# A weak Galerkin method and its two-grid algorithm for the quasi-linear
elliptic problems of non-monotone type

Peng Zhu College of Data Science, Jiaxing University, Jiaxing, Zhejiang
314001, China (pzh@zjxu.edu.cn). This research was supported in part by
Zhejiang Provincial Natural Science Foundation of China (LY19A010008) and
Natural Science Foundation of China (12071184). Shenglan Xie College of
Information Engineering, Jiaxing Nanhu University, Jiaxing, Zhejiang 314001,
China (slxie@jxnhu.edu.cn).

###### Abstract

In this article, a weak Galerkin method is firstly presented and analyzed for
the quasi-linear elliptic problem of non-monotone type. By using Brouwer’s
fixed point technique, the existence of WG solution and error estimates in
both the energy-like norm and the $L^{2}$ norm are derived. Then an efficient
two-grid WG method is introduced to improve the computational efficiency. The
convergence error of the two-grid WG method is analyzed in the energy-like
norm. Numerical experiments are presented to verify our theoretical findings.

###### keywords:

weak Galerkin, finite element method, quasi-linear elliptic problem, non-
monotone type, two-grid method

###### AMS:

Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50

##  1 Introduction

Consider the quasi-linear elliptic problem of the form

(1a) $\displaystyle-\nabla\cdot(a(x,u)u)$ $\displaystyle=$ $\displaystyle
f,\quad\mbox{in}\;\Omega,$ (1b) $\displaystyle u$ $\displaystyle=$
$\displaystyle g,\quad\mbox{on}\;\partial\Omega,$

where $\Omega$ is a bounded polytopal domain in $\mathbb{R}^{2}$. We suppose
that $a(x,u)$ is a twice continuously differentiable function in
$\overline{\Omega}\times\mathbb{R}$ and all the derivatives of
$a(\cdot,\cdot)$ through second order are bounded in
$\overline{\Omega}\times\mathbb{R}$. We also assume that there exist positive
constants $\alpha_{0},\alpha_{1}$ such that

$0<\alpha_{0}\leq a(x,u)\leq\alpha_{1}.$

For sake of notational simplicity, we denote $a(x,u)$ by $a(u)$ in the rest of
this paper.

The weak Galerkin (WG) method is a novel finite element framework for solving
partial differential equations. It was introduced in [29, 28] for the
numerical solution of second order elliptic problems. Since then, there has
been considerable interest in WG methods for the numerical solution of a wide
range of partial differential equations. We refer the reader to [14, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34], and the
references therein for details. However, there are few papers that are
concerned to the nonlinear elliptic problems. To the best of our knowledge,
only references [19, 14, 34] conducted their investigations in this direction.

For the second order elliptic quasi-linear PDEs, the existence of solutions of
the WG methods was shown in [19] by a Schauder’s fixed point argument.
However, the uniqueness and the error estimations of the numerical
approximations are restricted only to the linear PDEs, and have not been
addressed for the nonlinear ones. Recently in [14], the authors gave the well-
posedness and error estimate in the energy norm for the monotone quasi-linear
PDEs. Most recently, [34] investigated the stabilizer-free WG methods [31, 32]
on polytopal meshes for a class of quasi-linear elliptic problems of monotone
type.

Direct application of the techniques presented in [14, 34] to establish the a
priori error estimates of the WG solutions for non-monotone type quasi-linear
elliptic problem is not possible because the associated differential operators
may not satisfy monotonicity property. Non-monotone type quasi-linear elliptic
problems have been studied by conforming finite element methods [4], mixed
finite element methods [9, 13], discontinuous Galerkin methods [5, 6].

The first aim of this paper is to make an effort for conducting a priori error
analysis of the WG method for solving non-monotone quasi-linear elliptic
problems. Following the ideas in [10], we employ Brouwer’s fixed point theorem
to prove the existence of the discrete solution, and in turn derived the a
priori error estimates in a mesh-dependent energy norm and in the $L^{2}$
norm.

