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Convergence analysis for a finite element approximation of a steady model for electrorheological fluids

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Abstract

In this paper we study the finite element approximation of systems of \({p(\cdot )}\)-Stokes type, where \({p(\cdot )}\) is a (non constant) given function of the space variables. We derive—in some cases optimal—error estimates for finite element approximation of the velocity and of the pressure, in a suitable functional setting.

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Notes

  1. More precisely, on p and the John constant of G.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Via F. Buonarroti 1/c, 56127, Pisa, Italy

    Luigi C. Berselli

  2. Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, UK

    Dominic Breit

  3. Institut für Mathematik, Universität Osnabrück, Albrechtstr. 28a, 49076, Osnabrück, Germany

    Lars Diening

Authors
  1. Luigi C. Berselli
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  2. Dominic Breit
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  3. Lars Diening
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Corresponding author

Correspondence to Dominic Breit.

Appendix A: Orlicz spaces

Appendix A: Orlicz spaces

The following definitions and results are standard in the theory of Orlicz spaces and can for example be found in [30]. A continuous, convex function \(\rho :[0,\infty ) \rightarrow [0,\infty )\) with \(\rho (0)=0\), and \(\lim _{t \rightarrow \infty } \rho (t) = \infty \) is called a \(\phi \)-function.

We say that \(\rho \) satisfies the \(\Delta _2\)–condition, if there exists \(c > 0\) such that for all \(t \ge 0\) holds \(\rho (2t) \le c\, \rho (t)\). By \(\Delta _2(\rho )\) we denote the smallest such constant. Since \(\rho (t) \le \rho (2t)\) the \(\Delta _2\)-condition is equivalent to \(\rho (2t) \sim \rho (t)\) uniformly in t. For a family \(\rho _\lambda \) of \(\rho \)-functions we define \(\Delta _2({\{{\rho _\lambda }\}}) := \sup _\lambda \Delta _2(\rho _\lambda )\). Note that if \(\Delta _2(\rho ) < \infty \) then \(\rho (t) \sim \rho (c\,t)\) uniformly in \(t\ge 0\) for any fixed \(c>0\). By \(L^\rho \) and \(W^{k,\rho }\), \(k\in \mathbb {N}_0\), we denote the classical Orlicz and Orlicz-Sobolev spaces, i.e. \(f \in L^\rho \) iff \(\int \rho ({|{f}|})\,dx < \infty \) and \(f \in W^{k,\rho }\) iff \( \nabla ^j f \in L^\rho \), \(0\le j\le k\).

A \(\phi \)-function \(\rho \) is called a N-function iff it is strictly increasing and convex with

$$\begin{aligned} \lim _{t\rightarrow 0}\frac{\rho (t)}{t}= \lim _{t\rightarrow \infty }\frac{t}{\rho (t)}=0. \end{aligned}$$

By \(\rho ^*\) we denote the conjugate N-function of \(\rho \), which is given by \(\rho ^*(t) = \sup _{s \ge 0} (st - \rho (s))\). Then \(\rho ^{**} = \rho \).

Lemma 6.1

(Young’s inequality) Let \(\rho \) be an N-function. Then for all \(s,t\ge 0\) we have

$$\begin{aligned} st \le \rho (s)+\rho ^*(t). \end{aligned}$$

If \(\Delta _2(\rho ,\rho ^*)< \infty \), then additionally for all \(\delta >0\)

$$\begin{aligned} st&\le \delta \,\rho (s)+c_\delta \,\rho ^*(t), \\ st&\le c_\delta \,\rho (s)+\delta \,\rho ^*(t), \\ \rho '(s)t&\le \delta \,\rho (s)+c_\delta \,\rho (t), \\ \rho '(s)t&\le \delta \,\rho (t)+c_\delta \,\rho (s), \end{aligned}$$

where \(c_\delta = c(\delta , \Delta _2({\{{\rho ,\rho ^*}\}}))\).

Definition 6.2

Let \(\rho \) be an N-function. We say that \(\rho \) is elliptic, if \(\rho \) is \(C^1\) on \([0,\infty )\) and \(C^2\) on \((0,\infty )\) and assume that

$$\begin{aligned} \rho '(t)&\sim t\,\rho ''(t), \end{aligned}$$
(6.1)

uniformly in \(t > 0\). The constants hidden in \(\sim \) are called the characteristics of \(\rho \).

Note that (6.1) is stronger than \(\Delta _2(\rho ,\rho ^*)<\infty \). In fact, the \(\Delta _2\)-constants can be estimated in terms of the characteristics of \(\rho \).

Associated to an elliptic N-function \(\rho \) we define the tensors

$$\begin{aligned} \mathbf{A}^\rho ({\varvec{\xi }})&:=\frac{\rho '({|{{\varvec{\xi }}}|})}{{|{{\varvec{\xi }}}|}}{\varvec{\xi }},\quad {\varvec{\xi }}\in \mathbb R^{n\times n} \\ \mathbf{F}^\rho ({\varvec{\xi }})&:=\sqrt{\frac{\rho '({|{{\varvec{\xi }}}|})}{{|{{\varvec{\xi }}}|}}}\,{\varvec{\xi }},\quad {\varvec{\xi }}\in \mathbb R^{n\times n}. \end{aligned}$$

We define the shifted N-function \(\rho _a\) for \(a\ge 0\) by

$$\begin{aligned} \rho _a(t)&:= \int \limits _0^t \frac{\rho '(a+\tau )}{a+\tau } \tau \,d\tau . \end{aligned}$$
(6.2)

The following auxiliary result can be found in [17, 21].

