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Advances in Difference Equations

, 2019:374 | Cite as

A priori and a posteriori estimates of the stabilized finite element methods for the incompressible flow with slip boundary conditions arising in arteriosclerosis

  • Jian Li
  • Haibiao Zheng
  • Qingsong ZouEmail author
Open Access
Research
  • 103 Downloads

Abstract

In this paper, we develop the lower order stabilized finite element methods for the incompressible flow with the slip boundary conditions of friction type whose weak solution satisfies a variational inequality. The \(H^{1}\)-norm for the velocity and the \(L^{2}\)-norm for the pressure decrease with optimal convergence order. The reliable and efficient a posteriori error estimates are also derived. Finally, numerical experiments are presented to validate the theoretical results.

Keywords

Stokes equations Slip boundary condition Variational inequality Finite element methods A priori error estimates A posteriori error estimates Numerical experiments 

MSC

35L70 65N30 76D06 

1 Introduction

Incompressible viscous flow phenomena arise in numerous disciplines in science and engineering. The simplest viscous flow problems involve just one fluid in the laminar regime. Numerical methods for this model have been intensively studied during the past several decades [3, 9, 17, 45]. A critical problem in the construction of numerical methods is to establish the so-called Babus̆ka–Brezzi stability condition. It is known that if the finite element spaces for the velocity and pressure are of the lowest equal-order or lowest order pairs, the corresponding finite element schemes do not satisfy the Babus̆ka–Brezzi condition. For lower order schemes, some stabilized finite element methods are studied and developed in [7, 8, 10, 11, 14, 19, 32, 33, 44]. This numerical instability can be remedied by adding a so-called pressure projection term; see [7, 8, 11, 32].

In this paper, we are interested in the model of the Stokes problem with the slip boundary conditions of friction type [4, 6, 16, 24, 27, 39, 40, 41, 42]. This frictional boundary conditions appear in modeling of blood flow in vein of an arterial sclerosis patient and in that avalanche of water and rocks. This model has drawn many experts’ attention in the past several decades. However, even now, there are many difficulties arising from this model remain unsolved including a priori and posteriori estimates.

In this paper, we discretize the corresponding variational inequality with the stabilized finite element methods proposed in [8, 28, 29, 30, 31, 32, 33, 36, 38]. Since the discrete bilinear form is stable, our discrete variational inequality admits a unique finite element solution. An important contribution of the paper is to analyze the a priori estimates for the finite element solutions. The optimal result is first valid for both the unstable linear–constant and the linear–linear pairs. Another important contribution is the a posteriori estimates. Although a posteriori estimates have been intensively studied for the Stokes problems with Dirichlet boundary conditions [2, 5, 26, 31, 46, 47, 48], the corresponding result for the Stokes problems with nonlinear slip boundary conditions has not been completely solved. Here, we obtain the error estimators constitutes of the standard error indicators for the Stokes problem with Dirichlet boundary condition and error indicators on the slip boundary. With a rigorous proof, we show that the error estimators are reliable. The efficiency of the standard error indicators is also discussed. Numerical experiments show that the error indicators on the slip boundary are also efficient, its complete theoretical proof is still to be derived.

The paper is organized as follows. In Sect. 2, we introduce a Stokes problem with a slip boundary and the stabilized finite element methods for this problem. The optimal a priori error estimate is given in Sect. 3. While in Sect. 4, the a posteriori error estimate is provided. In the final section, numerical experiments are performed to validate our theory developed in previous sections.

Throughout this paper, “\(A\lesssim B\)” means that A can be bounded by B multiplied with a generic constant depending only on the shape regularity of the underlying grid or the given domain, “\(A\simeq B\)” stands for “\(A\lesssim B\)” and “\(B\lesssim A\)”. In addition, the spaces of consisting of vector-valued and scalar-valued functions will be denoted in bold face and usual face, respectively.

2 The Stokes problem with slip boundary conditions and its finite element solution

We consider the following Stokes equations with the slip boundary conditions of friction type:
$$\begin{aligned}& -\mu \Delta \mathbf{u}+\nabla p = \mathbf{f}, \qquad \operatorname{div}\mathbf{u}=0, \quad \mbox{in } \varOmega , \end{aligned}$$
(2.1)
$$\begin{aligned}& \mathbf{u} = 0, \quad \mbox{on } \varGamma _{D}, \end{aligned}$$
(2.2)
$$\begin{aligned}& \mathbf{u}_{n}=0, \qquad \vert \sigma _{\tau } \vert \le g, \qquad \sigma _{\tau } \mathbf{u}_{\tau }+g \vert \mathbf{u}_{\tau } \vert =0,\quad \mbox{on } \varGamma . \end{aligned}$$
(2.3)
Here Ω is a bounded convex polygon of \(R^{2}\), its boundary ∂Ω is flat and includes two connected portions \(\varGamma _{D}\) and Γ. The portions Γ and \(\varGamma _{D}\) are not empty. The functions \(\mathbf{u}(x)\), \(p(x)\) and \(\mathbf{f}(x)\) represent the velocity vector, the pressure, and the prescribed body force, respectively; the coefficient \(\mu >0\) stands for the viscosity and g is a given positive scalar function. Here and in what follows, the unit outward normal to the boundary is denoted by n and for a vector v defined on the boundary, we use \(\mathbf{v}_{n}= \mathbf{v}\cdot \mathbf{n}\) and \(\mathbf{v}_{\tau }=\mathbf{v}- \mathbf{v}_{n}\mathbf{n}\) to denote the normal and tangential component of v, respectively. The function \(\sigma _{\tau }=\mu \frac{ \partial \mathbf{u}_{\tau }}{\partial n}\) denotes tangential component of stress vector defined on Γ.
In the following, we present the weak formulation of (2.1)–(2.3). Let \(H^{m}(\varOmega )\) be the Sobolev space with the norm \(\|\cdot \|_{m,\varOmega }\). Note that in the case \(m=0\), we set \(H^{0}(\varOmega )=L^{2}(\varOmega )\). Let \(H^{1}_{0}(\varOmega )\) be the subspace of functions in \(H^{1}(\varOmega )\) with zero trace on ∂Ω. We define the following spaces [1]:
$$\begin{aligned}& \mathbf{V}=\bigl\{ \mathbf{v}\in \bigl[H^{1}(\varOmega ) \bigr]^{2}: {\mathbf{v}}|_{\varGamma _{D}}=0, \mathbf{v}_{n}|_{\varGamma }=0 \bigr\} , \\& Q=L_{0}^{2}(\varOmega )\equiv \biggl\{ q\in L^{2}(\varOmega ): \int _{\varOmega }q\,d\mathbf{x}=0\biggr\} . \end{aligned}$$
We denote the inner product and norm on Ω and its boundary Γ, respectively, by
$$\begin{aligned}& (\mathbf{u},\mathbf{v})_{\varOmega }= \int _{\varOmega }\mathbf{u}\cdot \mathbf{v}\, d \mathbf{x},\qquad \langle \mathbf{u},\mathbf{v}\rangle_{\varGamma }= \int _{\varGamma }\mathbf{u}\cdot \mathbf{v}\,ds, \\& \Vert \mathbf{u} \Vert _{0,\varOmega }=(\mathbf{u},\mathbf{u})^{1/2}, \qquad \Vert \mathbf{u} \Vert _{0,\varGamma }=\langle\mathbf{u}, \mathbf{u}\rangle_{\varGamma }. \end{aligned}$$
We introduce a norm \(|\!|\!|\cdot |\!|\!|\) in the coupled space \(\mathbf{V}\times Q\) by
$$ \bigl|\!\bigl|\!\bigl|(\mathbf{v},q) \bigr|\!\bigr|\!\bigr|= \bigl(\mu \Vert \nabla \mathbf{v} \Vert _{0}^{2}+ \Vert q \Vert _{0}^{2} \bigr)^{1/2},\quad \forall (\mathbf{v},q)\in \mathbf{V} \times Q, $$
where
$$ \Vert \nabla \mathbf{v} \Vert _{0}=(\nabla \mathbf{v},\nabla \mathbf{v})^{1/2},\quad \forall \mathbf{v}\in \mathbf{V}, $$
and
$$ \Vert p \Vert _{0}=(p,p)^{1/2}, \quad p\in Q. $$
We define
$$\begin{aligned}& a(\mathbf{u},\mathbf{v}) = 2\mu \bigl(\mathbb{D}(\mathbf{u}),\mathbb{D}( \mathbf{v})\bigr)=\mu (\nabla \mathbf{u},\nabla \mathbf{v}),\quad \forall \mathbf{u}, \mathbf{v}\in \mathbf{V}, \\& d(\mathbf{v},p) = (\operatorname{div}\mathbf{v},p),\quad (\mathbf{v},p)\in \mathbf{V} \times Q, \end{aligned}$$
where
$$ \mathbb{D}(\mathbf{v})=\frac{1}{2}\bigl(\nabla \mathbf{v}+\nabla \mathbf{v}^{T}\bigr) $$
represents the rate of deformation tensor. Let the bilinear form
$$ B\bigl((\mathbf{u},p);(\mathbf{v},q)\bigr)=a(\mathbf{u},\mathbf{v})-d( \mathbf{v},p)+d(\mathbf{u},q),\quad \forall (\mathbf{u},p),(\mathbf{v},q) \in \mathbf{V}\times Q. $$
Since the bilinear forms \(a(\cdot ,\cdot )\) and \(d(\cdot ,\cdot )\) are continuous, the bilinear form \(B(\cdot ,\cdot )\) is also continuous satisfying
$$ B\bigl((\mathbf{u},p);(\mathbf{v},q)\bigr)\lesssim \bigl|\!\bigl|\!\bigl|(\mathbf{u},p) \bigr|\!\bigr|\!\bigr|\bigl|\!\bigl|\!\bigl|( \mathbf{v},q) \bigr|\!\bigr|\!\bigr|,\quad \forall ( \mathbf{u},p), (\mathbf{v},q)\in \mathbf{V}\times Q. $$
(2.4)
Moreover, we have the following inf-sup property:
$$ \bigl|\!\bigl|\!\bigl|(\mathbf{u},p) \bigr|\!\bigr|\!\bigr|\lesssim \sup_{(\mathbf{v},q)\in (\mathbf{V},Q)} \frac{ \vert B((\mathbf{u},p);( \mathbf{v},q)) \vert }{ |\!|\!|(\mathbf{v},q) |\!|\!|}. $$
(2.5)
Here, the hidden constant is independent of \((\mathbf{u},p)\) and \((\mathbf{v},q)\).
Now we introduce the barrier term against slip on Γ as
$$ j(z)= \int _{\varGamma }g \vert z \vert \,ds, \quad z\in L^{2}(\varGamma ). $$
Obviously, j is a continuous functional defined on \(L^{2}(\varGamma )\).

