#  A Posteriori Error Estimates for A Modified Weak Galerkin Finite Element
Method Solving Linear Elasticity Problems ††thanks: Received***
Project supported by the National Natural Science Foundation of
China(No.11901189), the Natural Science Foundation of Hunan Province(No.
2022JJ30271).
${\dagger}$Corresponding author, E-mail<EMAIL_ADDRESS>
Chunmei Liu1, Liuqiang Zhong2, Yingying Xie, 3,†, Liping Zhou 1
1\. College of Science, Hunan University of Science and Engineering, Yongzhou
425199, China;
2\. School of Mathematical Sciences, South China Normal University, Guangzhou
510631, China;
3\. School of Mathematics and Information Science, GuangZhou University,
Guangzhou 510006, China

Abstract In this paper, a residual-type a posteriori error estimator is
proposed and analyzed for a modified weak Galerkin finite element method
solving linear elasticity problems. The estimator is proven to be both
reliable and efficient because it provides upper and lower bounds on the
actual error in a discrete energy norm. Numerical experiments are given to
illustrate the effectiveness of the this error estimator.

Keywords Weak Galerkin Methods; A Posteriori Error Estimates; Linear
Elasticity Problems

Mathematics Subject Classification 65N30, 35J20

##  1 Introduction

In this paper, we consider residual-type posteriori error estimates for a
modified weak Galerkin method about the following linear elasticity problems

$\displaystyle\left\\{\begin{array}[]{rl}-\mu\Delta\boldsymbol{u}-(\lambda+\mu)\nabla(\nabla\cdot\boldsymbol{u})=\boldsymbol{f},&(x,y)\in\Omega,\\\
\boldsymbol{u}=\boldsymbol{g},&(x,y)\in\partial\Omega,\end{array}\right.$
(1.3)

where $\Omega\subset\mathbb{R}^{d}(d=2,3)$ is a polytopal domain with the
boundary $\partial\Omega$, $\boldsymbol{f}$ si an external force,
$\boldsymbol{u}:\Omega\rightarrow\mathbb{R}^{d}$ is a displacement vector, the
Lamé constants $\mu=\frac{E}{2(1+\nu)}$ and
$\lambda=\frac{E\nu}{(1+\nu)(1-2\nu)}$ which can be composed by the elastic
modulus $E$ and Poisson’s ratio $\nu\in[0,0.5)$.

In this paper, we assume the solution $\boldsymbol{u}$ of the model1.3
satisfies the $H^{2}-$ regularity estimate according to [4, 5]

$\displaystyle\|\boldsymbol{u}\|_{2}+\lambda\|\nabla\cdot\boldsymbol{u}\|_{1}\leq
C\|\boldsymbol{f}\|,$

where $C$ is independent of $\lambda$.

The weak Galerkin finite element method(WGFEM) was first prosesed by Wang and
Ye [19] to solving a second order elliptic problem, then this method was
developed for various PDEs, such as elliptic problems [12, 14], parabolic
equations [35, 36], Stokes equations[20, 21, 22], Navier-Stokes equations[13,
16], Biharmonic equation[18, 30], and so on. Moreover, there is also a lot of
work in solving linear elastic problems by using the weak finite element
method[8, 11, 23, 24, 25, 29]. In [8] and [25], the WGFEMs in mixed form have
been developed, although their numerical schemes are different, the solutions
of the stress tensors with strong symmetry are achieved; In [11] and [29], the
linear elasticity problems are considered on the triangles or tetrahedrons and
the tetrahedrons or hexahedrons respectively. But the standary Raviart-Thomas
spaces are used to define the differential of approximate functions, and the
stable numerical schemes are obtained and the“locking-free” property of the
numerical schemes is proved. In particular, the two numerical schemes don’t
use stabilizers. In [23], a numerical scheme with“locking-free” property is
constructed for mesh generation which are shape regularity; In [24], the
hybrid technique is applied to the WGFEM for the linear elasticity problems,
and the optimal error estimates are obtained.

Recently, there have been quite visible research activities on a posteriori
error estimates of the WGFEM and the convergence for the adaptive WGFEM for
second order elliptic problems [7, 31, 32, 1, 15, 27, 28, 26]. A residual-type
a posteriori error estimator is designed firstly based on the triangle or
tetrahedron meshes in [7], and the reliability and efficiency of the estimator
are testified. Then a stabilizer is added to the variational problem of second
order elliptic problems in [31], a residual-type a posteriori error estimator
is constructed and the reliability and efficiency of the estimator are also
testified. A residual-type a posteriori error estimator is also construced in
[32], and the form of this estimator is different from the one in in [7].
After that, a posteriori error estimator with a simple form is presented in
[15] , and applied to general meshes such as hybrid, polytopal and meshes with
hanging nodes. An adaptive algorithm based on WG and modified WG method is
designed for the elliptic problem in [27] and [28], and the convergence of the
adaptive algorithm is proved, respectively. A residual-type a posteriori error
estimator is designed based on the weak Galerkin least-squares finite element
method applied to the reaction-diffusion equation in [1], the reliability and
efficiency of the estimator are also testified. A posteriori error estimator
of edge residual-type Weak Galerkin mixed finite element method solving
second-order elliptic problems in [26], where two different ways of a
posteriori error estimator are presented, both of which hold on polygonal
mesh. The posteriori error estimates of the weak Galerkin method for the
Stokes equation have also been studied such as [34, 2]. However, to our best
knowledge, there exists no work in the literature about the posteriori error
estimates for the linear elasticity problems. Our work is motivated by the
posteriori error estimates about second order elliptic problems and the Stokes
equations, we design the following the posteriori error estimator for the
linear elasticity problems

$\displaystyle\eta^{2}(\boldsymbol{v}_{h},\mathcal{T}_{h})$
$\displaystyle:=\sum\limits_{\tau\in\mathcal{T}_{h}}\left(\eta_{c}^{2}(\boldsymbol{v}_{h},\tau)+\eta_{nc}^{2}(\boldsymbol{v}_{h},\tau)+\mathrm{osc}^{2}(\boldsymbol{f},\tau)+s_{\tau}(\boldsymbol{v}_{h},\boldsymbol{v}_{h})\right),$

where

$\displaystyle\eta_{c}^{2}(\boldsymbol{v}_{h},\tau)$
$\displaystyle=h_{\tau}^{2}(\mu^{-1}+(\mu+\lambda)^{-1})\|\boldsymbol{f}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{v}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{v}_{h})\|^{2}_{\tau},$
$\displaystyle\eta_{nc}^{2}(\boldsymbol{v}_{h},\tau)$
$\displaystyle=\mu^{-1}\sum\limits_{e\in\partial\tau}h_{e}\|J_{e}(\mu\nabla_{w}\boldsymbol{v}_{h}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{v}_{h})\boldsymbol{I})\|^{2}_{e},$
$\displaystyle\mathrm{osc}^{2}(\boldsymbol{f},\tau)$
$\displaystyle=h^{2}_{\tau}(\mu^{-1}+(\mu+\lambda)^{-1})\|\boldsymbol{f}-\boldsymbol{f}_{h}\|^{2}_{\tau},$
$\displaystyle s_{\tau}(\boldsymbol{v}_{h},\boldsymbol{v}_{h})$
$\displaystyle=h_{\tau}^{-1}\langle
Q_{b}\boldsymbol{v}_{0}-\boldsymbol{v}_{b},Q_{b}\boldsymbol{v}_{0}-\boldsymbol{v}_{b}\rangle_{\partial\tau},$

with $h_{\tau}$ being the diameter of the element $\tau$, $h_{e}$ being the
length of edge or face $e$, $\nabla_{w}\boldsymbol{v}_{h}$ and
$\nabla_{w}\cdot\boldsymbol{v}_{h}$ is the weak gradient and the weak
divergence of $\boldsymbol{v}_{h}$, $J_{e}$ represents the jump across the
edge or face $e$, $Q_{b}$ is the $L^{2}$ projection operator to $V_{k-1}(e)$,
$\boldsymbol{f}_{h}$ is the projection of $\boldsymbol{f}$ to the weak
Galerkin finite element space.

In this work, we prove the reliability by the following upper bound

$\displaystyle\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|^{2}_{\mathcal{T}_{h}}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|^{2}_{\mathcal{T}_{h}}\leq
C_{1}^{2}\eta^{2}(\boldsymbol{u}_{h},\mathcal{T}_{h}),$

and the efficiency by the following lower bound

$\displaystyle\eta^{2}\lesssim\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|^{2}_{\tau}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|^{2}_{\tau}+\mathrm{osc}^{2}(\boldsymbol{f},\mathcal{T}_{h}).$

In this paper, in addition to a special constant, we always adopt the mark
$a\lesssim b$, which indicates that there is a constant $C$ such that $a\leq
Cb$.

The rest of this paper is organized as follow. In section 2, we description
the spaces of funcitons to be used , weak gradient operator, weak divergence
operator and present the modified weak finite element scheme. In section 3, we
introduce four modules of adaptive algorithm and the flow of adaptive
algorithm. Section 4 is devoted to the a posteriori error analysis. In section
5, we verify the theoretical results by two numerical examples.

##  2 Prelimimaries and Notations

In order to describe the modified weak Galerkin finite element method, we
recall the definions of weak gradient and weak divergence, the weak Galerkin
finite element spaces, the definions of discrete weak gradient and discrete
weak divergence, and the corresponding modified weak finite element scheme.