As it is well known, the two-grid method is an efficient algorithm for solving
nonlinear partial differential equations [12, 11]. Two-grid finite volume
element method [1] and two-grid discontinuous Galerkin method [2] were further
investigated for non-monotone quasi-linear elliptic problems. Another aim of
this paper is to propose an efficient WG method by adopting the two-grid idea
for the quasi-linear elliptic problem. Convergence of the two-grid WG method
is rigorously analyzed. As far as we know, the present work is the first
attempt to apply and analyze the two-grid technique in the setting of the WG
method.

The outline of this paper is as follows. Section 2 introduces the WG method
for the problem (1a)-(1b). In Section 3, we derive optimal order error
estimates of the WG method in both the energy-like norm and the $L^{2}$ norm.
The two-grid algorithm of the WG method is proposed in Section 4, which is
then followed by derivation of its convergence analysis in the energy-like
norm. Section 5 carries out numerical experiments to verify our theoretical
findings.

##  2 Weak Galerkin Finite Element Methods

First of all, let us introduce the concept of weak function. Let $K$ be any
polytopal domain with boundary $\partial K$. A weak function on $K$ refers to
a function $v=\\{v_{0},v_{b}\\}$ such that $v_{0}\in L^{2}(K)$ and $v_{b}\in
H^{1/2}(\partial K)$, where $v_{0}$ means the value of $v$ in $K$, and $v_{b}$
represents the boundary value of $v$. Note that $v_{b}$ may not necessarily be
related to the trace of $v_{0}$ on $\partial K$. Let $W(K)$ be the set of weak
functions on $K$, i.e.,

$W(K):=\\{v=\\{v_{0},v_{b}\\}:v_{0}\in L^{2}(K),v_{b}\in H^{1/2}(\partial
K)\\}.$

It is worth to point out that a function $v\in H^{1}(K)$ can be viewed as a
weak function $\\{v_{0},v_{b}\\}$ of $W(K)$ with $v_{0}=v|_{K}$ and
$v_{b}=v|_{\partial K}$.

Let ${\mathcal{T}}_{h}$ be a shape regular polytopal partition of the domain
$\Omega$ [28]. Denote by ${\cal E}_{h}$ the set of all edges in ${\cal
T}_{h}$, and let ${\cal E}_{h}^{0}={\cal E}_{h}\backslash\partial\Omega$ be
the set of all interior edges. For each element $K\in{\mathcal{T}}_{h}$, we
denote by $h_{K}$ the diameter of $K$, i.e., $h_{K}=\mathrm{diam}(K)$.
Similarly, let $h_{e}$ be the length of an edge $e\in{\mathcal{E}}_{h}$.

On this partition of ${\mathcal{T}}_{h}$, we introduce the following broken
Sobolev space

$\displaystyle H^{m}({\mathcal{T}}_{h}):=\\{v\in L^{2}(\Omega):v|_{K}\in
H^{1}(K),\;\forall K\in{\mathcal{T}}_{h}\\},$

for any integer $m\geq 0$.

Let $\mathbb{P}_{m}(D)$ denote the set of polynomials defined on $D$ with
degree no more than $m$, where $D$ may be an element $K$ of
${\mathcal{T}}_{h}$ or an edge $e$ of ${\mathcal{E}}_{h}$. In what follows, we
often consider the broken polynomial spaces

$\mathbb{P}_{m}({\mathcal{T}}_{h}):=\\{v\in
L^{2}(\Omega):v|_{K}\in\mathbb{P}_{m}(K),\,\,\forall
K\in{\mathcal{T}}_{h}\\},$

and

$\mathbb{P}_{m}({\mathcal{E}}_{h}):=\\{v\in
L^{2}({\mathcal{E}}_{h}):v|_{K}\in\mathbb{P}_{m}(e),\,\,\forall
e\in{\mathcal{E}}_{h}\\}.$

Then a WG finite element space $V_{h}$ associated with ${\mathcal{T}}_{h}$ for
$k\geq 1$ is defined by

