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
More precisely, on p and the John constant of G.
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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
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
If \(\Delta _2(\rho ,\rho ^*)< \infty \), then additionally for all \(\delta >0\)
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
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
We define the shifted N-function \(\rho _a\) for \(a\ge 0\) by
The following auxiliary result can be found in [17, 21].
Lemma 6.3
For all \(a,b, t \ge 0\) we have
Lemma 6.4
[17, Lemma 2.3] We have
uniformly in \(\mathbf{P}, \mathbf{Q}\in \mathbb {R}^{n \times n}\). Moreover, uniformly in \(\mathbf{Q}\in \mathbb {R}^{n \times n}\),
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
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
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
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
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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DOI: https://doi.org/10.1007/s00211-015-0735-4