For any bounded domain $K\subset\mathbb{R}^{d}(d=2,3)$ with Lipschitz
continuous boundary $\partial K$ , we use the standard definitions for the
Sololev spaces

$\displaystyle H^{m}(K)=\\{v\in L^{2}(K):D^{\alpha}v\in
L^{2}(K),\forall|\alpha|\leq m\\},$

where $D^{\alpha}v=\frac{\partial^{|\alpha|}v}{{\partial
x_{1}}^{\alpha_{1}}\cdots{\partial x_{d}}^{\alpha_{d}}}$, and
$|\alpha|=\alpha_{1}+\cdots+\alpha_{d}$. Let $H^{m}_{0}(K)$ be a subspace of
$H^{m}(K)$ such as

$\displaystyle H^{m}_{0}(K)=\\{v\in H^{m}(K):v=0~{}\mathrm{on}~{}\partial
K\\}.$

We also use the standard definition of norm $\|\cdot\|_{m,K}$ in these Sobolev
spaces $H^{m}(K)$, $[H^{m}(K)]^{d}$ and$[H^{m}(K)]^{d\times d}$. Specifically
when $m=0$, the space $H^{m}(K)=L^{2}(K)$. In addition, we denote
$<\cdot,\cdot>_{\partial K}$ to be the inner productor duality pairing in
$L^{2}(\partial K)$, and
$\mathbb{H}(\mathrm{div};K)=\\{\tau\in[L^{2}(K)]^{d\times
d}:\nabla\cdot\tau\in(L^{2}(K))^{d}\\}$ with the norm
$\|\tau\|_{\mathrm{div};K}=(\|\tau\|^{2}_{K}+\|\nabla\cdot\tau\|^{2}_{K})^{\frac{1}{2}}$.

###  2.1 Weak gradient and divergence operators

In this subsection, we review the definitions of weak gradient and weak
divergence operators which can be applied to descretize the linear elasticity
problems [29]. Let $K$ be any polygonal domain with boundary $\partial K$ and
$e\in\partial K$ be an edge ($d=2$) or a face $d=3$). Denote the space of weak
vector-valued funciton $\mathcal{V}(K)$ as follow

$\displaystyle\mathcal{V}(K)=\left\\{\boldsymbol{v}=\\{\boldsymbol{v}_{0},\boldsymbol{v}_{b}\\}:\boldsymbol{v}_{0}\in[L^{2}(K)]^{d},\boldsymbol{v}_{b}\cdot\boldsymbol{n}\in
H^{-\frac{1}{2}}(\partial K)\right\\},$

where $\boldsymbol{n}$ is the unite outward normal vector on $\partial K$, the
first compont $\boldsymbol{v}_{0}$ and the second component
$\boldsymbol{v}_{b}$ represent the vector $\boldsymbol{v}$ in $K$ and on the
boundary $\partial K$. Note that $\boldsymbol{v}_{b}$ may not necessarily be
related to the trace of $\boldsymbol{v}_{0}$ on $\partial K$, even if the
trace is well defined.

According to [29], we describe the definition of the weak divergence as
follow.

###### Definition 2.1

(Weak Divergence) For any weak vector-valued function
$\boldsymbol{v}\in\mathcal{V}(K)$, the weak divergence
$\nabla_{w,\tau}\cdot\boldsymbol{v}$ is defined as a linear function in the
Sobolev space $H^{1}(K)$

$\displaystyle(\nabla_{w,K}\cdot\boldsymbol{v},\phi)_{K}=-(\boldsymbol{v}_{0},\nabla\phi)_{K}+<\boldsymbol{v}_{b}\cdot\boldsymbol{n},\phi>_{\partial
K},\forall\phi\in H^{1}(K).$

In order to describe the weak gradient operator, we introduce the following
space of weak vector-valued funtions on $K$, such as

$\displaystyle\mathcal{W}(K)=\left\\{\boldsymbol{v}=\\{\boldsymbol{v}_{0},\boldsymbol{v}_{b}\\}:\boldsymbol{v}_{0}\in[L^{2}(K)]^{d},\boldsymbol{v}_{b}\in[H^{\frac{1}{2}}(\partial
K)]^{d}\right\\}.$

According to [29], we define the weak gradient as follow.

###### Definition 2.2

(Weak Gradient) For any weak vector-valued function
$\boldsymbol{v}\in\mathcal{W}(K)$, the weak gradient
$\nabla_{w,K}\boldsymbol{v}$ is defined as a linear function in the Sobolev
space $H[(div;K)]^{d\times d}$

$\displaystyle(\nabla_{w,K}\boldsymbol{v},\psi)_{\tau}=-(\boldsymbol{v}_{0},\nabla\cdot\psi)_{K}+<\boldsymbol{v}_{b},\psi\boldsymbol{n}>_{\partial
K},\forall\psi\in[H(\mathrm{div};K)]^{d},$

$\boldsymbol{n}$ is the unite outward normal vector on $\partial K$.

###  2.2 The modified weak finite element scheme

In this subsection, we introduce the modified weak finite element scheme, so
some notations are descripted fistly. Let $\mathcal{T}_{h}$ be a partition fo
the domain $\Omega$ consisiting of elements which are closed and simply
connected triangles or tetrahedrons, let $\mathcal{E}_{h}$ be the union of all
edges or faces of the mesh elements. For any element $\tau\in\mathcal{T}_{h}$,
$h_{\tau}$ denots the diameter of $T$,
$h=\max_{\tau\in\mathcal{T}_{h}}h_{\tau}$ denotes the mesh size of
$\mathcal{T}_{h}$.

For each element $\tau\in\mathcal{T}_{h}$, let rigid motion(RM) space be

$\displaystyle
RM(\tau)=\\{\boldsymbol{a}+\eta\boldsymbol{x}:\boldsymbol{a}\in\mathbb{R}^{d},\eta\in
so(d)\\},$

where $\boldsymbol{x}$ is the position vector on the element $\tau$, $so(d)$
is $d\times d-$dimensional skew-symmetric matrix space. A finite dimension
space is formed by traces of functions on each boundary $e\in\partial\tau$in
RM space as follows

$\displaystyle
P_{RM}(e)=\\{\boldsymbol{v}\in[L^{2}(e)]^{d}:\boldsymbol{v}=\tilde{\boldsymbol{v}}|_{e},\forall\tilde{\boldsymbol{v}}\in
RM(\tau),e\subset\partial\tau\\}.$

For any integer $k\geq 1$, the local weak finite element space on any element
$\tau$ is

$\displaystyle\mathcal{V}(\tau)=\left\\{\boldsymbol{v}=\\{\boldsymbol{v}_{0},\boldsymbol{v}_{b}\\}:\boldsymbol{v}_{0}\in[P_{k}(\tau)]^{d},\boldsymbol{v}_{b}\in
V_{k-1}(e),\forall e\subset\partial\tau\right\\},$

wher $V_{k-1}(e)=[P_{k-1}(e)]^{d}+P_{RM}(e)$, $P_{k}(\tau)$ is the set of
polynomials ofdegree no greater the $k$ on $\tau$, $P_{k-1}(e)$ is the set of
polynomials of degree no greater than $k-1$ on $e\subset\partial\tau$. Then,
we denote the global weak finite element space $\mathcal{V}_{h}$ and its
subspace $\mathcal{V}_{h}^{0}$ as follows

$\displaystyle\mathcal{V}_{h}$
$\displaystyle=\left\\{\boldsymbol{v}=\\{\boldsymbol{v}_{0},\boldsymbol{v}_{b}\\}:\boldsymbol{v}_{0}|_{\tau}\in[P_{k}(\tau)]^{d},\boldsymbol{v}_{b}|_{e}\in
V_{k-1}(e),\tau\in\mathcal{T}_{h},\forall e\in\mathcal{E}_{h}\right\\},$
$\displaystyle\mathcal{V}_{h}^{0}$
$\displaystyle=\left\\{\boldsymbol{v}=\\{\boldsymbol{v}_{0},\boldsymbol{v}_{b}\\}\in\mathcal{V}_{h}:\boldsymbol{v}_{b}=\boldsymbol{0}~{}\mathrm{on}~{}\partial\Omega\right\\},$

then, according to [5, 33], we denote the local matrix-valued function space
$\Sigma_{h}(\tau)$ and the global matrix-valued function space $\Sigma_{h}$ as
follows

$\displaystyle\Sigma(\tau)$
$\displaystyle=\left\\{\boldsymbol{w}\in[P_{k-1}(\tau)]^{d\times d}\right\\}.$
$\displaystyle\Sigma_{h}$
$\displaystyle=\left\\{\boldsymbol{w}\in[L^{2}(\tau)]^{d\times
d}:\boldsymbol{w}|_{\tau}\in\Sigma(\tau)\right\\}.$

Based on these definitions above, we now introduc the discrete weak gradient
operator, the discrete weak divergence operator and the modified weak finite
element scheme.