(2) $V_{h}=\\{v=\\{v_{0},v_{b}\\}:\
v_{0}\in\mathbb{P}_{k}({\mathcal{T}}_{h}),v_{b}\in\mathbb{P}_{k}({\mathcal{E}}_{h})\\}.$

Denote by $V_{h}^{0}$ a subspace of $V_{h}$ with vanishing traces,

(3) $\displaystyle V_{h}^{0}=\\{v=\\{v_{0},v_{b}\\}\in V_{h},\ v_{b}|_{e}=0,\
e\subset\partial K\cap\partial\Omega\\}.$

For convenience, we adopt the following notations, For the sake of simplicity,
we introduce the following notations: for any $v,w\in
H^{1}({\mathcal{T}}_{h})$,

$\displaystyle(v,w)_{{\mathcal{T}}_{h}}=\sum_{K\in{\mathcal{T}}_{h}}(v,w)_{K},\quad\mbox{where
}(v,w)_{K}=\int_{K}vw\mathrm{d}\bm{x},$

and

$\displaystyle\langle v,w\rangle_{\partial{\mathcal{T}}_{h}}$
$\displaystyle=\sum_{K\in{\mathcal{T}}_{h}}\langle v,w\rangle_{\partial
K},\quad\mbox{where }(v,w)_{\partial K}=\int_{\partial K}vw\mathrm{d}s.$

######  Definition 1 (Weak Gradient).

For any function $v=\\{v_{0},v_{b}\\}\in V_{h}$, its weak gradient
$\nabla_{w}v$, is piecewisely defined as the unique polynomial
$(\nabla_{w}v)|_{K}\in[\mathbb{P}_{k-1}(K)]^{2}$ such that

(4) $(\nabla_{w}v,\bm{\phi})_{K}=-(v_{0},\nabla\cdot\bm{\phi})_{K}+\langle
v_{b},\bm{\phi}\cdot\bm{n}\rangle_{\partial
K},\quad\forall\bm{\phi}\in[\mathbb{P}_{k-1}(K)]^{2},$

for any $K\in\mathcal{T}_{h}$.

Now, we are ready to present our weak Galerkin finite element method for the
problem (1a)-(1b).

######  Algorithm 1 (The WG method).

The weak Galerkin finite element scheme for solving problem (1a)-(1b) is
defined as follows: find $u_{h}=\\{u_{0},u_{b}\\}\in V_{h}$ such that
$u_{b}=\mathcal{Q}_{b}g$ on $\partial\Omega$ and the following equation

(5) $A_{h}(u_{h};u_{h},v_{h})=(f,\;v_{0}),\quad\forall\
v_{h}=\\{v_{0},v_{b}\\}\in V_{h}^{0},$

where

(6) $\displaystyle
A_{h}(u_{h};v_{h},w_{h})=(a(u_{0})\nabla_{w}v_{h},\nabla_{w}w_{h})_{{\mathcal{T}}_{h}}+s_{h}(v_{h},w_{h}),$

with the stabilization term $s_{h}(\cdot,\cdot)$ is defined by

$s_{h}(v_{h},w_{h}):=\sum_{K\in{\mathcal{T}}_{h}}h_{K}^{-1}\langle
v_{0}-v_{b},w_{0}-w_{b}\rangle_{\partial K},\quad\forall v_{h},w_{h}\in
V_{h}.$

##  3 Error analysis

In this section, we shall derive the a priori error estimates of the WG method
(5) for solving the quasi-linear elliptic problem (1a)-(1b). To this end, we
firstly introduce notations and some useful lemmas in Sect. 3.1 and then
derive an error equation for the WG method (5) in Sect. 3.2. Using this error
equation, a fixed point mapping is constructed and discussed in Sect. 3.3.
Finally, by the Brouwer’s fixed point theorem the error estimates in both
energy-like norm and $L^{2}$ norm are concluded in Sect. 3.4.

