁnite element method

applied to

(1.1)

, an optimal a priori error estimate was derived. In particular, this medius error

analysis reveals that the performances of the conforming Lagrange ﬁnite element method and

the non-conforming Crouzeix–Raviart ﬁnite element method applied to

(1.1)

are comparable.

However, for the case

p̸= 2

, to the best of the author’s knowledge, such results are still pending.

More precisely, there is neither a medius error analysis, i.e., a best-approximation result, available,

nor an optimal a priori error estimate, measured in the quasi-norm or natural distance, resp. It is

the purpose of this paper to ﬁll this lacuna.

1.2 New contribution

Deriving local eﬃciency estimates in terms of shifted

-functions and deploying the so-called

node-averaging quasi-interpolation operator, cf. [

], we generalize the medius error analysis

in [

] from

= 2 and

= 0 in

(1.2)

, i.e.,

idRd:Rd→Rd

, to general non-linear operators

A:Rd→Rd

having a (

p, δ

)-structure for

p∈

,∞

) and

δ≥

0, e.g.,

(1.2)

. This medius error analysis,

reveals that the performances of the conforming Lagrange ﬁnite element method applied to

(1.1)

and the non-conforming Crouzeix–Raviart ﬁnite element method applied to

(1.1)

are comparable.

As a result, we get a priori error estimates for the Crouzeix–Raviart ﬁnite element method applied to

(1.1)

, which are optimal for all

p∈

,∞

) and

δ≥

0. If

A:Rd→Rd

has a potential and, thus,

(1.1)

admits an equivalent formulation as a convex minimization problem, cf.

(1.3)

, then we have access

to a (discrete) convex duality theory, and

(1.1)

as well as the Crouzeix–Raviart approximation of

(1.1)

admit dual formulations with a dual solution and a discrete dual solution, resp., cf. [

We establish a priori error estimates for the error between the dual solution and the discrete dual

solution, measured in the so-called conjugate natural distance, which are optimal for all

p∈

,∞

)

and

δ≥

0. One further by-product of the medius error analysis consists in an eﬃciency type result,

which allows to establish the eﬃciency of a so-called primal-dual a posteriori error estimator,

which was recently derived in [8] and is also applicable if A:Rd→Rdhas a potential.

1.3 Outline

This article is organized as follows: In Section 2, we introduce the employed notation, the

basic

assumptions on the non-linear operator

A:Rd→Rd

and its corresponding properties, the relevant

ﬁnite element spaces, and give brief review of the continuous and the discrete

-Dirichlet problem.

In Section 3, we establish a medius error analysis, i.e., best-approximation result, for the Crouzeix–

Raviart ﬁnite element method applied to

(1.1)

. In Section 4, by means of this medius error analysis,

we derive a priori error estimates for the Crouzeix–Raviart ﬁnite element method applied to

(1.1)

which are optimal for all

p∈

,∞

) and

δ≥

0. In Section 5, we establish the eﬃciency of a

so-called

primal-dual a posteriori error estimator. In Section 6, we conﬁrm our theoretical ﬁndings via

numerical experiments.

Error analysis for a CR approximation of the p-Dirichlet problem 3

2. Preliminaries

Throughout the entire article, if not otherwise speciﬁed, we always denote by

Ω⊆Rd

d∈N

a bounded polyhedral Lipschitz domain, whose topological boundary

∂

Ω is disjointly divided into a

closed Dirichlet part Γ

, for which we always assume that

D|>

, and a Neumann part Γ

, i.e.,

∂Ω=ΓD∪ΓN

and

∅= ΓD∩ΓN

. We employ

c, C >

0 to denote generic constants, that may

change from line to line, but are not depending on the crucial quantities. Moreover, we write

f∼g

if and only if there exist constants c, C > 0 such that c f ≤g≤C f.

2.1 Standard function spaces

For p∈[1,∞] and l∈N, we employ the standard notations2

W1,p

D(Ω; Rl):=v∈Lp(Ω; Rl)| ∇v∈Lp(Ω; Rl×d),tr v= 0 in Lp(ΓD;Rl),

N(div; Ω):=y∈Lp(Ω; Rd)|div y∈Lp(Ω) ,⟨tr y·n, v⟩W1−1

p′,p′(∂Ω) = 0 for all v∈W1,p′

D(Ω),

W1,p(Ω; Rl):=W1,p

D(Ω; Rl)

ΓD=∅

, and

Wp(div; Ω) :=Wp

N(div; Ω)

ΓN=∅

, where we

denote

tr: W1,p(Ω; Rl)→Lp(∂Ω; Rl)

and by

(

)

·n:Wp(div; Ω) →W−1

p,p(∂Ω)

, the trace and normal

trace operator, resp. In particular, we

predominantly omit tr

(

) in this context. In addition, we em-

ploy the abbreviations

(Ω)