The weak formulation of the Stokes equations (2.1)–(2.3) reads (cf. [16])

Find \((\mathbf{u},p)\in \mathbf{V}\times Q\) such that
$$ B\bigl((\mathbf{u},\mathbf{p});(\mathbf{v}-\mathbf{u},q)\bigr)+ j(\mathbf{v} _{\tau })-j(\mathbf{u}_{\tau })\ge (f,\mathbf{v}-\mathbf{u}), \quad \forall ( \mathbf{v},q)\in \mathbf{V}\times Q. $$
(2.6)
Also from [16], we know that (2.6) admits a unique solution. Moreover, if \(\varGamma _{D}\in C^{2}\), \(\varGamma \in C^{4}\), \(\mathbf{f} \in [L^{2}(\varOmega )]^{2}\) and \(g\in H^{1}(\varGamma )\), then the solution \((\mathbf{u},p)\in [H^{2}(\varOmega )]^{2}\times H^{1}(\varOmega )\) (cf. [42]).

We next present the stabilized finite element methods for the present model. The stabilized finite element methods have been introduced in [8, 11, 32] for the Stokes equations with Dirichlet boundary conditions.

Let \(\mathcal{T}_{h}\) be a regular, quasi-uniform triangulation of the polygonal domain Ω into a union of triangles [12, 13]. Associated with \(\mathcal{T}_{h}=\{K\}\), we consider the finite element spaces for the velocity and pressure: \(V_{h}\subset V\) and \(Q_{h}\subset Q\) with \(h=\max \{h_{K}: K\in \mathcal{T}_{h}\}\).

We first define
$$\begin{aligned}& \mathbf{V}_{h}=\bigl\{ \mathbf{v}\in \mathbf{V}: \mathbf{v}|_{K}\in \bigl[P_{1}(K)\bigr]^{2}, \forall K\in \mathcal{T}_{h}\bigr\} , \\& Q_{h}= \textstyle\begin{cases} Q_{h}^{0}\equiv \{q\in Q : q|_{K}\in P_{0}(K), \forall K\in \mathcal{T}_{h} \}, \\ Q_{h}^{1}\equiv \{q\in Q : q|_{K}\in P_{1}(K), \forall K\in \mathcal{T}_{h} \}, \end{cases}\displaystyle \end{aligned}$$
where \(P_{1}\) and \(P_{0}\) denote the spaces of polynomials of degree 1 or 0 on set K, respectively. It is well known that, for all \((\mathbf{v},q)\in [W^{2,\infty }(\varOmega )]^{2}\times H^{1}(\varOmega )\), we have the approximation properties:
$$\begin{aligned}& \inf_{(\mathbf{v}_{h},q_{h})\in \mathbf{V}_{h}\times Q_{h}} \bigl|\!\bigl|\!\bigl|( \mathbf{v}-\mathbf{v}_{h},q-q_{h}) \bigr|\!\bigr|\!\bigr|\lesssim h\bigl( \Vert \mathbf{v} \Vert _{2}+ \Vert q \Vert _{1}\bigr), \end{aligned}$$
(2.7)
$$\begin{aligned}& \inf_{\mathbf{v}_{h}\in \mathbf{V}_{h}} \Vert \mathbf{v}-\mathbf{v}_{h} \Vert _{0,\varGamma } \lesssim h^{2} \Vert \mathbf{v} \Vert _{2,\infty }. \end{aligned}$$
(2.8)
Here, we aim to formulate the stabilized mixed finite element methods for the lowest equal-order pair \(\mathbf{V}_{h}\times Q_{h}^{1}\) and for the lowest order pair \(\mathbf{V}_{h}\times Q_{h}^{0}\). These two pairs are unstable and do not satisfy the so-called Babus̆ka–Brezzi stability condition. A local pressure projection method is recalled in [8, 11, 32]. Let \(\varPi _{h}\) be the elementwise \(L^{2}\)-projection
$$ \varPi _{h}= \textstyle\begin{cases} \varPi _{0} : L^{2}(\varOmega )\rightarrow Q_{h}^{1},& \mbox{for } P_{1}-P_{0}, \\ \varPi _{1} : L^{2}(\varOmega )\rightarrow Q_{h}^{0},& \mbox{for } P_{1}-P_{1}. \end{cases} $$
Here, the operators \(\varPi _{0}\) and \(\varPi _{1}\) are, respectively, applied to stabilize the lowest equal-order pair \(\mathbf{V}_{h}\times Q_{h} ^{1}\) and the lowest order pair \(\mathbf{V}_{h}\times Q_{h}^{0}\). Furthermore, the mapping satisfies the following properties [8, 32]:
$$\begin{aligned}& \Vert \varPi _{h} p \Vert _{0} \lesssim \Vert p \Vert _{0},\quad \forall p\in Q_{h}, \end{aligned}$$
(2.9)
$$\begin{aligned}& \Vert p-\varPi _{h} p \Vert _{0} \lesssim h \Vert p \Vert _{1} ,\quad \forall p\in H^{1}(\varOmega ) \cap Q_{h}. \end{aligned}$$
(2.10)
Now, we are in a position to introduce the stabilized bilinear form (cf. [8, 11, 32])
$$ G(p_{h},q_{h})= (p_{h}-\varPi _{h}p_{h},q_{h}-\varPi _{h}q_{h}). $$
(2.11)
We define the bilinear form for all \((\mathbf{u}_{h},p_{h}), ( \mathbf{v}_{h},q_{h})\in \mathbf{V}_{h}\times Q_{h}\) by
$$ \mathcal{B}_{h}\bigl((\mathbf{u}_{h},p_{h});( \mathbf{v}_{h},q_{h})\bigr)=B\bigl(( \mathbf{u}_{h},p_{h});(\mathbf{v}_{h},q_{h}) \bigr)+G(p_{h},q_{h}). $$
The finite element approximation to (2.6) is to find \((\mathbf{u}_{h},p_{h})\in \mathbf{V}_{h}\times Q_{h}\) such that, for all \((\mathbf{v}_{h},q_{h})\in \mathbf{V}_{h}\times Q_{h}\),
$$ \mathcal{B}_{h}\bigl((\mathbf{u}_{h},p_{h});( \mathbf{v}_{h}-\mathbf{u}_{h},q _{h}) \bigr)+j(\mathbf{v}_{h\tau })-j(\mathbf{u}_{h\tau }) \geq ( \mathbf{f}, \mathbf{v}_{h}-\mathbf{u}_{h}). $$
(2.12)
By the definition of \(\mathcal{B}_{h}(\cdot ,\cdot )\), (2.12) is for all \(\mathbf{v}_{h}\in \mathbf{V}_{h}\) equivalent to
$$ a(\mathbf{u}_{h},\mathbf{v}_{h}- \mathbf{u}_{h})- d(\mathbf{v}_{h}- \mathbf{u}_{h},p_{h})+j( \mathbf{v}_{h\tau })-j(\mathbf{u}_{h\tau }) \geq (\mathbf{f}, \mathbf{v}_{h}-\mathbf{u}_{h}), $$
(2.13)
and for all \(q_{h}\in Q_{h}\),
$$ d(\mathbf{u}_{h}, q_{h})+G(p_{h},q_{h})=0. $$
(2.14)