###### Definition 2.3

(Discrete Weak Divergence,[29, 33]) For each $\tau\in\mathcal{T}_{h}$, the
discret weak divergenc $\nabla_{w,\tau}\cdot\boldsymbol{v}\in P_{k-1}(\tau)$
of $\boldsymbol{v}\in\mathcal{V}(\tau)$ satisfies the following formula

$\displaystyle(\nabla_{w,\tau}\cdot\boldsymbol{v},\phi)_{\tau}=-(\boldsymbol{v}_{0},\nabla\phi)_{\tau}+<\boldsymbol{v}_{b}\cdot\boldsymbol{n},\phi>_{\partial\tau},\forall\phi\in
P_{k-1}(\tau),$ (2.4)

where $\boldsymbol{n}$ is the unite outward normal vector on $\partial\tau$ .

###### Definition 2.4

(Discrete Weak Gradient,[29, 33]) For each $\tau\in\mathcal{T}_{h}$, the
discret weak gradient $\nabla_{w,\tau}\boldsymbol{v}\in\Sigma(\tau)$ of
$\boldsymbol{v}\in\mathcal{V}(\tau)$, satisfies the following formula

$\displaystyle(\nabla_{w,\tau}\boldsymbol{v},\psi)_{\tau}=-(\boldsymbol{v}_{0},\nabla\cdot\psi)_{\tau}+<\boldsymbol{v}_{b},\psi\boldsymbol{n}>_{\partial\tau},\forall\psi\in\Sigma(\tau),$
(2.5)

where $\boldsymbol{n}$ is the unite outward normal vector on $\partial\tau$ .

Now, we turn to present the modified weak finite element scheme. Using the
finite element space of order $k$, we introduce the following discrete
variational problem of (1.3): Find
$\boldsymbol{u}_{h}=\\{\boldsymbol{u}_{0},\boldsymbol{u}_{b}\\}\in\mathcal{V}_{h}$,
$\boldsymbol{u}_{b}|_{\partial\Omega}=Q_{b}{\boldsymbol{g}}$, such that

$\displaystyle
a_{w}(\boldsymbol{u}_{h},\boldsymbol{v})=(\boldsymbol{f},\boldsymbol{v}_{0}),\forall\boldsymbol{v}=\\{\boldsymbol{v}_{0},\boldsymbol{v}_{b}\\}\in\mathcal{V}_{h}^{0},$
(2.6)

where the bilinear form $a_{w}(\cdot,\cdot)$ is defined by

$\displaystyle
a_{w}(\boldsymbol{w},\boldsymbol{v})=\mu\sum\limits_{\tau\in\mathcal{T}_{h}}(\nabla_{w}\boldsymbol{w},\nabla_{w}\boldsymbol{v})_{\tau}+(\mu+\lambda)\sum\limits_{\tau\in\mathcal{T}_{h}}(\nabla_{w}\cdot\boldsymbol{w},\nabla_{w}\cdot\boldsymbol{v})_{\tau}+s(\boldsymbol{w},\boldsymbol{v}),$
(2.7)

and

$\displaystyle
s(\boldsymbol{w},\boldsymbol{v})=\sum_{\tau\in\mathcal{T}_{h}}h_{\tau}^{-1}\langle
Q_{b}\boldsymbol{w}_{0}-\boldsymbol{w}_{b},Q_{b}\boldsymbol{v}_{0}-\boldsymbol{v}_{b}\rangle_{\partial\tau},$
(2.8)

here, $Q_{b}$ is the local $L^{2}-$ projection onto the space $V_{k-1}(e)$.

According to [29, 33], there exists a unique solution to the modifed weak
Galerkin finite element method defined in (2.7). In our paper, we will not
repeat this conclusion.

In the following section, we give a brief introduction of the adaptive
algorithm based on the modified weak Galerkin finite element method by
refering to the standard AFEM in [6, 17].

##  3 An adaptive modified weak Galerkin algorithm

Let $\mathcal{T}_{0}$ be a triangles grid or tetrahedrons grid on the bounded
domain $\textohm$, and let $\\{\mathcal{T}_{l}\\}_{l>0}$ be a sequence of
nested grids by a series of local refinement. The grid $\mathcal{T}_{l+1}$ is
generated from $\mathcal{T}_{l}$ by the following four algorithm modules :

${\mbox{\bf SOLVE}}\ \rightarrow\ {\mbox{\bf ESTIMATE}}\ \rightarrow\
{\mbox{\bf MARK}}\ \rightarrow\ {\mbox{\bf REFINE}}.$ (3.1)

The specific roles of these four modules are as follows:

(1)SOLVE

For the given functions $\boldsymbol{f}\in(L^{2}(\Omega))^{d}(d=2,3)$ and a
given grid $\mathcal{T}_{l}$, we assume that the algorithm module SOLVE
exactly outputs the discrete solution $\boldsymbol{u}_{l}$ of (2.7) as

$\displaystyle\boldsymbol{u}_{l}={\bf
SOLVE}(\mathcal{T}_{l},\boldsymbol{f},\boldsymbol{g})\in\mathcal{V}_{l}.$

(2) ESTIMATE

For a given grid $\mathcal{T}_{l}$ , let $e\in\mathcal{E}_{l}$ be shared by
two element $\tau_{1}$ and $\tau_{2}$,let $\boldsymbol{n}_{1}$ and
$\boldsymbol{n}_{2}$ be the unite outward normal vector on $e$ belong to
$\tau_{1}$ and $\tau_{2}$ respectively. For any $\boldsymbol{w}\in\Sigma_{h}$,
denote
$[\boldsymbol{w}]_{e}=\boldsymbol{w}|_{\tau_{1}}-\boldsymbol{w}|_{\tau_{2}}$,
$[\boldsymbol{w}\boldsymbol{n}]_{e}=\boldsymbol{w}|_{\tau_{1}}\boldsymbol{n}_{1}+\boldsymbol{w}|_{\tau_{2}}\boldsymbol{n}_{2}$.

We denote the jump across $e$ as follows

$\displaystyle
J_{e}(\mu\nabla_{w}\boldsymbol{v}_{l}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{v}_{l})\boldsymbol{I})$
$\displaystyle=\begin{cases}[(\mu\nabla_{w}\boldsymbol{v}_{l}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{v}_{l})\boldsymbol{I})\boldsymbol{n}]_{e},&\mbox{if}~{}e\in\mathcal{E}_{l}^{0},\\\
0,&\mbox{otherwise},\end{cases}$ $\displaystyle
J_{e}^{0}(\boldsymbol{v}^{l}_{0})$
$\displaystyle=\begin{cases}\boldsymbol{v}^{l}_{0}|_{\partial\tau_{1}}-\boldsymbol{v}^{l}_{0}|_{\partial\tau_{2}},&\mbox{if}~{}e\in\mathcal{E}_{l}^{0},\\\
0,&\mbox{otherwise}.\end{cases}$

For a given grid $\mathcal{T}_{l}$ and a given function
$\boldsymbol{u}_{k}\in\mathcal{V}_{l}$, the posteriori error estimator based
on $\tau$ is given by

$\displaystyle\eta^{2}(\boldsymbol{u}_{l},\mathcal{T}_{l})$
$\displaystyle:=\sum\limits_{\tau\in\mathcal{T}_{l}}\left(\eta_{c}^{2}(\boldsymbol{u}_{h},\tau)+\eta_{nc}^{2}(\boldsymbol{u}_{h},\tau)+\mathrm{osc}^{2}(\boldsymbol{f},\tau)+s_{\tau}(\boldsymbol{u}_{h},\boldsymbol{u}_{h})\right),$
(3.2)

where

$\displaystyle\eta_{c}^{2}(\boldsymbol{u}_{l},\tau)$
$\displaystyle=h_{\tau}^{2}(\mu^{-1}+(\mu+\lambda)^{-1})\|\boldsymbol{f}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{l})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{l})\|^{2}_{\tau},$
$\displaystyle\eta_{nc}^{2}(\boldsymbol{u}_{h},\tau)$
$\displaystyle=\mu^{-1}\sum\limits_{e\in\partial\tau}h_{e}\|J_{e}(\mu\nabla_{w}\boldsymbol{u}_{l}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{u}_{l})\boldsymbol{I})\|^{2}_{e},$
$\displaystyle\mathrm{osc}^{2}(\boldsymbol{f},\tau)$
$\displaystyle=h^{2}_{\tau}(\mu^{-1}+(\mu+\lambda)^{-1})\|\boldsymbol{f}-\boldsymbol{f}_{l}\|^{2}_{\tau},$
$\displaystyle s_{\tau}(\boldsymbol{u}_{l},\boldsymbol{u}_{l})$
$\displaystyle=h_{\tau}^{-1}\langle
Q_{b}\boldsymbol{u}_{l,0}-\boldsymbol{u}_{l,b},Q_{b}\boldsymbol{u}_{l,0}-\boldsymbol{u}_{l,b}\rangle_{\partial\tau},$

with $h_{\tau}$ being the diameter of the element $\tau$, $h_{e}$ being the
length of edge or face $e$, $\nabla_{w}\boldsymbol{u}_{l}$ and
$\nabla_{w}\cdot\boldsymbol{u}_{l}$ is the weak gradient and the weak
divergence of $\boldsymbol{u}_{l}$, $J_{e}$ represents the jump across the
edge or face $e$, $Q_{b}$ is the $L^{2}$ projection operator to $V_{l-1}(e)$,
$\boldsymbol{f}_{l}$ is the projection of $\boldsymbol{f}$ to the weak
Galerkin finite element space.