Lemma 6.3

For all \(a,b, t \ge 0\) we have

$$\begin{aligned} \rho _a(t)&\sim {\left\{ \begin{array}{ll} \rho ''(a) t^2 &{}\quad \text {if }\,\, t \lesssim a \\ \rho (t) &{}\quad \text {if }\,\, t \gtrsim a, \end{array}\right. }\\ (\rho _a)_b(t)&\sim \rho _{a+b}(t). \end{aligned}$$

Lemma 6.4

[17, Lemma 2.3] We have

$$\begin{aligned} \big ({\mathbf{A}^\rho }(\mathbf{P}) - {\mathbf{A}^\rho }(\mathbf{Q})\big ) \cdot \big (\mathbf{P}-\mathbf{Q}\big )&\sim {\big |{ \mathbf{F}^\rho (\mathbf{P}) - \mathbf{F}^\rho (\mathbf{Q})}\big |}^2 \\&\sim \rho _{{|{\mathbf{P}}|}}({|{\mathbf{P}- \mathbf{Q}}|}) \\&\sim \rho ''\big ( {|{\mathbf{P}}|} + {|{\mathbf{Q}}|} \big ){|{\mathbf{P}- \mathbf{Q}}|}^2, \end{aligned}$$

uniformly in \(\mathbf{P}, \mathbf{Q}\in \mathbb {R}^{n \times n}\). Moreover, uniformly in \(\mathbf{Q}\in \mathbb {R}^{n \times n}\),

$$\begin{aligned} \mathbf{A}^\rho (\mathbf{Q}) \cdot \mathbf{Q}&\sim {|{\mathbf{F}^\rho (\mathbf{Q})}|}^2\sim \rho ({|{\mathbf{Q}}|}) \\ {|{{\mathbf{A}^\rho }(\mathbf{P}) - {\mathbf{A}^\rho }(\mathbf{Q})}|}&\sim \big (\rho _{{|{\mathbf{P}}|}}\big )'({|{\mathbf{P}- \mathbf{Q}}|}). \end{aligned}$$

The constants depend only on the characteristics of \(\rho \).

Lemma 6.5

(Change of shift) Let \(\rho \) be an elliptic N-function. Then for each \(\delta >0\) there exists \(C_\delta \ge 1\) (only depending on \(\delta \) and the characteristics of \(\rho )\) such that

$$\begin{aligned} \rho _{{|{\mathbf{a}}|}}(t)&\le C_\delta \, \rho _{{|{\mathbf{b}}|}}(t) +\delta \, \rho _{{|{\mathbf{a}}|}}({|{\mathbf{a}- \mathbf{b}}|}), \\ (\rho _{{|{\mathbf{a}}|}})^*(t)&\le C_\delta \, (\rho _{{|{\mathbf{b}}|}})^*(t) +\delta \, \rho _{{|{\mathbf{a}}|}}({|{\mathbf{a}- \mathbf{b}}|}), \end{aligned}$$

for all \(\mathbf{a},\mathbf{b}\in \mathbb {R}^n\) and \(t\ge 0\).

The case \(\mathbf{a}=0\) or \(\mathbf{b}=0\) implies the following corollary.

Corollary 6.6

(Removal of shift) Let \(\rho \) be an elliptic N-function. Then for each \(\delta >0\) there exists \(C_\delta \ge 1\) (only depending on \(\delta \) and the characteristics of \(\rho )\) such that

$$\begin{aligned} \rho _{{|{\mathbf{a}}|}}(t)&\le C_\delta \, \rho (t) +\delta \, \rho ({|{\mathbf{a}}|}), \\ \rho (t)&\le C_\delta \, \rho _{{|{\mathbf{a}}|}}(t) +\delta \, \rho ({|{\mathbf{a}}|}), \end{aligned}$$

for all \(\mathbf{a}\in \mathbb {R}^n\) and \(t\ge 0\).

Lemma 6.7

Let \(\rho \) be an elliptic N-function. Then \((\rho _a)^*(t) \sim (\rho ^*)_{\rho '(a)}(t)\) uniformly in \(a,t \ge 0\). Moreover, for all \(\lambda \in [0,1]\) we have

$$\begin{aligned} \rho _a(\lambda a)&\sim \lambda ^2 \rho (a) \sim (\rho _a)^*(\lambda \rho '(a)). \end{aligned}$$

Lemma 6.8

Let \(\rho (t) := \int \nolimits _0^t (\kappa +s)^{q-2} s\,ds\) with \(q \in (1,\infty )\) and \(t\ge 0\). Then

$$\begin{aligned} \rho _a(\lambda t)&\le c\, \max {\{{\lambda ^q, \lambda ^2}\}} \rho (t), \\ (\rho _a)^*(\lambda t)&\le c\, \max {\{{\lambda ^{q'}, \lambda ^2}\}} \rho (t), \end{aligned}$$

uniformly in \(a,\lambda \ge 0\).

Remark 6.9

Let \(p\in \mathcal {P}(\Omega )\) with \(p^{-}>1\) and \(p^+<\infty \). The results above extend to the function \(\phi (x,t)=\int _0^t (\kappa +s)^{p(x)-2}s\,ds\) uniformly in \(x\in \Omega \), where the constants only depend on \(p^-\) and \(p^+\).

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Berselli, L.C., Breit, D. & Diening, L. Convergence analysis for a finite element approximation of a steady model for electrorheological fluids. Numer. Math. 132, 657–689 (2016). https://doi.org/10.1007/s00211-015-0735-4

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