3 A priori error estimates

In this section, we estimate the \(|\!|\!|\cdot |\!|\!|\) norm of the error between the exact solution \((\mathbf{u}, p)\) of (2.6) and the finite element solution \((\mathbf{u}_{h},p_{h})\) of (2.12).

To this end, we first study the properties of the bilinear form \(\mathcal{B}_{h}(\cdot ,\cdot )\). By a similar approach in [8, 32], it is easy to verify the following continuity: for all \((\mathbf{v}_{h},q_{h})\), \((\mathbf{w}_{h},\lambda _{h})\in \mathbf{V} _{h}\times Q_{h}\),
$$ \bigl\vert \mathcal{B}_{h}\bigl((\mathbf{v}_{h},q_{h});( \mathbf{w}_{h},\lambda _{h})\bigr) \bigr\vert \lesssim \bigl|\!\bigl|\!\bigl|(\mathbf{v}_{h},q_{h}) \bigr|\!\bigr|\!\bigr|\bigl|\!\bigl|\!\bigl|(\mathbf{w}_{h},\lambda _{h}) \bigr|\!\bigr|\!\bigr|, $$
(3.1)
where the hidden constant is independent of the mesh size h. Moreover, by similar arguments in [20, 25, 33, 39, 40], it is not hard to establish the following inf-sup property: for all \((\mathbf{v}_{h},q _{h})\in \mathbf{V}_{h}\times Q_{h}\),
$$ \bigl|\!\bigl|\!\bigl|(\mathbf{v}_{h},q_{h}) \bigr|\!\bigr|\!\bigr|\lesssim \sup_{(\mathbf{w}_{h},\lambda _{h})\in \mathbf{V}_{h}\times Q_{h}} \frac{\mathcal{B}_{h}((\mathbf{v}_{h},q_{h});(\mathbf{w}_{h},\lambda _{h}))}{ |\!|\!|( \mathbf{w}_{h},\lambda _{h}) |\!|\!|}. $$
(3.2)

Due to (3.1), (3.2) and the fact that \(j(\cdot )\) is a continuous functional defined on the convex set \(\mathbf{V}_{h}\times Q_{h}\), the discrete problem (2.12) has a unique solution [4, 40].

For completeness, we are ready to prove the following theorem for the model approximated by the lower order finite element pairs.

Theorem 3.1

Let\((\mathbf{u},p)\in \mathbf{\mathbf{V}}\times Q\)and\((\mathbf{u} _{h},p_{h})\in \mathbf{V}_{h}\times Q_{h}\)be solutions of (2.6) and (2.12), respectively. If\((\mathbf{u},p)\in [H^{2}(\varOmega )]^{2} \times H^{1}(\varOmega )\), then
$$ \bigl|\!\bigl|\!\bigl|(\mathbf{u}-\mathbf{u}_{h},p-p_{h}) \bigr|\!\bigr|\!\bigr|\lesssim h\bigl( \Vert \mathbf{u} \Vert _{2}+ \Vert p \Vert _{1}+ \Vert {\mathbf{u}} \Vert ^{\frac{1}{2}}_{2}\bigr), $$
(3.3)
where the hidden constant only depends on the data\((\mu , \mathbf{f}, g, \varOmega )\).