###  3.1 Preliminary

In order to analyze the WG method (5), we introduce the energy norm
${|||}\cdot{|||}$ over $V_{h}^{0}$ by

(7) ${|||}v{|||}=(\sum_{K\in{\mathcal{T}}_{h}}{|||}v{|||}_{K}^{2})^{1/2},$

with

$\displaystyle{|||}v{|||}_{K}=(\|\nabla_{w}v\|_{L^{2}(K)}^{2}+h_{K}^{-1}\|v_{0}-v_{b}\|_{L^{2}(\partial
K)}^{2})^{1/2},$

and we also need the $H^{1}$-like norm $\|\cdot\|_{1,h}$

(8) $\|v\|_{1,h}=(\sum_{K\in{\mathcal{T}}_{h}}\|\nabla
v_{h}\|_{1,h,K}^{2})^{1/2},$

for all $v\in V_{h}+H^{1}({\mathcal{T}}_{h})$, where

$\displaystyle\|\nabla v_{h}\|_{1,h,K}=(\|\nabla
v_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}+h_{K}^{-1}\|v_{0}-v_{b}\|_{L^{2}(\partial
K)}^{2})^{1/2}.$

It is easy to see that $\|\cdot\|_{1,h}$ is indeed a norm on the finite
element space $V_{h}^{0}$. We claim that the semi-norm ${|||}\cdot{|||}$ is
also a norm on $V_{h}^{0}$ since these two norms are equivalent.

######  Lemma 2.

There exist two positive constants $C_{1}$ and $C_{2}$ such that for any
$v=\\{v_{0},v_{b}\\}\in V_{h}$, we have

$C_{1}\|v\|_{1,h}\leq{|||}v{|||}\leq C_{2}\|v\|_{1,h}.$

###### Proof.

The proof is trivial. The interesting reader is referred to the proof of Lemma
5.3 in [20]. ∎

In what follows, the trace inequality and inverse inequality are frequently
used in our analysis, which state as [28]: for any $p,q\in[1,\infty)$, there
holds

(9) $\|\phi\|_{L^{p}(\partial K)}\leq
C\left(h_{K}^{-1/p}\|\phi\|_{L^{p}(K)}+h_{K}^{1-1/p}\|\nabla\phi\|_{L^{p}(K)}\right),\quad\forall\phi\in
W_{p}^{1}(K),$

and

(10) $\|\phi_{h}\|_{L^{p}(\partial K)}\leq
Ch_{K}^{-1/p}\|\phi_{h}\|_{L^{p}(K)},\quad\forall\phi_{h}\in\mathbb{P}_{k}(K),$

and

(11) $\|\phi_{h}\|_{L^{p}(K)}\leq
Ch_{K}^{2(1/p-1/q)}\|\phi_{h}\|_{L^{q}(K)},\quad\forall\phi_{h}\in\mathbb{P}_{k}(K).$

In [7, 8], the following Poincaré type inequality has been proved for
discontinuous functions in the broken Sobolev space
$H^{1}({\mathcal{T}}_{h})$.

######  Lemma 3 (Poincaré type inequality).

For any $v\in H^{1}({\mathcal{T}}_{h})$, there exists a constant $C_{P}>0$
independent of mesh size $h$ and $v$ such that for any $p\in[1,\infty)$

$\displaystyle\|v\|_{L^{p}(\Omega)}\leq C_{P}\|v\|_{dg},$

with the norm $\|\cdot\|_{dg}$ is defined by

(12) $\displaystyle\|v\|_{dg}=\\{\|\nabla
v\|_{L^{2}({\mathcal{T}}_{h})}^{2}+\sum_{e\in{\mathcal{E}}_{h}}h_{e}^{-1}\|{[[}{v}{]]}\|_{L^{2}(e)}^{2}\\}^{1/2},$

where ${[[}{v}{]]}$ is the value jump of the function $v$ across an interior
edge $e$, which is defined as: if $e$ is an interior edge shared by two
elements $K_{1}$ and $K_{2}$, i.e. $e=\partial{K}_{1}\cap\partial{K}_{2}$, we
set ${[[}{v}{]]}|_{e}=v|_{K_{1}}-v|_{K_{2}}$. In case $e$ is a boundary edge,
i.e., $e=\partial K\cap\partial\Omega$, we define ${[[}{v}{]]}|_{e}=v|_{K}$.