For any $\mathcal{W}_{l}\subset\mathcal{T}_{l}$ and
$\boldsymbol{u}_{k}\in\mathcal{V}_{l}$, define the following sets by

$\displaystyle\eta^{2}(\boldsymbol{u}_{l},\mathcal{W}_{l})=\sum\limits_{\tau\in\mathcal{W}_{l}}\eta_{c}^{2}(\boldsymbol{u}_{l},\tau),\
\mathrm{osc}^{2}(\boldsymbol{f},\mathcal{W}_{l})=\sum\limits_{\tau\in\mathcal{W}_{l}}\mathrm{osc}^{2}(\boldsymbol{f},\tau).$

For any given grid $\mathcal{T}_{l}$ and the corresponding discrete exact
$\boldsymbol{u}_{l}\in\mathcal{V}_{l}$ of (2.7), we can obtain the posteriori
error estimator $\eta^{2}_{\mathcal{T}_{l}}(\boldsymbol{u}_{l},\tau)$ of any
element $\tau\in\mathcal{T}_{l}$ by the following algorithm module

$\displaystyle\eta^{2}_{\mathcal{T}_{l}}(\boldsymbol{u}_{l},\tau)=\mathrm{{\bf
ESTIMATE}}(\mathcal{T}_{l},\boldsymbol{u}_{l},\boldsymbol{f},\boldsymbol{g}).$

(3) MARK

In this paper, we utilize the Dörfler marking way([10]) to mark elements which
will be refined. Given a grid $\mathcal{T}_{l}$, a set of posteriori error
estimators
$\\{\eta^{2}_{\mathcal{T}_{l}}(\boldsymbol{u}_{l},\tau)\\}_{\tau\in\mathcal{T}_{l}}$
and a Dörfler marking parameter $\vartheta\in(0,1)$, we can get a marked
element set $\mathcal{M}_{l}\subset\mathcal{T}_{l}$ by the following algorithm
module

$\displaystyle\mathcal{M}_{l}=\mathrm{{\bf
MARK}}(\eta^{2}(\boldsymbol{u}_{l},\mathcal{T}_{l}),\mathcal{T}_{l},\vartheta),$

in addition, the set $\mathcal{M}_{l}$ satisfies

$\displaystyle\eta^{2}_{\mathcal{T}_{l}}(\boldsymbol{u}_{l},\mathcal{M}_{l})\geq\vartheta\eta^{2}(\boldsymbol{u}_{l},\mathcal{T}_{l})$

and has a minimal cardinality.

(4) REFINE

We assume that a module REFINE implements an iterative or a recursive
bisection (see[27]). For a given number $l>1$, any grid
$\mathcal{T}_{k}\in\mathcal{L}(\mathcal{T}_{0})$ and a subset
$\mathcal{M}_{l}\subset\mathcal{T}_{l}$, we can obtain a conforming grid
$\mathcal{T}_{l+1}\in\mathcal{L}(\mathcal{T}_{0})$ by the algorithm module
REFINE as

$\displaystyle\mathcal{T}_{l+1}=\mathrm{{\bf
REFINE}}(\mathcal{T}_{l},\mathcal{M}_{l}).$

Using the above four algorithm modules, we design an adaptive modified weak
Galerkin finite element method(AMWG-FEM) as follow.

######  Algorithm 3.1 (AMG-FEM)

For given functions $\boldsymbol{f}$, $\boldsymbol{g}$, choosing a Dörfler
marking parameter $\vartheta\in(0,1)$ and a error control constant $tol$, the
modules of AMWG-FEM algorithm is

  1. 1.

Give an initial conforming grid $\mathcal{T}_{0}$ and set $l=0$.

  2. 2.

$\boldsymbol{u}_{l}=\mathrm{{\bf
SOLVE}}(\mathcal{T}_{l},\boldsymbol{f},\boldsymbol{g})$.

  3. 3.

$\eta^{2}_{\mathcal{T}_{l}}(\boldsymbol{u}_{l},\tau)=\mathrm{{\bf
ESTIMATE}}(\mathcal{T}_{l},\boldsymbol{u}_{l},\boldsymbol{f},\boldsymbol{g})$.
If $\eta^{2}(\boldsymbol{u}_{l},{\mathcal{T}_{l}})<tol$, then the algorithm
stops.

  4. 4.

$\mathcal{M}_{l}=\mathrm{{\bf
MARK}}(\eta^{2}(\boldsymbol{u}_{l},\mathcal{T}_{l}),\mathcal{T}_{l},\vartheta)$.

  5. 5.

$\mathcal{T}_{l+1}=\mathrm{{\bf REFINE}}(\mathcal{T}_{l},\mathcal{M}_{l})$.

  6. 6.

Set $l=l+1$ and go to 2.

##  4 A posteriori error analysis for the MWG method

This section is devoted to a study of reliability and efficiency for the error
estimator $\eta(\boldsymbol{u}_{h},\mathcal{T}_{k})$ defined in (3.2).
Firstly, we give the followinn three lemmas.

###### Lemma 4.1

For any
$\boldsymbol{v}=\\{\boldsymbol{v}_{0},\boldsymbol{v}_{b}\\}\in\mathcal{V}_{h}$,
we have

$\displaystyle\|\nabla_{w}\boldsymbol{v}-\nabla\boldsymbol{v}_{0}\|_{\mathcal{T}_{h}}^{2}\lesssim
s(\boldsymbol{v},\boldsymbol{v}).$ (4.1)

Proof:  By the definition 2.2 and Green formula, we will get the relationship
between the weak gradient and classical gradient as follows

$\displaystyle(\nabla_{w,\tau}\boldsymbol{v},\boldsymbol{\varphi})_{\tau}=(\nabla\boldsymbol{v}_{0},\boldsymbol{\varphi})_{\tau}-\langle\boldsymbol{v}_{0}-\boldsymbol{v}_{b},\boldsymbol{\varphi}\boldsymbol{n}\rangle_{\partial\tau},\forall\boldsymbol{\varphi}\in[H^{1}(\mathrm{div};\tau)]^{d}.$

Let $\boldsymbol{\varphi}=\nabla_{w}\boldsymbol{v}-\nabla\boldsymbol{v}_{0}$,
by using trace inequality, we have

$\displaystyle\|\nabla_{w}\boldsymbol{v}-\nabla\boldsymbol{v}_{0}\|_{\mathcal{T}_{h}}^{2}$
$\displaystyle=\sum_{\tau\in\mathcal{T}_{h}}\langle\boldsymbol{v}_{0}-\boldsymbol{v}_{b},(\nabla_{w,\tau}\boldsymbol{v}-\nabla\boldsymbol{v}_{0})\boldsymbol{n}\rangle_{\partial\tau}$
$\displaystyle\leqslant(s(\boldsymbol{v},\boldsymbol{v}))^{1/2}\cdot\sum_{\tau\in\mathcal{T}_{h}}h_{\tau}^{1/2}\|\nabla_{w,\tau}\boldsymbol{v}_{h}-\nabla\boldsymbol{v}_{0}\|_{\partial\tau}$
$\displaystyle\lesssim(s(\boldsymbol{v},\boldsymbol{v}))^{1/2}\cdot\|\nabla_{w}\boldsymbol{v}-\nabla\boldsymbol{v}_{0}\|_{\mathcal{T}_{h}},$

Dividing
$\|\nabla_{w}\boldsymbol{v}-\nabla\boldsymbol{v}_{0}\|_{\mathcal{T}_{h}}$ on
both sides of the above equation, then we obtain (4.1).

###### Lemma 4.2

For any
$\boldsymbol{v}=\\{\boldsymbol{v}_{0},\boldsymbol{v}_{b}\\}\in\mathcal{V}_{h}$,
we have

$\displaystyle\sum_{e\in\mathcal{E}_{h}}h_{e}^{-1}\|[\boldsymbol{v}_{0}]\|_{e}^{2}\lesssim
s(\boldsymbol{v},\boldsymbol{v}).$ (4.2)

Proof:  Noting that $\|[\boldsymbol{v}_{b}]\|_{e}=0,\forall
e\in\mathcal{E}_{h}$, we obtain

$\displaystyle\sum_{e\in\mathcal{E}_{h}}h_{e}^{-1}\|[\boldsymbol{v}_{0}]\|_{e}^{2}=\sum_{e\in\mathcal{E}_{h}}h_{e}^{-1}\|[\boldsymbol{v}_{0}-\boldsymbol{v}_{b}]\|_{e}^{2},$
(4.3)

For any edge or face $e\in\mathcal{E}_{h}^{0}$, there exists
$\tau_{1}\in\mathcal{T}_{h}$ and $\tau_{2}\in\mathcal{T}_{h}$, such that
$e=\partial\tau_{1}\cap\partial\tau_{2}$. By using of Cauchy-Schwarz
inequality, we get