Proof

First, we estimate \(\|\nabla (\mathbf{u}-\mathbf{u}_{h})\|_{0}\). By the triangle inequality, for all \(\mathbf{v}_{h}\in \mathbf{V}_{h}\),
$$ \bigl\Vert \nabla (\mathbf{u}-\mathbf{u}_{h}) \bigr\Vert _{0}\le \bigl\Vert \nabla (\mathbf{u}-\mathbf{v} _{h}) \bigr\Vert _{0} + \bigl\Vert \nabla (\mathbf{u}_{h}- \mathbf{v}_{h}) \bigr\Vert _{0}. $$
By the definition of the bilinear \(\mathcal{B}_{h}\), for all \((\mathbf{v} _{h},q_{h})\in \mathbf{V}_{h}\times Q_{h}\),
$$ \mu \bigl\Vert \nabla (\mathbf{u}_{h}- \mathbf{v}_{h}) \bigr\Vert _{0}^{2} \leq \mathcal{B}_{h}\bigl(( \mathbf{u}_{h}-\mathbf{v}_{h},p_{h}-q_{h}),( \mathbf{u}_{h}-\mathbf{v} _{h},p_{h}-q_{h}) \bigr)=I_{1}-I_{2}, $$
(3.4)
where
$$ I_{1}=\mathcal{B}_{h}\bigl((\mathbf{u}_{h},p_{h}),( \mathbf{u}_{h}-\mathbf{v}_{h},p _{h}-q_{h}) \bigr) $$
and
$$ I_{2}=\mathcal{B}_{h}(\mathbf{v}_{h},q_{h}),( \mathbf{u}_{h}-\mathbf{v}_{h},p _{h}-q_{h})). $$
By (2.12), we have
$$ I_{1}\le (f,\mathbf{u}_{h}- \mathbf{v}_{h})+j(\mathbf{v}_{h\tau })-j( \mathbf{u}_{h\tau }). $$
(3.5)
Taking \((\mathbf{v},q)=(\mathbf{u}_{h}-\mathbf{v}_{h}+\mathbf{u},p _{h}-q_{h})\) in (2.6), we have
$$\begin{aligned} (f,\mathbf{u}_{h}-\mathbf{v}_{h}) \le & B\bigl(( \mathbf{u},p),(\mathbf{u} _{h}-\mathbf{v}_{h},p_{h}-q_{h}) \bigr)+j\bigl((\mathbf{u}_{h}-\mathbf{v}_{h}+ \mathbf{u})_{\tau }\bigr)-j(\mathbf{u}_{\tau }) \\ \le & \mathcal{B}_{h}\bigl((\mathbf{u},p),(\mathbf{u}_{h}- \mathbf{v}_{h},p_{h}-q _{h}) \bigr)-G(p,p_{h}-q_{h}) \\ &{}+j\bigl((\mathbf{u}_{h}-\mathbf{v}_{h}+ \mathbf{u})_{\tau }\bigr)-j(\mathbf{u} _{\tau }). \end{aligned}$$
(3.6)
Using the Hölder inequality, it is easy to verify that, for all \(\mathbf{v}\in {\mathbf{V}}\),
$$\begin{aligned} j(\mathbf{v}_{h\tau })-j(\mathbf{u}_{\tau }) =& \int _{\varGamma }g \vert \mathbf{v}_{h\tau } \vert \,ds- \int _{\varGamma }g \vert \mathbf{u}_{\tau } \vert \,ds \\ \leq & \int _{\varGamma }g \vert \mathbf{v}_{h\tau }- \mathbf{u}_{\tau } \vert \,ds \\ \lesssim &h_{\varGamma }^{1/2} \Vert g \Vert _{\infty ,\varGamma } \Vert \mathbf{v}_{h\tau }- \mathbf{u}_{\tau } \Vert _{0,\varGamma }, \end{aligned}$$
(3.7)
where the mesh size \(h_{\varGamma }\) is the largest scale of the set \(K\cap \varGamma \).
Moreover, using the definition of \(G(\cdot ,\cdot )\) and the Hölder inequality, yields
$$\begin{aligned} G(p,p_{h}-q_{h}) =&\bigl(p-\varPi _{h}p,(p_{h}-q_{h})- \varPi _{h}(p_{h}-q_{h})\bigr) \\ \leq& \Vert p-\varPi _{h} p \Vert _{0} \Vert p_{h}-q_{h} \Vert _{0}. \end{aligned}$$
(3.8)
Substituting (3.6) and (3.7) into (3.5), we have
$$\begin{aligned} I_{1} \le & \mathcal{B}_{h}\bigl((\mathbf{u},p),( \mathbf{u}_{h}-\mathbf{v}_{h},p _{h}-q_{h}) \bigr)+2h_{\varGamma }^{1/2} \Vert g \Vert _{\infty ,\varGamma } \Vert \mathbf{u}_{\tau }- \mathbf{v}_{h\tau } \Vert _{0,\varGamma } \\ &{} + \Vert p-\varPi _{h} p \Vert _{0} \Vert p_{h}-q_{h} \Vert _{0}. \end{aligned}$$
Therefore,
$$\begin{aligned} I_{1}-I_{2} \le &\mathcal{B}_{h}\bigl(( \mathbf{u}-\mathbf{v}_{h},p-q_{h}),( \mathbf{u}_{h}-\mathbf{v}_{h},p_{h}-q_{h}) \bigr) \\ &{}+ \Vert p-\varPi _{h} p \Vert _{0} \Vert p_{h}-q_{h} \Vert _{0}+2h_{\varGamma }^{1/2} \Vert g \Vert _{\infty ,\varGamma } \Vert \mathbf{u}_{\tau }- \mathbf{v}_{h\tau } \Vert _{0,\varGamma } \\ \lesssim & \bigl|\!\bigl|\!\bigl|(\mathbf{u}-\mathbf{v}_{h},p-q_{h}) \bigr|\!\bigr|\!\bigr|\bigl|\!\bigl|\!\bigl|( \mathbf{u}_{h}-\mathbf{v}_{h},p_{h}-q_{h}) \bigr|\!\bigr|\!\bigr|\\ &{}+ \Vert p-\varPi _{h} p \Vert _{0} \Vert p_{h}-q_{h} \Vert _{0}+h_{\varGamma }^{1/2} \Vert g \Vert _{\infty ,\varGamma } \Vert \mathbf{u}_{\tau }- \mathbf{v}_{h\tau } \Vert _{0,\varGamma }. \end{aligned}$$
Then, by (3.4) and the Young inequality,
$$\begin{aligned} \bigl\Vert \nabla (\mathbf{u}_{h}-\mathbf{v}_{h}) \bigr\Vert _{0}^{2} \lesssim& \bigl|\!\bigl|\!\bigl|( \mathbf{u}- \mathbf{v}_{h},p-q_{h}) \bigr|\!\bigr|\!\bigr|^{2}+ \Vert p_{h}-q_{h} \Vert _{0} ^{2} \\ &{}+ \Vert p-\varPi _{h} p \Vert _{0}^{2}+h_{\varGamma }^{1/2} \Vert \mathbf{u}_{\tau }- \mathbf{v}_{h\tau } \Vert _{0,\varGamma }. \end{aligned}$$
Consequently
$$ \bigl\Vert \nabla (\mathbf{u}-\mathbf{u}_{h}) \bigr\Vert _{0}^{2}\lesssim \bigl|\!\bigl|\!\bigl|( \mathbf{u}- \mathbf{v}_{h},p-q_{h}) \bigr|\!\bigr|\!\bigr|^{2}+ \Vert p_{h}-q_{h} \Vert _{0} ^{2}+h^{2} \Vert p \Vert _{1}^{2}+h^{1/2} \Vert \mathbf{u}_{\tau }-\mathbf{v}_{h\tau } \Vert _{0,\varGamma }, $$
(3.9)
where we have used the fact that \(\|p-\varPi _{h} p\|_{0}\lesssim h\|p\| _{1}\). On the other hand, set that
$$ \bar{\mathbf{V}}_{h}=\bigl\{ \mathbf{v}\in \bigl[H_{0}^{1}( \varOmega )\bigr]^{2}: \mathbf{v}|_{K} \in \bigl[P_{1}(K) \bigr]^{2}\bigr\} \subset \mathbf{V}_{h}. $$
Let \(\mathbf{w}_{h}\in \bar{\mathbf{V}}_{h}\) and taking \(\mathbf{v}= \mathbf{u}\pm \mathbf{w}_{h}\) separately in (2.6), we obtain
$$ a(\mathbf{u},\mathbf{w}_{h})-d(\mathbf{w}_{h},p)=( \mathbf{f}, \mathbf{w}_{h}). $$
Also, taking \(\mathbf{v}_{h}=\mathbf{u}_{h}\pm \mathbf{w}_{h}\) separately in (2.12), we obtain
$$ a(\mathbf{u}_{h},\mathbf{w}_{h})-d( \mathbf{w}_{h},p_{h})=(\mathbf{f}, \mathbf{w}_{h}). $$
Then, for all \((\mathbf{w}_{h},\lambda _{h})\in \mathbf{V}_{h}\times Q_{h}\), noting that
$$ d(\mathbf{u-u_{h}}, \lambda _{h})+G(p-p_{h},\lambda _{h})=0, $$
we can derive from the above two equalities that
$$ a(\mathbf{u-u_{h}},\mathbf{w}_{h})-d( \mathbf{w}_{h},p-p_{h})+d( \mathbf{u-u_{h}}, \lambda _{h})+G(p-p_{h},\lambda _{h})=0. $$
(3.10)
Then, we can derive the following result from the weak coercivity property in [19]:
$$\begin{aligned}& \Vert p_{h}-q_{h} \Vert _{0} \\& \quad \lesssim \sup_{(\mathbf{w}_{h},\lambda _{h})\in (\bar{\mathbf{V}}_{h},Q _{h})}\frac{\mathcal{B}_{h}((u_{h}-v_{h},p_{h}-q_{h});(\mathbf{w}_{h},\lambda _{h}))}{ |\!|\!|(\mathbf{w}_{h},\lambda _{h}) |\!|\!|} \\& \quad \lesssim \sup_{(\mathbf{w}_{h},\lambda _{h})\in (\bar{\mathbf{V}}_{h},Q _{h})}\frac{\mathcal{B}_{h}((u_{h}-u,p_{h}-p);(\mathbf{w}_{h},\lambda _{h}))+\mathcal{B}_{h}((u-v_{h},p-q_{h});(\mathbf{w}_{h},\lambda _{h}))}{ |\!|\!|(\mathbf{w} _{h},\lambda _{h}) |\!|\!|} \\& \quad \lesssim \biggl(\sup_{(\mathbf{w}_{h},\lambda _{h})\in (\bar{\mathbf{V}}_{h},Q _{h})}\frac{G(p,\lambda _{h})}{ |\!|\!|(\mathbf{w}_{h},\lambda _{h}) |\!|\!|}+ \sup _{(\mathbf{w}_{h},\lambda _{h})\in (\bar{\mathbf{V}}_{h},Q_{h})}\frac{\mathcal{B}_{h}((u-v_{h},p-q_{h});(\mathbf{w}_{h},\lambda _{h}))}{ |\!|\!|(\mathbf{w} _{h},\lambda _{h}) |\!|\!|} \biggr) \\& \quad \lesssim \bigl( \bigl|\!\bigl|\!\bigl|(u-v_{h},p-q_{h}) \bigr|\!\bigr|\!\bigr|+ \Vert p-\varPi _{h}p \Vert _{0}\bigr), \end{aligned}$$
(3.11)
where the hidden constant only depends on Ω.
Thus, we obtain, for all \((\mathbf{v}_{h},q_{h})\in \mathbf{V}_{h} \times Q_{h}\),
$$ \bigl|\!\bigl|\!\bigl|(\mathbf{u}-\mathbf{u}_{h}, p-p_{h}) \bigr|\!\bigr|\!\bigr|^{2} \lesssim \bigl|\!\bigl|\!\bigl|( \mathbf{u}- \mathbf{v}_{h},p-q_{h}) \bigr|\!\bigr|\!\bigr|^{2}+h^{2} \Vert p \Vert _{1}^{2}+h ^{1/2} \Vert \mathbf{u}_{\tau }-\mathbf{v}_{h\tau } \Vert _{0,\varGamma }. $$
Noticing estimates (2.7) and (2.8), the proof of (3.3) is completed. □

4 A posteriori error estimates

In this section, we will derive the a reliable and efficient a posteriori estimates for the error \(|\!|\!|(\mathbf{u}-\mathbf{u} _{h},p-p_{h})|\!|\!|\).