(Ω;

W1,p(Ω) :=W1,p(Ω; R1)

and

W1,p

D(Ω) :=W1,p

D(Ω; R1)

2.2 N-functions

A (real) convex function

ψ:R≥0→R≥0

is called

-function, if

ψ(0) = 0

ψ(t)>0

for all

t > 0

limt→0ψ

(

)

= 0, and

limt→∞ ψ

(

)

∞

. If, in addition,

ψ∈C1

(

R≥0

)

∩C2

(

R>0

) and

ψ′′(t)>0

for all

t >

0, we call

aregular

-function. For a regular

-function

ψ:R≥0→R≥0

, we have that

(0) =

ψ′

(0) = 0,

ψ′:R≥0→R≥0

is increasing and

limt→∞ ψ′

(

) =

∞

. For a given

N-function

ψ:R≥0→R≥0

, we deﬁne the (Fenchel) conjugate

N-function ψ∗:R≥0→R≥0

, for every

t≥

0, by

ψ∗(t):= sups≥0(st −ψ(s))

, which satisﬁes (

ψ∗

)

′

= (

ψ′

)

−1

R≥0

. An

-function

satisﬁes the

∆

-condition (in short,

ψ∈

∆

), if there exists

K >

2 such that for all

t≥0

, it holds

ψ(2 t)≤K ψ(t)

Then, we denote the smallest such constant by ∆

(

)

0. We say that an

-function

ψ:R≥0→R≥0

satisﬁes the

∇2

-condition (in short,

ψ∈ ∇2

), if its (Fenchel) conjugate

ψ∗:R≥0→R≥0

is an

-function satisfying the ∆

-condition. If

ψ:R≥0→R≥0

satisﬁes the ∆

- and the

∇2

-condition

(in short,

ψ∈

∆

2∩∇2

), then, there holds the following reﬁned version of the

-Young inequality:

for every

ε >

0, there exists a constant

cε>

0, depending only on ∆

(

)

∆

(

ψ∗

)

<∞

, such that

for every s, t ≥0, it holds

t s ≤ε ψ(t) + cεψ∗(s).(2.1)

The mean value of a locally integrable function

Ω

→R

over a (Lebesgue) measurable set

M⊆Ω

is denoted by

ﬄMfdx:

|M|´Mfdx

. Furthermore, we employ the notations (

f, g

)

´Mfg dx

and

ρψ,M

(

)

´Mψ

(

·,|f|

)

, for (Lebesgue) measurable functions

f, g :

Ω

→R

, a (Lebesgue)

measurable set

M⊆

Ω and a generalized

-function

M×R≥0→R≥0

, i.e.,

is a Carath´eodory

function and

(

x, ·

) an

-function for a.e.

x∈M

, whenever the right-hand side is

well-deﬁned

2.3 Basic properties of the non-linear operator

Throughout the entire paper, we assume that the non-linear operator

has a (

p, δ

)-structure,

which will be deﬁned now. A detailed discussion and full proofs can be found, e.g., in [

For p∈(1,∞) and δ≥0, we deﬁne a special N-function φ:=φp,δ :R≥0→R≥0by

φ(t):=ˆt

φ′(s) ds, where φ′(t):= (δ+t)p−2t , for all t≥0.(2.2)

Then,

φ:R≥0→R≥0

satisﬁes, independent of

δ≥

0, the ∆

-condition with

∆2(ϕ)≤c2max{2,p}

In addition, the (Fenchel) conjugate function

φ∗:R≥0→R≥0

satisﬁes, uniformly in

t≥0

and

δ≥0

φ∗(t)∼(δp−1+t)p′−2t2as well as the ∆2-condition with ∆2(φ∗)≤c2max{2,p′}.

For a (Lebesgue) measurable set

M⊆Rd

d∈N

, we denote by

|M|

its

-dimensional Lebesgue measure. For

a (

d−

1)-dimensional submanifold

M⊆Rd

d∈N

, we denote by

|M|

its (

d−

1)-dimensional Hausdorﬀ measure.

2Here, W−1

p,p(∂Ω) := (W1−1

p′,p′(∂Ω))∗.

A. Kaltenbach 4

For an

-function

ψ:R≥0→R≥0

, we deﬁne shifted

-functions

ψa:R≥0→R≥0

a≥0

, by

ψa(t):=ˆt

ψ′

a(s) ds , where ψ′

a(t):=ψ′(a+t)t

a+t,for all a, t ≥0.(2.3)

Remark 2.1. For the above deﬁned

-function

φ:R≥0→R≥0

, cf.