$\displaystyle\|[\boldsymbol{v}_{0}-\boldsymbol{v}_{b}]\|_{e}^{2}$
$\displaystyle=\langle[\boldsymbol{v}_{0}-\boldsymbol{v}_{b}],\boldsymbol{v}_{0}-\boldsymbol{v}_{b}\rangle_{\partial\tau_{1}\cap
e}+\langle[\boldsymbol{v}_{0}-\boldsymbol{v}_{b}],\boldsymbol{v}_{0}-\boldsymbol{v}_{b}\rangle_{\partial\tau_{2}\cap
e}$
$\displaystyle\leqslant\|[\boldsymbol{v}_{0}-\boldsymbol{v}_{b}]\|_{e}(\|\boldsymbol{v}_{0}-\boldsymbol{v}_{b}\|_{\partial\tau_{1}\cap
e}+\|\boldsymbol{v}_{0}-\boldsymbol{v}_{b}\|_{\partial\tau_{2}\cap e});$

Similarly, for any boundary edge or face $e\in\mathcal{E}_{h}^{\partial}$, a
similar conclusion can be proved . We now sum over $e\in\mathcal{E}_{h}$ and
the following estimate is true

$\displaystyle\sum_{e\in\mathcal{E}_{h}}h_{e}^{-1}\|[\boldsymbol{v}_{0}-\boldsymbol{v}_{b}]\|_{e}^{2}\lesssim\sum_{e\in\mathcal{E}_{h}}h_{e}^{-1/2}\|[\boldsymbol{v}_{0}-\boldsymbol{v}_{b}]\|_{e}\cdot(s(\boldsymbol{v},\boldsymbol{v}))^{1/2},$

that is

$\displaystyle\sum_{e\in\mathcal{E}_{h}}h_{e}^{-1}\|[\boldsymbol{v}_{0}-\boldsymbol{v}_{b}]\|_{e}^{2}\lesssim
s(\boldsymbol{v},\boldsymbol{v}).$ (4.4)

Combine (4.3) and (4.4), we obtain the conclusion (4.2).

Let
$\tilde{\mathcal{V}}_{h}=\\{\boldsymbol{v},\boldsymbol{v}\in[P_{0}(\tau)]^{d},\forall\tau\in\mathcal{T}_{h}\\}$,
$\mathcal{V}^{c}_{h}=(H_{0}^{1}(\Omega))^{d}\cap(P_{1}(\tau))^{2}$, we
introduce the following estimats by refer to [3].

###### Lemma 4.3

For any $\tau\in\mathcal{T}_{h}$, there exists an interpolation operator
$I_{\mathcal{T}_{h}}^{c}:\tilde{\mathcal{V}}_{h}\rightarrow\mathcal{V}^{c}_{h}$,
such that

$\|v_{\mathcal{T}_{h}}-I_{\mathcal{T}}^{c}v_{\mathcal{T}_{h}}\|_{\mathcal{T}_{h}}\lesssim
h\|\nabla v_{\mathcal{T}_{h}}\|_{L^{2}(\Omega)},\forall v_{\mathcal{T}_{h}}\in
H_{0}^{1}(\Omega),$ (4.5)

where the constant is only dependent on the shape regular of mesh
$\mathcal{T}_{h}$. For any $|a|=0,1$, we have

$\|D^{a}(v_{\mathcal{T}}-I_{\mathcal{T}}^{c}v_{\mathcal{T}_{h}})\|_{\mathcal{T}}^{2}\lesssim\sum_{e\in\mathcal{E}_{\mathcal{T}}}h_{\tau}^{1-2|a|}\|[v_{\mathcal{T}}]_{e}\|^{2}_{e},\forall
v_{\mathcal{T}}\in\tilde{\mathcal{V}}_{h}.$ (4.6)

where the constant is only dependent on the shape regular of mesh
$\mathcal{T}_{h}$.

Now, we shall present the reliability for the error estimator defined in (3.2)
by the following upper bound estimate.

###### Theorem 4.4

Let $\boldsymbol{u}$ be the solution of (1.3) and
$\boldsymbol{u}_{h}=\\{\boldsymbol{u}_{0}^{h},\boldsymbol{u}_{b}^{h}\\}\in\mathcal{V}_{h}$
be the solution of (2.6), respectively. There exists a constant $C_{1}>0$,
such that such

$\displaystyle\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|^{2}_{\mathcal{T}_{h}}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|^{2}_{\mathcal{T}_{h}}\leq
C_{1}^{2}\eta^{2}(\boldsymbol{u}_{h},\mathcal{T}_{h}),$ (4.7)

when the constant $C_{1}>0$, only depends on the shape regularity of
$\mathcal{T}_{h}$.

Proof:  Let
$\boldsymbol{e}_{1}=\mu(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})$,
$e_{2}=(\mu+\lambda)(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})$.
By Lemma 4.3, we have
$\boldsymbol{u}_{h}^{c}=I_{\mathcal{T}_{h}}^{c}\boldsymbol{u}_{0}^{h}\in\mathcal{V}^{c}(\mathcal{T}_{h})$
and

$\displaystyle\nabla_{w}\boldsymbol{u}_{h}^{c}=\nabla\boldsymbol{u}_{h}^{c},$
(4.8)
$\displaystyle\nabla_{w}\cdot\boldsymbol{u}_{h}^{c}=\nabla\cdot\boldsymbol{u}_{h}^{c}.$
(4.9)

By using the above notations, the following estimate is ture

$\displaystyle E_{h}^{2}$
$\displaystyle=\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|^{2}_{\tau}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|^{2}_{\tau}$
$\displaystyle=(\boldsymbol{e}_{1},\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})_{\tau}+(e_{2},\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})_{\tau}$
$\displaystyle=(\boldsymbol{e}_{1},\nabla\boldsymbol{u}-\nabla\boldsymbol{u}_{h}^{c})_{\tau}+(\boldsymbol{e}_{1},\nabla\boldsymbol{u}_{h}^{c}-\nabla_{w}\boldsymbol{u}_{h})_{\tau}+(e_{2},\nabla\cdot\boldsymbol{u}-\nabla\cdot\boldsymbol{u}_{h}^{c})_{\tau}+(e_{2},\nabla\cdot\boldsymbol{u}_{h}^{c}-\nabla_{w}\cdot\boldsymbol{u}_{h})_{\tau}$
$\displaystyle=I_{1}+I_{2},$ (4.10)

where
$I_{1}=(\boldsymbol{e}_{1},\nabla\boldsymbol{u}-\nabla\boldsymbol{u}_{h}^{c})_{\tau}+(e_{2},\nabla\cdot\boldsymbol{u}-\nabla\cdot\boldsymbol{u}_{h}^{c})_{\tau}$,
$I_{2}=(\boldsymbol{e}_{1},\nabla\boldsymbol{u}_{h}^{c}-\nabla_{w}\boldsymbol{u}_{h})_{\tau}+(e_{2},\nabla\cdot\boldsymbol{u}_{h}^{c}-\nabla_{w}\cdot\boldsymbol{u}_{h})_{\tau}$.

Firstly, we shall estimate $I_{1}$. Let
$\boldsymbol{w}=\boldsymbol{u}-\boldsymbol{u}_{h}^{c}\in(H_{0}^{1}(\Omega))^{2}$,
by refer to[7], we know that there exists an interpolation operator
$\boldsymbol{w}_{h}$ which satisfies

$\displaystyle(\boldsymbol{e}_{1},\nabla\boldsymbol{w}_{h})_{\tau}+(e_{2},\nabla\cdot\boldsymbol{w}_{h})_{\tau}=\boldsymbol{0}.$
(4.11)

Using (4.11), Green formula, the continuity of $\nabla\boldsymbol{u}$ and
$\nabla\cdot\boldsymbol{u}$ on the edge or face of the unit $\tau$, the
estimate in [7], we obtain