From Lemma 3, we can easily establish the analogue of Poincaré type inequality
for weak functions.

######  Lemma 4.

For any weak function $v=\\{v_{0},v_{b}\\}\in
V_{h}^{0}+H^{1}({\mathcal{T}}_{h})$, there exists a constant $C_{P}>0$
independent of mesh size $h$ and $v$ such that for any $p\in[1,\infty)$

$\displaystyle\|v_{0}\|_{L^{p}(\Omega)}\leq C_{P}\|v\|_{1,h}.$

###### Proof.

Let $v=\\{v_{0},v_{b}\\}$ be any weak function in the space
$V_{h}^{0}+H^{1}({\mathcal{T}}_{h})$. For any edge $e=\partial
K_{1}\cap\partial K_{2}$, since

$\displaystyle{[[}v_{0}{]]}|_{e}=(v_{0}|_{K_{1}}-v_{b})+(v_{b}-v_{0}|_{K_{2}}),$

then by the triangle inequality we deduce that

$\displaystyle\|{[[}v_{0}{]]}|\|_{L^{2}(e)}\leq\|v_{0}-v_{b}\|_{L^{2}(\partial
K_{1}\cap e)}+\|v_{b}-v_{0}\|_{L^{2}(\partial K_{2}\cap e)}.$

Thus, under the assumption of shape regularity of the partition
${\mathcal{T}}_{h}$, we have

$\displaystyle h_{e}^{-1}\|{[[}v_{0}{]]}|\|_{L^{2}(e)}^{2}\leq
C(h_{K_{1}}^{-1}\|v_{0}-v_{b}\|_{L^{2}(\partial K_{1}\cap
e)}^{2}+h_{K_{2}}^{-1}\|v_{b}-v_{0}\|_{L^{2}(\partial K_{2}\cap e)}^{2}).$

From the definitions of $\|\cdot\|_{dg}$ and $\|\cdot\|_{1,h}$, it is easy to
see that

$\displaystyle\|v_{0}\|_{dg}\leq C\|v\|_{1,h},$

which together with Lemma 3 completes the proof. ∎

Let us introduce the projection operators $\mathcal{Q}_{h}$ and $\Pi_{h}$. For
each element $K\in{\mathcal{T}}_{h}$, we denote by $\mathcal{Q}_{0}$ and
$\Pi_{h}$ the $L^{2}$-orthogonal projections
$\mathcal{Q}_{0}:L^{2}(K)\rightarrow\mathbb{P}_{k}(K)$ and
$\Pi_{h}:[L^{2}(K)]^{2}\rightarrow[\mathbb{P}_{k-1}(K)]^{2}$, respectively,
that is,

$\displaystyle(\mathcal{Q}_{0}v-v,w)_{K}$ $\displaystyle=0,\quad\forall
w\in\mathbb{P}_{k}(K),$
$\displaystyle(\Pi_{h}\bm{\sigma}-\bm{\sigma},\bm{\tau})_{K}$
$\displaystyle=0,\quad\forall\bm{\tau}\in[\mathbb{P}_{k-1}(K)]^{2}(K).$

For each edge $e\in{\mathcal{E}}_{h}$, the $L^{2}$-orthogonal projection
$\mathcal{Q}_{b}:L^{2}(e)\rightarrow\mathbb{P}_{k}(e)$ is defined as follows:

(13) $\langle\mathcal{Q}_{b}v-v,w\rangle_{e}=0,\quad\forall
w\in\mathbb{P}_{k}(e).$

Finally, for a smooth function $v\in H^{1}({\mathcal{T}}_{h})$, we introduce
the projection $\mathcal{Q}_{h}:H^{1}({\mathcal{T}}_{h})\rightarrow V_{h}$
such that

$\displaystyle(\mathcal{Q}_{h}v)|_{K}:=\\{\mathcal{Q}_{0}(v|_{K}),\mathcal{Q}_{b}(v|_{\partial
K})\\}.$

In [19], it was proved that the projections $\mathcal{Q}_{h}$ and $\Pi_{h}$
has the following commutative property.