Denote by \(\mathcal{E}_{h}\) the set of all of the interior element-edges. For \(K\in \mathcal{T}_{h}\) and \(E\in \mathcal{E}_{h}\), we define the element residual and jump residual by
$$\begin{aligned}& \mathbf{R}_{K}=\mathbf{f}+\mu \Delta \mathbf{u}_{h}-\nabla p_{h}, \\& J_{E}=\biggl[\frac{\partial \mathbf{u}_{h}}{\partial {\mathbf{n}}}-p_{h}{\mathbf{n}} \biggr]_{E}=[\nabla \mathbf{u}_{h}-p_{h} \mathbf{I}]_{E}\cdot \mathbf{n}. \end{aligned}$$
Hereafter, \([q]_{E}\) denotes the jump of q across an interior side \(E\in \mathcal{E}_{h}\), n denotes a unit normal of \(E\in \mathcal{E}_{h}\), and I denotes the identity matrix. We define the elementwise local error indicator for all \(K\in \mathcal{T}_{h}\) by
$$ \eta _{K}^{2} = h_{K}^{2} \Vert \mathbf{R}_{K} \Vert _{0,K}^{2}+ \Vert \operatorname{div}\mathbf{u}_{h} \Vert ^{2}_{0,K}+ \sum_{E\in \mathcal{E}_{h}\cap \partial K}h_{E} \Vert J _{E} \Vert _{0,E}^{2}+\sum _{E\in \varGamma \cap \partial K}h_{E}^{2} \Vert g \Vert _{\infty ,E} ^{2}. $$
Then, the global error estimator can be written by
$$ \eta = \biggl(\sum_{K\in \mathcal{T}_{h}}\eta _{K}^{2} \biggr)^{1/2}. $$

In the following, we analyze the reliability and efficiency of η. Let \(\mathbf{u}_{i}\) and \(p_{i}\) be the Scott–Zhang interpolation (cf. [43]) of u and p, respectively. For simplicity, we denote \((\mathbf{e}_{i},\epsilon _{i})=( \mathbf{u}_{i}-\mathbf{u}_{h},p_{i}-p_{h})\) and \((\mathbf{e},\epsilon )=( \mathbf{u}-\mathbf{u}_{h},p-p_{h})\).

4.1 Reliability

In this subsection, we show that the error \(|\!|\!|(\mathbf{e}, \epsilon )|\!|\!|\) can be bounded by the estimator η.

Theorem 4.1

Let\((\mathbf{u},p)\in \mathbf{V}\times Q\)and\((\mathbf{u}_{h},p_{h}) \in \mathbf{V}_{h}\times Q_{h}\)be solutions of (2.6) and (2.12), respectively. Then
$$ \bigl|\!\bigl|\!\bigl|(\mathbf{e},\epsilon ) \bigr|\!\bigr|\!\bigr|\lesssim \eta , $$
(4.1)
where the hidden constant is independent ofh.