$\displaystyle I_{1}$
$\displaystyle=(\boldsymbol{e}_{1},\nabla\boldsymbol{u}-\nabla\boldsymbol{u}_{h}^{c})_{\tau}+(e_{2},\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h}^{c})_{\tau}$
$\displaystyle=(\boldsymbol{e}_{1},\nabla\boldsymbol{w})_{\tau}+(e_{2},\nabla\cdot\boldsymbol{w})_{\tau}$
$\displaystyle=(\boldsymbol{e}_{1},\nabla(\boldsymbol{w}-\boldsymbol{w}_{h}))_{\tau}+(e_{2},\nabla\cdot(\boldsymbol{w}-\boldsymbol{w}_{h}))_{\tau}$
$\displaystyle=-(\nabla\cdot\boldsymbol{e}_{1},\boldsymbol{w}-\boldsymbol{w}_{h})_{\tau}+(\boldsymbol{e}_{1}\boldsymbol{n},\boldsymbol{w}-\boldsymbol{w}_{h})_{\partial\tau}-(\nabla
e_{2},\boldsymbol{w}-\boldsymbol{w}_{h})_{\tau}+(e_{2},(\boldsymbol{w}-\boldsymbol{w}_{h})\boldsymbol{n})_{\partial\tau}$
$\displaystyle=-(\nabla\cdot\boldsymbol{e}_{1}+\nabla
e_{2},\boldsymbol{w}-\boldsymbol{w}_{h})_{\tau}+(\boldsymbol{e}_{1}\boldsymbol{n},\boldsymbol{w}-\boldsymbol{w}_{h})_{\partial\tau}+(e_{2},(\boldsymbol{w}-\boldsymbol{w}_{h})\cdot\boldsymbol{n})_{\partial\tau}$
$\displaystyle=(\boldsymbol{f}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}),\boldsymbol{w}-\boldsymbol{w}_{h})_{\tau}+(\mu\nabla_{w}\boldsymbol{u}_{h}\boldsymbol{n},\boldsymbol{w}-\boldsymbol{w}_{h})_{\partial\tau}$
$\displaystyle~{}~{}~{}+(((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h})\boldsymbol{n},\boldsymbol{w}-\boldsymbol{w}_{h})_{\partial\tau}$
$\displaystyle=(\boldsymbol{f}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h}+(\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}\boldsymbol{I}),\boldsymbol{w}-\boldsymbol{w}_{h})_{\tau}$
$\displaystyle~{}~{}~{}+((\mu\nabla_{w}\boldsymbol{u}_{h}+(\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}\boldsymbol{I})\boldsymbol{n},\boldsymbol{w}-\boldsymbol{w}_{h})_{\partial\tau}$
$\displaystyle\lesssim\|\boldsymbol{f}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h}+(\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}\boldsymbol{I})\|_{\tau}\|\boldsymbol{w}-\boldsymbol{w}_{h}\|_{\tau}$
$\displaystyle~{}~{}~{}+\|[(\mu\nabla_{w}\boldsymbol{u}_{h}+(\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}\boldsymbol{I})\boldsymbol{n}]\|_{\partial\tau}\|\boldsymbol{w}-\boldsymbol{w}_{h}\|_{\partial\tau}$
$\displaystyle\lesssim\|\boldsymbol{f}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h}+(\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}\boldsymbol{I})\|_{\tau}h_{\tau}\|\nabla_{w}\boldsymbol{w}\|_{\tau}$
$\displaystyle~{}~{}~{}+(\|[(\mu\nabla_{w}\boldsymbol{u}_{h}+(\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}\boldsymbol{I})\boldsymbol{n}]\|_{\partial\tau})h_{e}^{\frac{1}{2}}\|\nabla_{w}\boldsymbol{w}\|_{\tau}$
$\displaystyle\lesssim\eta(\boldsymbol{u}_{h},\mathcal{T}_{h})\|\nabla\boldsymbol{u}-\nabla\boldsymbol{u}_{h}^{c}\|_{\tau}.$
(4.12)

By using the Lemma 4.1 and Lemma4.2, we shall estimate
$\|\nabla\boldsymbol{u}-\nabla\boldsymbol{u}_{h}^{c}\|_{\tau}$.

$\displaystyle\|\nabla\boldsymbol{u}-\nabla\boldsymbol{u}_{h}^{c}\|_{\tau}$
$\displaystyle\leq\|\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h}\|_{\tau}+\|\nabla_{w}\boldsymbol{u}_{h}-\nabla_{w}\boldsymbol{u}_{0}^{h}\|_{\tau}+\|\nabla_{w}\boldsymbol{u}_{0}^{h}-\nabla\boldsymbol{u}_{h}^{c}\|_{\tau}$
$\displaystyle\leq\|\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h}\|_{\tau}+\|\nabla_{w}\boldsymbol{u}_{h}-\nabla\boldsymbol{u}_{0}^{h}\|_{\tau}+\|\nabla\boldsymbol{u}_{0}^{h}-\nabla\boldsymbol{u}_{h}^{c}\|_{\tau}$
$\displaystyle\lesssim\|\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h}\|_{\tau}+s(\boldsymbol{u}_{h},\boldsymbol{u}_{h})+h_{e}^{-\frac{1}{2}}\|[\boldsymbol{u}_{0}^{h}]\|_{e}$
$\displaystyle\lesssim E_{h}+s(\boldsymbol{u}_{h},\boldsymbol{u}_{h})\lesssim
E_{h}+\eta^{2}(\boldsymbol{u}_{h},\mathcal{T}_{h}),$ (4.13)

Combine (4.12 ) and (4.13 ), we obtain

$\displaystyle
I_{1}\lesssim\eta(\boldsymbol{u}_{h},\mathcal{T}_{h})E_{h}+\eta^{2}(\boldsymbol{u}_{h},\mathcal{T}_{h}).$
(4.14)

Secondly, we shall estimate $I_{2}$. By using the Lemma 4.1 and Lemma 4.2, the
relationship between the weak gradient and classical gradient, the norm of the
gradient is less than the norm of divergence, the relationship between the
weak divergence and classical divergence, we get

$\displaystyle I_{2}$
$\displaystyle=(\boldsymbol{e}_{1},\nabla\boldsymbol{u}_{h}^{c}-\nabla_{w}\boldsymbol{u}_{h})_{\tau}+(e_{2},\nabla\cdot\boldsymbol{u}_{h}^{c}-\nabla_{w}\cdot\boldsymbol{u}_{h})_{\tau}$
$\displaystyle=(\boldsymbol{e}_{1},\nabla\boldsymbol{u}_{h}^{c}-\nabla\boldsymbol{u}_{0}^{h})_{\tau}+(\boldsymbol{e}_{1},\nabla\boldsymbol{u}_{0}^{h}-\nabla_{w}\boldsymbol{u}_{h})_{\tau}$
$\displaystyle+(e_{2},\nabla\cdot\boldsymbol{u}_{h}^{c}-\nabla\cdot\boldsymbol{u}_{0}^{h})_{\tau}+(e_{2},\nabla\cdot\boldsymbol{u}_{0}^{h}-\nabla_{w}\cdot\boldsymbol{u}_{h})_{\tau}$
$\displaystyle\leq\|\boldsymbol{e}_{1}\|_{\tau}s(\boldsymbol{u}_{h},\boldsymbol{u}_{h})+\|\boldsymbol{e}_{1}\|_{\tau}s(\boldsymbol{u}_{h},\boldsymbol{u}_{h})+\|e_{2}\|_{\tau}\|\nabla\boldsymbol{u}_{h}^{c}-\nabla_{w}\boldsymbol{u}_{0}^{h}\|_{\tau}+\|e_{2}\|_{\tau}s(\boldsymbol{u}_{h},\boldsymbol{u}_{h})$
$\displaystyle\lesssim E_{h}s(\boldsymbol{u}_{h},\boldsymbol{u}_{h})\leq
E_{h}\eta(\boldsymbol{u}_{h},\mathcal{T}_{h}).$ (4.15)

Combine (4.14) and (4.15), we have

$\displaystyle E_{h}^{2}=I_{1}+I_{2}\lesssim
E_{h}\eta(\boldsymbol{u}_{h},\mathcal{T}_{h})+\eta^{2}(\boldsymbol{u}_{h},\mathcal{T}_{h})+E_{h}\eta(\boldsymbol{u}_{h},\mathcal{T}_{h}).$

Using the inequality $2ab\leq\varepsilon a^{2}+\frac{1}{\varepsilon}b^{2}$,
mergeing items with the same form, we have completed the proof.

Next, we shall use the standard bubble function technique to prove the
efficiency estimate(see [7]). Let $\omega_{e}=\tau_{1}\cup\tau_{2}$, where
$\tau_{1}$ and $\tau_{2}$ share the edge or face $e$. We present the following
lemma.

###### Lemma 4.5

There exists a constant $C>0$, such that

$\displaystyle
h_{\tau}\|\boldsymbol{f}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{v}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{v}_{h})\|_{\tau}$
$\displaystyle\leq
C(\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|_{\tau}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\tau}+h_{\tau}\|\boldsymbol{f}-\boldsymbol{f}_{h}\|_{\tau}).$
(4.16) $\displaystyle
h_{e}^{1/2}\|[(\mu\nabla_{w}\boldsymbol{v}_{h}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{v}_{h})\boldsymbol{I})\boldsymbol{n}]\|_{e}$
$\displaystyle\leq
C\left(\mathrm{osc}(\boldsymbol{f},\omega_{e})+\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|_{\omega_{e}}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\omega_{e}}\right).$
(4.17)

Proof:  Let
$\boldsymbol{w}_{\tau}=(\boldsymbol{f}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{v}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{v}_{h}))\phi_{\tau}(\boldsymbol{x})$,
where $\phi_{\tau}(\boldsymbol{x})=27\lambda_{1}\lambda_{2}\lambda_{3}$ is a
bubble function defined on $\tau$, we have

$\displaystyle(\boldsymbol{f},\boldsymbol{w}_{\tau})_{\tau}=(\mu\nabla\boldsymbol{u},\nabla\boldsymbol{w}_{\tau})_{\tau}+((\mu+\lambda)\nabla\cdot\boldsymbol{u},\nabla\cdot\boldsymbol{w}_{\tau})_{\tau}.$

Subtracting and adding $(\boldsymbol{f}_{h},\boldsymbol{w}_{\tau})_{\tau}$ ,
$(\mu\nabla_{w}\boldsymbol{u}_{h},\boldsymbol{w}_{\tau})_{\tau}$ and
$((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h},\nabla\cdot\boldsymbol{w}_{\tau})_{\tau}$
from both sides of the above equation, we get

$\displaystyle(\boldsymbol{f}-\boldsymbol{f}_{h},\boldsymbol{w}_{\tau})_{\tau}+(\boldsymbol{f}_{h},\boldsymbol{w}_{\tau})_{\tau}-(\mu\nabla_{w}\boldsymbol{u}_{h},\nabla\boldsymbol{w}_{\tau})_{\tau}-((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h},\nabla\cdot\boldsymbol{w}_{\tau})_{\tau}$
$\displaystyle=(\mu(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h}),\nabla\boldsymbol{w}_{\tau})_{\tau}+((\mu+\lambda)(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h}),\nabla\cdot\boldsymbol{w}_{\tau})_{\tau}.$