(2.2)

, uniformly in

a, t ≥0

we have that

φa

(

)

∼

(

)

p−2t2

and (

φa

)

∗

(

)

∼

((

)

p−1

)

p′−2t2

. Apart from that, the

families

{φa}a≥0,{

(

φa

)

∗}a≥0:R≥0→R≥0

satisfy, uniformly in

a≥

0, the ∆

-condition, i.e.,

for every a≥0, it holds ∆2(φa)≤c2max{2,p}and ∆2((φa)∗)≤c2max{2,p′}, respectively.

Assumption 2.2. We assume that

A∈C0

(

;

)

∩C1

(

Rd\ {

}

;

)satisﬁes

(0) = 0 and

has a (p, δ)-structure, i.e., there exist p∈(1,∞),δ≥0, and constant C0, C1>0such that

((∇A)(a)b)·b≥C0(δ+|a|)p−2|b|2,

|(∇A)(a)| ≤ C1(δ+|a|)p−2,

are satisﬁed for all

a, b ∈Rd

with

a̸

= 0 and

i, j

= 1

, . . . , d

. The constants

C0, C1>

0and

p∈(1,∞)are called the characteristics of A.

Remark 2.3. An example of a non-linear operator

A:Rd→Rd

satisfying Assumption 2.2 for

some p∈(1,∞)and δ≥0, for every a∈Rd, is given via

A(a) = φ′(|a|)

|a|a= (δ+|a|)p−2a , (2.4)

where the characteristics of

A:Rd→Rd

depend only on

p∈

,∞

)and are independent of

δ≥0

Closely related to the non-linear operator

A:Rd→Rd

with (

p, δ

)-structure, where

p∈

,∞

)

and δ≥0, are the non-linear operators F, F ∗:Rd→Rd, for every a∈Rddeﬁned by

F(a):= (δ+|a|)p−2

2a , F ∗(a):= (δp−1+|a|)p′−2

2a . (2.5)

The connections between

A, F, F ∗:Rd→Rd

and

φa,

(

φ∗

)

(

φa

)

∗:R≥0→R≥0

a≥0

, are

best explained by the following proposition.

Proposition 2.4. Let

A:Rd→Rd

satisfy Assumption 2.2 for

p∈

,∞

)and

δ≥

0. Moreover,

let

φ:R≥0→R≥0

be deﬁned by

(2.2)

and let

F, F ∗:Rd→Rd

be deﬁned by

(2.5)

, each for the

same p∈(1,∞)and δ≥0. Then, uniformly with respect to a, b ∈Rd, we have that

(A(a)−A(b)) ·(a−b)∼ |F(a)−F(b)|2∼φ|a|(|a−b|)

∼(φ|a|)∗(|A(a)−A(b)|)∼(φ∗)|A(a)|(|A(a)−A(b)|) (2.6)

∼ |F∗(A(a)) −F∗(A(b))|2,

|F∗(a)−F∗(b)|2∼(φ∗)|a|(|a−b|).(2.7)

The constants in (2.6)and (2.7)depend only on the characteristics of A.

Proof.

For the ﬁrst two equivalences in

(2.6)

and the equivalence

(2.7)

, we refer to [

, Lemma 6.16].

For the last three equivalences in (2.6), we refer to [24, Lemma 2.8] and [22, Lemma 26].

In addition, we need the following auxiliary result.

Lemma 2.5 (Change of shift).Let

φ:R≥0→R≥0

be deﬁned by

(2.2)

for

p∈

,∞

)and

δ≥

and let

F:Rd→Rd

be deﬁned by

(2.5)

for the same

p∈

,∞

)and

δ≥

0. Then, for every

ε > 0

there exists

cε≥

1, depending only on

ε >

p∈

,∞

), and

δ≥

0, such that for every

a, b ∈Rd

and t≥0, it holds

φ|a|(t)≤cεφ|b|(t) + ε|F(a)−F(b)|2,(2.8)

(φ|a|)∗(t)≤cε(φ|b|)∗(t) + ε|F(a)−F(b)|2.(2.9)

Proof. See [23, Corollary 26, (5.5) & Corollary 28, (5.8)].

Error analysis for a CR approximation of the p-Dirichlet problem 5

Remark 2.6 (Natural distance).If

A:Rd→Rd

satisﬁes Assumption 2.2 for

p∈(1,∞)

and

δ≥0

then, due to (2.6), uniformly in u, v ∈W1,p(Ω), it holds

(A(∇u)−A(∇v),∇u− ∇v)Ω∼ ∥F(∇u)−F(∇v)∥2

L2(Ω;Rd)∼ρφ|∇u|,Ω(∇u− ∇v).

In the context of the

-Dirichlet problem, the quantity

F:Rd→Rd

was ﬁrst introduced in [

], while

the last expression equals the quasi-norm introduced in [

] if raised to the power of

ρ= max{p, 2}

We refer to all three equivalent quantities as the natural distance.