Proof

A direct calculation yields
$$ B\bigl((\mathbf{e},\epsilon ),(\mathbf{e}_{i},\epsilon _{i}) \bigr)=B\bigl((\mathbf{u},p),( \mathbf{e}_{i},\epsilon _{i}) \bigr)-\mathcal{B}_{h}\bigl((\mathbf{u}_{h},p_{h}),( \mathbf{e} _{i},\epsilon _{i})\bigr)+G(p_{h},\epsilon _{i}). $$
Then
$$ B\bigl((\mathbf{e},\epsilon ),(\mathbf{e}_{i},\epsilon _{i}) \bigr)=I_{1}+I_{2}, $$
where
$$ I_{1}=(\mathbf{f},\mathbf{e}_{i})-\mathcal{B}_{h} \bigl((\mathbf{u}_{h},p_{h}),( \mathbf{e}_{i}, \epsilon _{i})\bigr)-(\mathbf{f},\mathbf{e})+B\bigl((\mathbf{u},p),( \mathbf{e},\epsilon )\bigr), $$
and
$$ I_{2}=-(\mathbf{f},\mathbf{e}_{i}-\mathbf{e})+B\bigl(( \mathbf{u},p),( \mathbf{e}_{i}-\mathbf{e},\epsilon _{i}-\epsilon ) \bigr)+G(p_{h},\epsilon _{i}). $$
Taking \((\mathbf{v}_{h},q_{h})=(\mathbf{u}_{i},p_{i}-p_{h})\) in (2.12), we obtain
$$ (\mathbf{f},\mathbf{e}_{i})-\mathcal{B}_{h}\bigl(( \mathbf{u}_{h},p_{h}),( \mathbf{e}_{i},\epsilon _{i})\bigr) \leq j(\mathbf{u}_{i\tau })-j( \mathbf{u}_{h\tau }). $$
Similarly, taking \((\mathbf{v},q)=(\mathbf{u}_{h},p_{h}-p)\) in (2.6) gives
$$ -(\mathbf{f},e)+B\bigl((\mathbf{u},p),(\mathbf{e},\epsilon )\bigr) \leq j( \mathbf{u}_{h\tau })-j(\mathbf{u}_{\tau }). $$
Thus
$$\begin{aligned} I_{1} \le & j(\mathbf{u}_{i\tau })-j(\mathbf{u}_{\tau }) \\ =& \int _{\varGamma }g \vert \mathbf{u}_{i\tau }- \mathbf{u}_{\tau } \vert \,ds \\ \le & j(\mathbf{e}_{i\tau })-j(\mathbf{e}_{\tau }). \end{aligned}$$
We next calculate \(I_{2}\). By the definition of the bilinear form \(B(\cdot ,\cdot )\) and the Green’s formula,
$$\begin{aligned} B\bigl((\mathbf{u}_{h},p_{h}),(\mathbf{e}- \mathbf{e}_{i},\epsilon -\epsilon _{i})\bigr) =& \sum _{K\in \mathcal{T}_{h}}(-\mu \Delta \mathbf{u}_{h}+\nabla p_{h},\mathbf{e}- \mathbf{e}_{i}) \\ &{}+\sum_{E\in \mathcal{E}_{h}}\langle J_{E}, \mathbf{e}-\mathbf{e}_{i}\rangle +\sum_{K\in \mathcal{T}_{h}}( \operatorname{div}\mathbf{u}_{h},\epsilon -\epsilon _{i}). \end{aligned}$$
(4.2)
Then, we have
$$\begin{aligned} I_{2} =&-(\mathbf{f},\mathbf{e}_{i}-\mathbf{e})+B \bigl((\mathbf{u},p),( \mathbf{e}_{i}-\mathbf{e},\epsilon _{i}-\epsilon )\bigr)+G(p_{h},\epsilon _{i}) \\ =&-(\mathbf{f},\mathbf{e}_{i}-\mathbf{e})+B\bigl(( \mathbf{u}_{h},p_{h}),( \mathbf{e}_{i}- \mathbf{e},\epsilon _{i}-\epsilon )\bigr) \\ &{}+G(p_{h},\epsilon _{i})+B\bigl((\mathbf{e},\epsilon ), ( \mathbf{e}_{i}-\mathbf{e},\epsilon _{i}-\epsilon )\bigr) \\ =&\sum_{K\in \mathcal{T}_{h}}(\mathbf{R}_{K}, \mathbf{e}-\mathbf{e}_{i})- \sum_{E\in \mathcal{E}_{h}} \langle J_{E},\mathbf{e}-\mathbf{e}_{i}\rangle - \sum _{K\in \mathcal{T}_{h}}(\operatorname{div}\mathbf{u}_{h},\epsilon _{i}-\epsilon ) \\ &{}+G(p_{h},\epsilon _{i})+B\bigl((\mathbf{e},\epsilon ), ( \mathbf{e}_{i}-\mathbf{e},\epsilon _{i}-\epsilon )\bigr). \end{aligned}$$
Noticing (2.14), we obtain
$$\begin{aligned} I_{2}+B\bigl((\mathbf{e},\epsilon ), (\mathbf{e}-\mathbf{e}_{i}, \epsilon -\epsilon _{i})\bigr) =& \sum_{K\in \mathcal{T}_{h}}( \mathbf{R}_{K},\mathbf{e}-\mathbf{e}_{i})- \sum _{E\in \mathcal{E}_{h}}\langle J_{E},\mathbf{e}- \mathbf{e}_{i}\rangle \\ &{}-\sum_{K\in \mathcal{T}_{h}}(\operatorname{div}\mathbf{u}_{h}, \epsilon ). \end{aligned}$$
Therefore,
$$\begin{aligned} B\bigl((\mathbf{e},\epsilon ),(\mathbf{e},\epsilon )\bigr) =&B\bigl((\mathbf{e},\epsilon ),( \mathbf{e} _{i},\epsilon _{i})\bigr)+B\bigl((\mathbf{e},\epsilon ), (\mathbf{e}-\mathbf{e}_{i},\epsilon -\epsilon _{i})\bigr) \\ =&I_{1}+I_{2}+B\bigl((\mathbf{e},\epsilon ), (\mathbf{e}- \mathbf{e}_{i},\epsilon -\epsilon _{i})\bigr) \\ \le &j(\mathbf{e}_{i\tau })-j(\mathbf{e}_{\tau })+\sum _{K\in \mathcal{T}_{h}}( \mathbf{R}_{K},\mathbf{e}- \mathbf{e}_{i}) \\ &{}+\sum_{E\in \mathcal{E}_{h}}\langle J_{E}, \mathbf{e}-\mathbf{e}_{i}\rangle -\sum_{K\in \mathcal{T}_{h}}( \operatorname{div}\mathbf{u}_{h},\epsilon ). \end{aligned}$$
Due to the trace inequality and by straightforward calculation,
$$\begin{aligned} j(\mathbf{e}_{i\tau })-j(\mathbf{e}_{\tau }) \leq & \sum _{E\in \varGamma \cap \partial K} \Vert g \Vert _{\infty ,E} \int _{E} \vert \mathbf{e}_{\tau }- \mathbf{e}_{i\tau } \vert \,ds \\ \leq &\sum_{E\in \varGamma \cap \partial K}h_{E}^{1/2} \Vert g \Vert _{\infty ,E} \Vert \mathbf{e}_{\tau }- \mathbf{e}_{i\tau } \Vert _{0,E} \\ \leq &\sum_{E\in \varGamma \cap \partial K}h_{E} \Vert g \Vert _{\infty ,E} \Vert \nabla \mathbf{e} \Vert _{0,K} \\ \leq & \biggl(\sum_{E\in \varGamma \cap \partial K}h_{E}^{2} \Vert g \Vert _{\infty ,E}^{2} \biggr) ^{1/2} \Vert \nabla \mathbf{e} \Vert _{0}. \end{aligned}$$
Using the Cauchy–Schwarz inequality and the property of the Scott–Zhang interpolation, we obtain
$$ \mu \Vert \nabla e \Vert _{0}^{2} = B\bigl((\mathbf{e}, \epsilon );(\mathbf{e},\epsilon )\bigr)\lesssim \eta \bigl|\!\bigl|\!\bigl|(\mathbf{e},\epsilon ) \bigr|\!\bigr|\!\bigr|. $$
(4.3)
By the same approach as for Theorem 3.1, we next estimate \(\|\epsilon \| _{0}\). Let \(\mathbf{w}_{h}\in \bar{\mathbf{V}}_{h}\) be arbitrary and take \(\mathbf{v}=\mathbf{u}\pm \mathbf{w}_{h}\) separately in (2.6) to obtain
$$ a(\mathbf{u},\mathbf{w}_{h})-d(\mathbf{w}_{h},p)=( \mathbf{f}, \mathbf{w}_{h}). $$
Also, taking \(\mathbf{v}=\mathbf{u}_{h}\pm \mathbf{w}_{h}\) separately in (2.12), we have
$$ a(\mathbf{u}_{h},\mathbf{w}_{h})-d( \mathbf{w}_{h},p_{h})=(\mathbf{f}, \mathbf{w}_{h}). $$
Then, for all \(\mathbf{w}_{h}\in \bar{\mathbf{V}}_{h}\), we have
$$ a(\mathbf{e},\mathbf{w}_{h})-d(\mathbf{w}_{h},\epsilon )=0. $$
(4.4)
Since \(\epsilon \in Q\), there exists \(\mathbf{w}\in [H_{0}^{1}(\varOmega )]^{2}\) such that
$$ d(\mathbf{w},\epsilon )= \Vert \epsilon \Vert _{0}^{2}, \qquad \Vert \nabla \mathbf{w} \Vert _{0} \lesssim \Vert \epsilon \Vert _{0}, $$
(4.5)
where the hidden constant is only dependent of Ω. Thus, by (4.4), for all \(\mathbf{w}_{h}\in \bar{\mathbf{V}}_{h}\),
$$\begin{aligned} \Vert \epsilon \Vert _{0}^{2} =&d(\mathbf{w}_{h}, \epsilon )+d(\mathbf{w}-\mathbf{w}_{h},\epsilon ) \\ =&a(\mathbf{e},\mathbf{w})+a(\mathbf{e},\mathbf{w}_{h}- \mathbf{w})+d( \mathbf{w}-\mathbf{w}_{h},\epsilon ). \end{aligned}$$
(4.6)
A straightforward calculation yields
$$ a(\mathbf{e},\mathbf{w}_{h}-\mathbf{w})+d( \mathbf{w}-\mathbf{w}_{h},\epsilon )= \sum_{K\in \mathcal{T}_{h}}(R_{K}, \mathbf{w}-\mathbf{w}_{h})+\sum_{E\in \mathcal{E}_{h}} \langle J_{E},\mathbf{w}-\mathbf{w}_{h}\rangle . $$
(4.7)
Now we choose \(\mathbf{w}_{h}\in \bar{\mathbf{V}}_{h}\) such that
$$ \Vert {\mathbf{w}}-\mathbf{w}_{h} \Vert _{0}\le C \Vert \nabla {\mathbf{w}} \Vert _{0}. $$
Therefore, using (4.5), (4.6) and (4.7), we deduce
$$ \Vert \epsilon \Vert _{0}\le C\bigl( \Vert \nabla \mathbf{e} \Vert _{0}+\eta \bigr). $$
(4.8)
Combining (4.3) and (4.8), we obtain (4.1). □

4.2 Efficiency

We notice that our reliable estimator derived in the previous section includes several terms. Among them, the terms \(\|h_{K}R_{K}\|_{0,K}\), \(\|\operatorname{div} {\mathbf{u}}_{h}\|_{0,K}\) and \(\|h_{E}^{1/2}J_{E}\| _{0,E}\) appear even for Stokes problem with pure Dirichlet boundary conditions. By a standard argument (cf., [46, 47]), we can prove that the following efficiency results in conjecture 4.2 are also valid for Dirichlet boundary problems. The novelty in our estimator is that we have the new term \(\|h_{E}^{\frac{1}{2}}g\|_{0,E}\) which comes from the slip boundary condition. In fact, the presented methods are still open on the efficiency in theory although numerical experiments reported in Sect. 5 is reliable and efficient. We have not been able to prove this fact yet.

Conjecture 4.2

Let\((\mathbf{u},p)\)and\((\mathbf{u}_{h},p_{h})\)be the solutions of (2.6) and (2.12), respectively. For each\(K\in \mathcal{T}_{h}\),
$$\begin{aligned}& \Vert h_{K}R_{K} \Vert _{0,K} \lesssim \bigl|\!\bigl|\!\bigl|(\mathbf{u}-\mathbf{u}_{h},p-p _{h}) \bigr|\!\bigr|\!\bigr|_{K}+ \bigl\Vert h_{K}(R_{K}- \bar{R}_{K}) \bigr\Vert _{0,K}, \end{aligned}$$
(4.9)
$$\begin{aligned}& \Vert \operatorname{div} {\mathbf{u}}_{h} \Vert _{0,K} \lesssim \bigl|\!\bigl|\!\bigl|( \mathbf{u}-\mathbf{u}_{h},p-p_{h}) \bigr|\!\bigr|\!\bigr|_{K}. \end{aligned}$$
(4.10)
For each\(E\in \mathcal{E}_{h}\),
$$ \bigl\Vert h_{E}^{1/2}J_{E} \bigr\Vert _{0,E} \lesssim \bigl|\!\bigl|\!\bigl|(\mathbf{u}-\mathbf{u} _{h},p-p_{h}) \bigr|\!\bigr|\!\bigr|_{\omega _{E}}+\sum _{K\subset \omega _{E}} \mathrm{osc}_{K}, $$
(4.11)
where the oscillation is defined for all\(K\in \mathcal{T}_{h}\)by
$$\begin{aligned} \mathrm{osc}_{K}^{2} =& \bigl\Vert h_{K}(R_{K}-\bar{R}_{K}) \bigr\Vert ^{2}_{0,K}+ \sum_{E\in \mathcal{E}_{h}\cap \partial K} h_{E}^{1/2} \Vert J_{E}- \bar{J}_{E} \Vert _{0,E}^{2} \\ &{}+\sum_{E\in \varGamma \cap \partial K} \bigl\Vert h_{E}^{\frac{1}{2}}(g- \bar{g}_{E}) \bigr\Vert ^{2}_{0,E}, \end{aligned}$$
with\(\bar{R}_{K}=\frac{1}{|K|}\int _{K} R_{K}\), \(\bar{J}_{E}= \frac{1}{|E|}\int _{E} J_{E}\)and\(\bar{g}_{E}=\frac{1}{|E|}\int _{E} g\)denote the averages of\(R_{K}\)onK, \(J_{E}\)andgonE, respectively.