Using the integration by parts, inverse inequality and
$\boldsymbol{w}_{\tau}|_{\partial\tau}=0$, the above equation becomes

$\displaystyle(\boldsymbol{f}_{h}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}),\boldsymbol{w}_{\tau})_{\tau}$
$\displaystyle=(\mu(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h}),\nabla\boldsymbol{w}_{\tau})_{\tau}+((\mu+\lambda)(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h}),\nabla\cdot\boldsymbol{w}_{\tau})_{\tau}-(\boldsymbol{f}-\boldsymbol{f}_{h},\boldsymbol{w}_{\tau})_{\tau}.$

Using the properties of the bubble function $\phi_{\tau}(\boldsymbol{x})$, we
obtain

$\displaystyle\|\boldsymbol{f}_{h}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h})\|^{2}_{\tau}$
$\displaystyle\leq\mu^{1/2}\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|_{\tau}\|\nabla\boldsymbol{w}_{\tau}\|_{\tau}+(\mu+\lambda)^{1/2}\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})_{\tau}\|\nabla\cdot\boldsymbol{w}_{\tau}\|_{\tau}+\|\boldsymbol{f}-\boldsymbol{f}_{h}\|_{\tau}\|\boldsymbol{w}_{\tau}\|_{\tau}$
$\displaystyle\leq
C(\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|_{\tau}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\tau}h_{\tau}^{-1}\|\boldsymbol{w}_{\tau}\|_{\tau}+\|\boldsymbol{f}-\boldsymbol{f}_{h}\|_{\tau}\|\boldsymbol{w}_{\tau}\|_{\tau}$
$\displaystyle\leq
C(\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|_{\tau}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\tau}+h_{\tau}\|\boldsymbol{f}-\boldsymbol{f}_{h}\|_{\tau})h_{\tau}^{-1}\|\boldsymbol{w}_{\tau}\|_{\tau}.$

Notice that
$\|\boldsymbol{w}_{\tau}\|_{\tau}=\|\boldsymbol{f}_{h}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\tau}$,
we have

$\displaystyle
h_{\tau}\|\boldsymbol{f}_{h}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\tau}$
$\displaystyle\leq
C(\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|_{\tau}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\tau}+h_{\tau}\|\boldsymbol{f}-\boldsymbol{f}_{h}\|_{\tau}).$

Then we have completed the proof of (4.16) .

Let
$\boldsymbol{v}_{e}=[(\mu\nabla_{w}\boldsymbol{v}_{h}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{v}_{h})\boldsymbol{I})\boldsymbol{n}]_{e}\phi_{e}(\boldsymbol{x})$,
where $\phi_{e}(\boldsymbol{x})$ is the bubble function defined on the edge or
face $e$, we arrive at

$\displaystyle\sum\limits_{\tau\in\omega_{e}}(\boldsymbol{f},\boldsymbol{v}_{e})_{\tau}=\sum\limits_{\tau\in\omega_{e}}(\mu\nabla\boldsymbol{u},\nabla\boldsymbol{v}_{e})_{\tau}+\sum\limits_{\tau\in\omega_{e}}((\mu+\lambda)\nabla\cdot\boldsymbol{u},\nabla\cdot\boldsymbol{v}_{e})_{\tau}.$

Subtracting
$\sum\limits_{\tau\in\omega_{e}}(\mu\nabla_{w}\boldsymbol{u}_{h},\nabla\boldsymbol{v}_{e})_{\tau}$
and
$\sum\limits_{\tau\in\omega_{e}}((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h},\nabla\cdot\boldsymbol{v}_{e})_{\tau}$
from both sides of the above equation , then we get

$\displaystyle\sum\limits_{\tau\in\omega_{e}}(\boldsymbol{f},\boldsymbol{v}_{e})_{\tau}-\sum\limits_{\tau\in\omega_{e}}(\mu\nabla_{w}\boldsymbol{u}_{h},\nabla\boldsymbol{v}_{e})_{\tau}-\sum\limits_{\tau\in\omega_{e}}((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h},\nabla\cdot\boldsymbol{v}_{e})_{\tau}$
$\displaystyle=\sum\limits_{\tau\in\omega_{e}}(\mu(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h}),\nabla\boldsymbol{v}_{e})_{\tau}+\sum\limits_{\tau\in\omega_{e}}((\mu+\lambda)(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h}),\nabla\cdot\boldsymbol{v}_{e})_{\tau}.$

Using the properties of the bubble function $\phi_{\tau}(\boldsymbol{x})$ and
the integration by parts, we obtain

$\displaystyle\sum\limits_{\tau\in\omega_{e}}(\boldsymbol{f},\boldsymbol{v}_{e})_{\tau}+\sum\limits_{\tau\in\omega_{e}}(\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h}),\boldsymbol{v}_{e})_{\tau}-<\mu\nabla_{w}\boldsymbol{u}_{h}\boldsymbol{n}_{1},\boldsymbol{v}_{e}>_{e}-<\mu\nabla_{w}\boldsymbol{u}_{h}\boldsymbol{n}_{2},\boldsymbol{v}_{e}>_{e}$
$\displaystyle+\sum\limits_{\tau\in\omega_{e}}(\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}),\boldsymbol{v}_{e})_{\tau}-<(\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h},\boldsymbol{v}_{e}\cdot\boldsymbol{n}_{1}>_{e}-<(\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h},\boldsymbol{v}_{e}\cdot\boldsymbol{n}_{2}>_{e}$
$\displaystyle=\sum\limits_{\tau\in\omega_{e}}(\mu(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h}),\nabla\boldsymbol{v}_{e})_{\tau}+\sum\limits_{\tau\in\omega_{e}}((\mu+\lambda)(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h}),\nabla\cdot\boldsymbol{v}_{e})_{\tau}.$

Merging the inner product of the edges, we have

$\displaystyle\sum\limits_{\tau\in\omega_{e}}(\boldsymbol{f},\boldsymbol{v}_{e})_{\tau}+\sum\limits_{\tau\in\omega_{e}}(\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h}),\boldsymbol{v}_{e})_{\tau}+\sum\limits_{\tau\in\omega_{e}}(\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}),\boldsymbol{v}_{e})$
$\displaystyle-<(\mu\nabla_{w}\boldsymbol{u}_{h}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{u}_{h})\boldsymbol{I})\boldsymbol{n}_{1},\boldsymbol{v}_{e}>_{e}-<(\mu\nabla_{w}\boldsymbol{u}_{h}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{u}_{h})\boldsymbol{I})\boldsymbol{n}_{2},\boldsymbol{v}_{e}>_{e}$
$\displaystyle=\sum\limits_{\tau\in\omega_{e}}(\mu(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h}),\nabla\boldsymbol{v}_{e})_{\tau}+\sum\limits_{\tau\in\omega_{e}}((\mu+\lambda)(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h}),\nabla\cdot\boldsymbol{v}_{e})_{\tau}.$

Subtracting and adding $(\boldsymbol{f}_{h},\boldsymbol{v}_{e})_{\tau}$ and
merging, we obtain

$\displaystyle<[(\mu\nabla_{w}\boldsymbol{u}_{h}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{u}_{h})\boldsymbol{I})\boldsymbol{n}],\boldsymbol{v}_{e}>_{e}$
$\displaystyle=\sum\limits_{\tau\in\omega_{e}}(\boldsymbol{f}-\boldsymbol{f}_{h},\boldsymbol{v}_{e})_{\tau}+\sum\limits_{\tau\in\omega_{e}}(\boldsymbol{f}_{h}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}),\boldsymbol{v}_{e})_{\tau}$
$\displaystyle~{}~{}~{}~{}-\sum\limits_{\tau\in\omega_{e}}(\mu(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h}),\nabla\boldsymbol{v}_{e})_{\tau}-\sum\limits_{\tau\in\omega_{e}}((\mu+\lambda)(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h}),\nabla\cdot\boldsymbol{v}_{e})_{\tau}.$