Remark 2.7 (Conjugate natural distance).If

A:Rd→Rd

satisﬁes Assumption 2.2 for

p∈(1,∞)

and

δ≥

0, then, it is readily seen that

A:Rd→Rd

is continuous, strictly monotone, and coercive,

so that from the theory of monotone operators, cf. [

], it follows that

A:Rd→Rd

is bijective

and

A−1:Rd→Rd

continuous. In addition, due to

(2.6)

, uniformly in

z, y ∈Lp′(Ω; Rd)

, it holds

(A−1(z)−A−1(y), z −y)Ω∼ ∥F∗(z)−F∗(y)∥2

L2(Ω;Rd)∼ρ(φ∗)|z|,Ω(z−y).

We refer to all three equivalent quantities as the conjugate natural distance.

2.4 Triangulations and standard ﬁnite element spaces

Throughout the entire paper, we denote by

h >

0, a family of regular, i.e., uniformly

shape regular and conforming, triangulations of Ω

⊆Rd

d∈N

, cf. [

]. Here,

h > 0

refers to the

average mesh-size, i.e.,

h:= (|Ω|/card(Th)) 1

. For every element

T∈ Th

, we denote by

ρT>

the supremum of diameters of inscribed balls. We assume that there exists a constant

ω0>

independent of

h >

0, such that

maxT∈ThhTρ−1

T≤ω0

. The smallest such constant is called

the chunkiness of (

)

h>0

. Also note that, in what follows, all constants may depend on the

chunkiness, but are independent of

h > 0

. For every

T∈ Th

, let

ωT

denote the patch of

, i.e.,

the union of all elements of

touching

. We assume that

int

(

ωT

) is connected for all

T∈ Th

Under these assumptions,

|T| ∼ |ωT|

uniformly in

T∈ Th

and

h >

0, and the number of elements

ωT

and patches to which an element

belongs to are uniformly bounded with respect to

T∈ Th

and

h >

0. We deﬁne the sides of

in the

following

way: an interior side is the closure

of the non-empty relative interior of

∂T ∩∂T ′

, where

T, T ′∈ Th

are adjacent elements. For an

interior side

∂T ∩∂T ′∈ Sh

, where

T, T ′∈ Th

, we employ the notation

ωS:

T∪T′

. A

boundary side is the closure of the non-empty relative interior of

∂T ∩∂

Ω, where

T∈ Th

denotes a

boundary element of

. For a boundary side

∂T ∩∂

Ω, we employ the notation

ωS:

. By

and

, we denote the sets of all interior sides and the set of all sides, respectively. Eventually,

we deﬁne hS:= diam(S) for all S∈ Shand hT:= diam(T) for all T∈ Th.

For

k∈N∪{

}

and

T∈ Th

, let

(

) denote the set of polynomials of maximal degree

Then, for

k∈N∪ {

}

and

l∈N

, the sets of continuous and

element-wise

polynomial functions

or vector ﬁelds, respectively, are deﬁned by

Sk(Th)l:=vh∈C0(Ω; Rl)|vh|T∈ Pk(T)lfor all T∈ Th,

Lk(Th)l:=vh∈L∞(Ω; Rl)|vh|T∈ Pk(T)lfor all T∈ Th.

The element-wise constant mesh-size function

hT∈ L0

(

) is deﬁned by

hT|T:=hT

for all

T∈ Th

The side-wise constant mesh-size function

hS∈ L0

(

) is deﬁned by

hS|S:=hS

for all

S∈ Sh

For every

T∈ Th

and

S∈ Sh

, we denote by

xT:=1

d+1 Pz∈Nh∩Tz

and

xS:=1

dPz∈Nh∩Sz

the midpoints (barycenters) of

and

respectively

. The (local)

-projection operator onto

element-wise constant functions or vector ﬁelds, respectively, is denoted by

Πh:L1(Ω; Rl)→ L0(Th)l.

For every

vh∈ L1(Th)l

, it holds Π

hvh|T

(

) in

for all

T∈ Th

. The element-wise gradient

operator

∇h:L1

(

)

l→ L0

(

)

l×d

, for every

vh∈ L1

(

)

, is deﬁned by

∇hvh|T:

∇

(

vh|T

) in

for all T∈ Th.

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ErroranalysisforaCrouzeix–Raviartapproximationofthep-DirichletproblemAlexKaltenbach∗11DepartmentofAppliedMathematics,UniversityofFreiburg,Ernst–Zermelo–Straße1,79104FreiburgimBreisgau,GermanyOctober21,2022AbstractInthepresentpaper,weexamineaCrouzeix–Raviartapproximationfornon-linearpartialdifferenti...

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Error analysis for a CrouzeixRaviart approximation of the p-Dirichlet problem Alex Kaltenbach1.pdf

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Error analysis for a CrouzeixRaviart approximation of the p-Dirichlet problem Alex Kaltenbach1

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