5 Numerical experiments

In this section, we verify the theoretical findings in previous sections by the following three numerical examples: the first one is for a problem with a known smooth solution, the second one is for a problem on L-shape domain with a singular source f, the final one is for the well-known driven cavity problem. All experiments are implemented by using the public free finite element software FreeFem++ [15]. Moreover, for a given mesh \(\mathcal{T} _{h}\), the stabilized finite element system (2.12) is solved with Uzawa iteration by the lowest order pair \(\mathbf{P}_{1}-P_{0}\) and the lowest equal-order pair \(\mathbf{P}_{1}-P_{1}\), respectively.

In the sequel, we always use Dof to denote the cardinality of the triangulation \(\mathcal{T}_{h}\). Let \(\eta _{g}= (\sum_{E\in \varGamma \cap \partial K}h_{E}^{2}\|g\|_{\infty ,E}^{2} ) ^{1/2}\). Moreover, let \(E_{\eta }= \frac{\eta }{|\!|\!|(\mathbf{e},\epsilon )|\!|\!|}\) be the effective index of our error estimator η.

5.1 A smooth problem

We consider (2.1) to (2.3) on the square \(\varOmega =[0,1]^{2}\) with \(\mu =1\), and
f = ( 6 x 2 6 x 3 + y ( 3 y 2 ) ( 6 x 2 ) + 4 y 2 6 y 3 6 y 2 x ( 3 x 2 ) ( 6 y 2 ) + 4 x 2 ) . Open image in new window
The boundary of Ω is split into \(\partial \varOmega =\varGamma _{D} \cup \varGamma _{1}\cup \varGamma _{2}\), where the Dirichlet boundary \(\varGamma _{D}=(\{0\}\times [0,1])\cup ([0,1]\times \{0\})\), the slip boundary \(\varGamma =\varGamma _{1}\cup \varGamma _{2}\) with \(\varGamma _{1}=\{1\} \times [0,1]\) and \(\varGamma _{2}=[0,1]\times \{1\}\). On Γ,
$$ g(x,y)= \textstyle\begin{cases} 4y^{2}(1-y),& (x,y)\in \varGamma _{1}, \\ 4x^{2}(1-x),&(x,y)\in \varGamma _{2}. \end{cases} $$
This problem admits a smooth solution
u = ( x 2 y ( x 1 ) ( 3 y 2 ) x y 2 ( y 1 ) ( 3 x 2 ) ) , p = ( 2 x 1 ) ( 2 y 1 ) . Open image in new window
Let \(\mathcal{T}_{1}\) be an initial mesh obtained by partitioning the unit square into \(10\times 10\) equal squares and then dividing each square by connecting its diagonal into two triangles. Moreover, we let \(\mathcal{T}_{n}\), \(n=2,\ldots ,7\) be a sequence of nested meshes satisfying \(\#\mathcal{T}_{n}\simeq \#\mathcal{T}_{n-1}\), \(n=2,\ldots ,7\). We list the errors \(|\!|\!|(\mathbf{e},\epsilon )|\!|\!|\), the special error estimator \(\eta _{g}\), the error estimator η and the effectivity index \(E_{\eta }\) corresponding to these 7 meshes in Tables 1 and 2. From these tables, we find that the effectivity indices almost equal 5.0 and 6.0, respectively. These results imply that our error estimator is reliable and efficient for these examples. And this fact supports our analysis in Sect. 4.
Table 1

Errors, estimators and effectivity ratio: \(\mathbf{P}_{1}-P_{0}\)

Meshes

Dof

\(\|\nabla \mathbf{e}\|_{0}\)

\(\|\epsilon \|_{0}\)

⦀(e,ϵ)⦀

η

\(\eta _{g}\)

\(E_{\eta }\)

\(\mathcal{T}_{1}\)

200

0.0781549

0.0827345

0.113812

0.48791

0.0836507

4.2870

\(\mathcal{T}_{2}\)

392

0.0555608

0.0583028

0.0805371

0.360565

0.0598029

4.4770

\(\mathcal{T}_{3}\)

800

0.0387081

0.0399877

0.0556537

0.257753

0.0418828

4.6314

\(\mathcal{T}_{4}\)

1800

0.0257007

0.0260654

0.036605

0.174117

0.0279351

4.7566

\(\mathcal{T}_{5}\)

3698

0.0178858

0.0178874

0.0252955

0.12227

0.0194876

4.8337

\(\mathcal{T}_{6}\)

7200

0.0127973

0.0126668

0.0180061

0.0879465

0.0139675

4.8843

\(\mathcal{T}_{7}\)

14,450

0.00902263

0.00885705

0.0126434

0.062228

0.00985919

4.9218

Table 2

Errors, estimators and effectivity ratio: \(\mathbf{P}_{1}-P_{1}\)

Meshes

Dof

\(\|\nabla \mathbf{e}\|_{0}\)

\(\|\epsilon \|_{0}\)

⦀(e,ϵ)⦀

η

\(\eta _{g}\)

\(E_{\eta }\)

\(\mathcal{T}_{1}\)

200

0.0775394

0.0357616

0.0853888

0.475923

0.0836507

5.5736

\(\mathcal{T}_{2}\)

392

0.0554011

0.0212961

0.0593532

0.342517

0.0598029

5.7708

\(\mathcal{T}_{3}\)

800

0.0387005

0.0122152

0.0405825

0.24037

0.0418828

5.9230

\(\mathcal{T}_{4}\)

1800

0.0257246

0.00648786

0.0265301

0.160261

0.0279351

6.0407

\(\mathcal{T}_{5}\)

3698

0.0179059

0.00370698

0.0182856

0.111726

0.0194876

6.1101

\(\mathcal{T}_{6}\)

7200

0.0128105

0.00221413

0.0130004

0.080006

0.0139675

6.1541

\(\mathcal{T}_{7}\)

14,450

0.00903022

0.00129542

0.00912266

0.0564323

0.00985919

6.1859

In order to present precisely the convergence order of the errors, we plot the errors \(\|\nabla {\mathbf{e}}\|_{0}\), \(\|\epsilon \|_{0}\), \(|\!|\!|( \mathbf{e},\epsilon )|\!|\!|\), the especial estimator \(\eta _{g}\) and the estimator η in Figs. 1 and 2. From the figures, we clearly observe that the errors \(\|\nabla {\mathbf{e}}\|_{0}\), \(\|\epsilon \|_{0}\), and \(|\!|\!|(\mathbf{e},\epsilon )|\!|\!|\) converge to 0 with the order 1. This fact nicely support our theoretic result in Theorem 3.1. The convergence order of the pressure \(\|\epsilon \|_{0}\) is between 1 and 2 in Fig. 2. This phenomenon may be surprising because we have used \(P_{1}\) finite elements to approximate the pressure. Since the estimators \(\eta _{g}\) and η decrease also with the order 1, the analysis in Sect. 4 is also validated.
Figure 1

The convergence order on uniform-type mesh: \(\mathbf{P}_{1}-P_{0}\)

Figure 2

The convergence order on uniform-type mesh: \(\mathbf{P}_{1}-P_{1}\)