Using Schwarz inequality and inverse inequality, the above equation becomes

$\displaystyle\|[(\mu\nabla_{w}\boldsymbol{v}_{h}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{v}_{h})\boldsymbol{I})\boldsymbol{n}]\|^{2}_{e}$
$\displaystyle\leq\left|\sum\limits_{\tau\in\omega_{e}}(\boldsymbol{f}-\boldsymbol{f}_{h},\boldsymbol{v}_{e})_{\tau}+\sum\limits_{\tau\in\omega_{e}}(\boldsymbol{f}_{h}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}),\boldsymbol{v}_{e})_{\tau}\right|$
$\displaystyle~{}~{}~{}~{}+\left|\sum\limits_{\tau\in\omega_{e}}(\mu(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h}),\nabla\boldsymbol{v}_{e})_{\tau}+\sum\limits_{\tau\in\omega_{e}}((\mu+\lambda)(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h}),\nabla\cdot\boldsymbol{v}_{e})_{\tau}\right|$
$\displaystyle\lesssim\left(\|\boldsymbol{f}-\boldsymbol{f}_{h}\|_{\omega_{e}}+\|\boldsymbol{f}_{h}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}\|_{\omega_{e}}\right)\|\boldsymbol{v}_{e}\|_{\omega_{e}}$
$\displaystyle~{}~{}~{}~{}+\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|_{\omega_{e}}\|\nabla\boldsymbol{v}_{e}\|_{\omega_{e}}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\omega_{e}}\|\nabla\cdot\boldsymbol{v}_{e}\|_{\omega_{e}}$
$\displaystyle\lesssim\left(\|\boldsymbol{f}-\boldsymbol{f}_{h}\|_{\omega_{e}}+\|\boldsymbol{f}_{h}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}\|_{\omega_{e}}\right)\|\boldsymbol{v}_{e}\|_{\omega_{e}}$
$\displaystyle~{}~{}~{}~{}+\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|_{\omega_{e}}h_{e}^{-1}\|\boldsymbol{v}_{e}\|_{\omega_{e}}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\omega_{e}}h_{e}^{-1}\|\boldsymbol{v}_{e}\|_{\omega_{e}}$
$\displaystyle\lesssim\left(h_{e}^{1/2}\|\boldsymbol{f}-\boldsymbol{f}_{h}\|_{\omega_{e}}+h_{e}^{1/2}\|\boldsymbol{f}_{h}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}\|_{\omega_{e}}\right.$
$\displaystyle~{}~{}~{}~{}\left.+h_{e}^{-1/2}\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|_{\omega_{e}}+h_{e}^{-1/2}\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\omega_{e}}\right)\|\boldsymbol{v}_{e}\|_{e}$
$\displaystyle\lesssim\left(h_{e}^{1/2}\|\boldsymbol{f}-\boldsymbol{f}_{h}\|_{\omega_{e}}+h_{e}^{1/2}\|\boldsymbol{f}_{h}+\nabla\cdot(\mu\nabla_{w}\boldsymbol{u}_{h})+\nabla((\mu+\lambda)\nabla_{w}\cdot\boldsymbol{u}_{h}\|_{\omega_{e}}\right.$
$\displaystyle~{}~{}~{}~{}\left.+h_{e}^{-1/2}\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|_{\omega_{e}}+h_{e}^{-1/2}\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\omega_{e}}\right)$
$\displaystyle~{}~{}~{}~{}\|(\mu\nabla_{w}\boldsymbol{v}_{h}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{v}_{h})\boldsymbol{I})\boldsymbol{n}\|_{e}.$

Dividing
$\|(\mu\nabla_{w}\boldsymbol{v}_{h}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{v}_{h})\boldsymbol{I})\boldsymbol{n}\|^{2}_{e}$
on both sides of the above inequality, and using (4.16) and the definition
$\mathrm{osc}(\boldsymbol{f},\omega_{e})$, we obtain

$\displaystyle
h_{e}^{1/2}\|[(\mu\nabla_{w}\boldsymbol{v}_{h}+(\mu+\lambda)(\nabla_{w}\cdot\boldsymbol{v}_{h})\boldsymbol{I})\boldsymbol{n}]\|_{e}$
$\displaystyle\leq
C\left(\mathrm{osc}(\boldsymbol{f},\omega_{e})+\|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|_{\omega_{e}}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|_{\omega_{e}}\right).$

Then we have completed the proof of (4.17) .

Using (4.16) and (4.17), then summing over all $e\in\mathcal{E}_{h}$ and all
$\tau\in\mathcal{T}_{h}$, we arrive at the following lower bound for the error
estimator.

######  Theorem 4.6 (Lower Bound)

Let $\boldsymbol{u}$ be the solution of (1.3) and
$\boldsymbol{u}_{h}=\\{\boldsymbol{u}_{0}^{h},\boldsymbol{u}_{b}^{h}\\}\in\mathcal{V}_{h}$
be the solution of (2.6), respectively. There exists a constant $C_{1}>0$,
such that such

$\displaystyle\eta^{2}(\boldsymbol{u}_{h},\mathcal{T}_{h})\leq
C_{2}\left(|\mu^{1/2}(\nabla\boldsymbol{u}-\nabla_{w}\boldsymbol{u}_{h})\|^{2}_{\tau}+\|(\mu+\lambda)^{1/2}(\nabla\cdot\boldsymbol{u}-\nabla_{w}\cdot\boldsymbol{u}_{h})\|^{2}_{\tau}+\mathrm{osc}^{2}(\boldsymbol{f},\mathcal{T}_{h})\right).$

wher the constant $C_{2}>0$, only depends on the shape regularity of
$\mathcal{T}_{h}$ .

##  5 Numeriacl Experiments

In this section, we give two experiments to verify the theoretical result.
During these experiments, we adopt the lowest order($k=1$) during the weak
finite element space $\mathcal{T}_{h}$ and the energy norm
$\|\cdot\|_{A}=a_{w}(\cdot,\cdot)^{\frac{1}{2}}$ to do the error analysis, the
Lamé constants $\mu=0.5$ and $\lambda=1.0$, and the error control constant
$tol=10^{-8}$.

###### Example 5.1

In this example, we examine the ’L-shape’ problem in two dimension. Let
$\Omega=(-1,1)^{2}\backslash(0,1)\times(-1,0)$, the proper vector source
$\boldsymbol{f}$ and the boundary function $\boldsymbol{g}$ are chosen to
ensure the solution

$\displaystyle\boldsymbol{u}=(u_{1},u_{2})^{T},$

where $u_{1}(x,y)=u_{2}(x,y)=r^{\frac{2}{3}}\sin(\frac{2}{3}\theta)$ in polar
coordinates.

During this example, we adopt the initial mesh like the left figure of Figure
1. After performing the AMWG-FEM, we can see that the refinement elements are
concentrated with singular of the solution $\boldsymbol{u}$. The right figure
of Figure 1 shows the 11st refinement meshes with $\vartheta=0.5$.

In the left figure of Figure 2, the abscissa value represents the number of
unknowns of the mesh $\mathcal{T}_{l}$, and the ordinate value
$\|\boldsymbol{u}-\boldsymbol{u}_{l}\|_{A}$ represents the energy norm of the
error between the solution $\boldsymbol{u}$ and the modified weak Galerkin
finite element solution $\boldsymbol{u}_{l}$. we present the error curve about
the errors of the solution $\boldsymbol{u}$ and the modified weak Galerkin
finite element solution $\boldsymbol{u}_{l}$ under the energy norm with the
D’́offler parameter $\vartheta=0.1,0.3,0.5$ . In the right of of Figure 2, the
ordinate value represents the estimator
$\eta(\boldsymbol{u}_{l},\mathcal{T}_{l})$. All straight lines with slope
$\frac{1}{2}$ can be moved vertically in Figure 2.

Figure 1: The initial mesh (left) and the 11st refinement mesh(right) of
Exmaple 5.1.

Figure 2: The curves of $\|\boldsymbol{u}-\boldsymbol{u}_{l}\|_{A}$ (left) and
$\eta(\boldsymbol{u}_{l},\mathcal{T}_{l})$ (right) for $\vartheta=0.1,0.3,0.5$
of Exmaple 5.1.

###### Example 5.2

In this example, we examine the ’L-shape’ problem in three dimension. Let
$\Omega=(-1,1)^{3}\backslash(0,1)\times(0,1)\times(-1,1)$, the proper vector
source $\boldsymbol{f}$ and the boundary function $\boldsymbol{g}$ are chosen
to ensure the solution

$\displaystyle\boldsymbol{u}=(u_{1},u_{2},u_{3})^{T},$

where
$u_{1}(x,y,z)=u_{2}(x,y,z)=u_{3}(x,y,z)=r^{\frac{2}{3}}\sin(\frac{2}{3}\theta)$,
in polar coordinates.

During this example, we adopt the initial mesh like the left figure of Figure
3. After performing the AMWG-FEM, we can see that the refinement elements are
concentrated with singular of the solution $\boldsymbol{u}$. The right figure
of Figure 3 shows the 12nd refinement meshes with $\vartheta=0.5$.

In the left figure of Figure 4, the abscissa value represents the number of
unknowns of the mesh $\mathcal{T}_{l}$, and the ordinate value
$\|\boldsymbol{u}-\boldsymbol{u}_{l}\|_{A}$ represents the energy norm of the
error between the solution $\boldsymbol{u}$ and the modified weak Galerkin
finite element solution $\boldsymbol{u}_{l}$. we present the error curve about
the errors of the solution $\boldsymbol{u}$ and the modified weak Galerkin
finite element solution $\boldsymbol{u}_{l}$ under the energy norm with the
D’́offler parameter $\vartheta=0.1,0.3,0.5$ . In the right of of Figure 4, the
ordinate value represents the estimator
$\eta(\boldsymbol{u}_{l},\mathcal{T}_{l})$. All straight lines with slope
$\frac{1}{3}$ can be moved vertically in Figure 4.

Figure 3: The initial mesh (left) and the 12nd refinement mesh(right) of
Exmaple 5.2.

Figure 4: The curves of $\|\boldsymbol{u}-\boldsymbol{u}_{l}\|_{A}$ (left) and
$\eta(\boldsymbol{u}_{l},\mathcal{T}_{l})$ (right) for $\vartheta=0.1,0.3,0.5$
of Exmaple 5.2.

From the above numerical examples, we verify the relibility and efficiency of
the estimator in (3.2), and the convergence of the AMWG-FEM.

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