5.2 A problem on L-shape domain

We consider (2.1) to (2.3) on an L-shape domain \(\varOmega = [-1,1]^{2}\setminus [0,1]^{2}\). Let the resource function
f = ( 6 r 0.5 cos ( 0.5 θ ) 6 r 0.5 sin ( 0.5 θ ) ) , Open image in new window
where \((r,\theta )\) is a polar representation of a point on the L-shape domain. Apparently f is singular at the corner of domain. The boundary ∂Ω is split into the Dirichlet boundary \(\varGamma _{D}=([-1,1]\times \{-1\})\cup (\{1\}\times [-1,0])\cup ([-1,0] \times \{1\})\cup (\{-1\}\times [-1,1])\) and slip boundary \(\varGamma = \partial \varOmega \setminus \varGamma _{D}\).
Unlike the first example, the exact solution of this problem is not known. In this example, some singularities arise at the L-corner due to the singular source and the slip boundary conditions. We consider two choices for the scale function g. In the first example, g is, respectively, set as Sect. 5.1. In the second example,
$$ g(x,y)= \textstyle\begin{cases} 1,& (x,y)\in \varGamma _{1}, \\ 0.5,&(x,y)\in \varGamma _{2}, \end{cases} $$
where \(\varGamma =\varGamma _{1}\cup \varGamma _{2}\) with \(\varGamma _{1}=[0,1] \times \{0\}\) and \(\varGamma _{2}=\{0\}\times [0,1]\). Obviously, g is discontinuous at point \((0,0)\) for the latter choice.
Observed from Figs. 3 and 4, the convergence results based on the a posteriori error estimates are shown. The adaptive refinement clearly leads to the optimal convergence rate of \(O(h)\) approximated by the lowest finite element pair and equal-order finite element pair for the two different g. We note that the adaptive and uniform approaches can both leads to the optimal order rate of \(O(h)\) if g is continuous at \((0,0)\). However, lots of unknowns are saved by the adaptive refinement to obtain the same accuracy. In the second example, the prominent advantage appears that the adaptive stabilized finite element methods have achieved the same convergence result as theoretical expectation for that lack of convexity of domains, but the convergence rate of the uniform refinement is reduced.
Figure 3

The convergence rate of adaptive and uniform refinements for the problem on L-shape domain with slip boundary condition: g is continuous at \((0,0)\). (a): \(\mathbf{P}_{1}-P_{0}\); (b): \(\mathbf{P} _{1}-P_{1}\)

Figure 4

The convergence rate of adaptive and uniform refinements for the problem on L-shape domain with slip boundary condition: g is discontinuous at \((0,0)\). (c): \(\mathbf{P}_{1}-P_{0}\); (d): \(\mathbf{P}_{1}-P_{1}\)

5.3 Lid driven cavity

Cavity flows have been widely used as test cases for validating the incompressible fluid dynamics algorithm. It is a popular benchmark problem for testing numerical schemes. In the classical test, the upper corners where the moving surface meets the stationary walls are singular points of the flow at which the horizontal velocity is multi-valued. Moreover, the lower corners are also weakly singular points. The fluid is enclosed in a square box with an imposed velocity of unity in the horizontal direction on the top boundary, and a no slip condition on the remaining walls. In our test, we impose a slip boundary condition to the bottom boundary, in fact, we let \(u_{2}=0\), and set \(g=1.0, 10.0, 100.0\) and \(\mu =1.0\). We are thus able to study how meshes adapt to various effects on singularities by changing the slip boundary conditions.

Figures 510 show the adaptive refinements by the \(\mathbf{P}_{1}-P_{0}\) and \(\mathbf{P}_{1}-P_{1}\) stabilized methods, with different g. From Figs. 510, the choice of g has important effect on mesh distribution for the presented model since g is a major factor in the estimator, which is different from the Stokes equations with the Dirichlet boundary condition. Both singularities at upper and bottom definitely have important impact on the model presented here but the degree of impaction may be different. As observed from Fig. 5 with smallest \(g=1.0\), adaptive refinement is only generated in the two upper corners of the cavity. The affection of the slip boundary condition seems weak, only the influence of two singularities arising at the top corners of the square box are obviously found. As seen from Fig. 6 with \(g=10.0\), the influence of two singularities also increases at the two bottom corners, but it is a little weaker than that of the upper two singularities. As for Fig. 7 with \(g=100.0\), the influence of the singularities derived from the slip boundary condition increases and becomes a dominative factor as the given scalar function g increases. The adaptive refinements for the two stabilized methods are similar, which means both stabilized methods with adaptive strategy are effective to solve the incompressible flow with the slip boundary conditions of friction type.
Figure 5

Adaptive refinements by \(\mathbf{P}_{1}-P_{0}\) stabilized method for the cavity flow with slip boundary condition: \(g=1.0\). (a): \(\mathcal{T}_{1}\); (b): \(\mathcal{T}_{2}\); (c): \(\mathcal{T}_{3}\); (d): \(\mathcal{T}_{4}\); (e): \(\mathcal{T}_{5}\); (f): \(\mathcal{T}_{6}\)

Figure 6

Adaptive refinements by \(\mathbf{P}_{1}-P_{0}\) stabilized method for the cavity flow with slip boundary condition: \(g=10.0\). (a): \(\mathcal{T}_{1}\); (b): \(\mathcal{T}_{2}\); (c): \(\mathcal{T}_{3}\); (d): \(\mathcal{T}_{4}\); (e): \(\mathcal{T}_{5}\); (f): \(\mathcal{T}_{6}\)

Figure 7

Adaptive refinements by \(\mathbf{P}_{1}-P_{0}\) stabilized method for the cavity flow with slip boundary condition: \(g=100.0\). (a): \(\mathcal{T}_{1}\); (b): \(\mathcal{T}_{2}\); (c): \(\mathcal{T}_{3}\); (d): \(\mathcal{T}_{4}\); (e): \(\mathcal{T}_{5}\); (f): \(\mathcal{T}_{6}\)

Figure 8

Adaptive refinements by \(\mathbf{P}_{1}-P_{1}\) stabilized method for the cavity flow with slip boundary condition: \(g=1.0\). (a): \(\mathcal{T}_{1}\); (b): \(\mathcal{T}_{2}\); (c): \(\mathcal{T}_{3}\); (d): \(\mathcal{T}_{4}\); (e): \(\mathcal{T}_{5}\); (f): \(\mathcal{T}_{6}\)

Figure 9

Adaptive refinements by \(\mathbf{P}_{1}-P_{1}\) stabilized method for the cavity flow with slip boundary condition: \(g=10.0\). (a): \(\mathcal{T}_{1}\); (b): \(\mathcal{T}_{2}\); (c): \(\mathcal{T}_{3}\); (d): \(\mathcal{T}_{4}\); (e): \(\mathcal{T}_{5}\); (f): \(\mathcal{T}_{6}\)

Figure 10

Adaptive refinements by \(\mathbf{P}_{1}-P_{1}\) stabilized method for the cavity flow with slip boundary condition: \(g=100.0\). (a): \(\mathcal{T}_{1}\); (b): \(\mathcal{T}_{2}\); (c): \(\mathcal{T}_{3}\); (d): \(\mathcal{T}_{4}\); (e): \(\mathcal{T}_{5}\); (f): \(\mathcal{T}_{6}\)

6 Conclusion

In this paper, we consider the lower order stabilized finite element methods for the incompressible flow arising in arteriosclerosis. The velocity in \(H^{1}\)-norm and the pressure in \(L^{2}\)-norm decrease with optimal convergence orde. The reliable and efficient a posteriori error estimates for our finite element solutions are also derived. Moreover, we achieve a series of numerical results for the presented problem [21, 22, 23]. Furthermore, we will then use the efficient methods [18, 34, 35, 37] to solve the similar practical problem for the coupling problem.

Notes

Acknowledgements

The authors would like to thank the editor and reviewers for the constructive comments and useful suggestions, which improved the manuscript.

Availability of data and materials

Not applicable.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Funding

The authors acknowledge financial supported in part by NSF of China (Grant Nos. 11771259, 11571384, and 11971174), special support program to develop innovative talents in the region of Shaanxi province, NSF of Shanghai (Grant No. 19ZR1414300), Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000), the special project high performance computing of National Key Research and Development Program 2016YFB0200604, and by Guangdong Natural Science Foundation through grant 2017B030311001.

Competing interests

The authors declare that they have no competing interests.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics, School of Arts and SciencesShaanxi University of Science and TechnologyXianP.R. China
  2. 2.School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical PracticeEast China Normal UniversityShanghaiP.R. China
  3. 3.School of Data and Computational Science and Guangdong Province Key Laboratory of Computational ScienceSun Yat-sen UniversityGuangzhouChina

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