######  Lemma 5.

[19, Lemma 5.1] For any $v\in H^{1}(K)$ and $K\in{\mathcal{T}}_{h}$, there
holds

$\displaystyle\nabla_{w}(\mathcal{Q}_{h}v)=\Pi_{h}(\nabla v).$

It is well known that (cf.[3]) the projections $\mathcal{Q}_{0}$ and $\Pi_{h}$
have the standard approximation properties:

######  Lemma 6.

For any $K\in{\mathcal{T}}_{h}$ and any $w\in W_{p}^{k+1}(K)$ and
$\bm{\sigma}\in[W_{p}^{k}(K)]^{2}$ with $1\leq p\leq q\leq\infty$, there
exists positive constant $C$ independent of $h_{K}$ such that,

(14) $\displaystyle|w-\mathcal{Q}_{0}w|_{W_{p}^{r}(K)}\leq
Ch_{K}^{k+1-r}|w|_{W_{p}^{k+1}(K)},\quad 0\leq r\leq k+1$

and

(15) $\displaystyle|\bm{\sigma}-\Pi_{h}\bm{\sigma}|_{W_{q}^{r}(K)}\leq
Ch_{K}^{k-r-2/p+2/q}|\bm{\sigma}|_{W_{p}^{k}(K)},\quad 0\leq r\leq k.$

In the analysis below, we shall use the Taylor series expansion: for any
$s,t\in\mathbb{R}$, we have

(16) $\displaystyle a(s)=a(t)+\widetilde{a}_{u}(s,t)(s-t),$

where $\widetilde{a}_{u}(s,t)=\int_{0}^{1}a_{u}(t+\tau(s-t))\mathrm{d}\tau$,
and

(17) $\displaystyle a(s)=a(t)+a_{u}(t)(s-t)+\widetilde{a}_{uu}(s,t)(s-t)^{2},$

where
$\widetilde{a}_{uu}(s,t)=\int_{0}^{1}(1-\tau)a_{uu}(t+\tau(s-t))\mathrm{d}\tau$.

Denote by

$M_{a}=\max\\{\|a\|_{L^{\infty}(\Omega\times\mathbb{R})},\|a_{u}\|_{L^{\infty}(\Omega\times\mathbb{R})},\|a_{uu}\|_{L^{\infty}(\Omega\times\mathbb{R})}\\}.$

For the sake of convenience, denote by $Lu:=-\nabla\cdot(a(u)\nabla u)$. For
any $w\in W^{1,p}$, the linearized operator $L$ at $w$ (namely, the Fréchet
derivative of $L$ at $w$) is then given by

$\displaystyle L^{\prime}[w]\phi:=-\nabla\cdot(a(w)\nabla\phi+a_{u}(w)\nabla
w\phi).$

Introducing the bilinear form (induced by $L^{\prime}[w]$)

(18) $\displaystyle
D_{h}(w;\phi_{h},v_{h})=(a(w)\nabla_{w}\phi_{h},\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}+(a_{u}(w)\nabla
w\phi_{0},\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}+s_{h}(\phi_{h},v_{h})$

for any $\phi_{h}=\\{\phi_{0},\phi_{b}\\},v_{h}=\\{v_{0},v_{b}\\}\in V_{h}$.
It is easy to see that

$D_{h}(w;\phi_{h},v_{h})=A_{h}(w;\phi_{h},v_{h})+(a_{u}(w)\nabla
w\phi_{0},\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}.$

First of all, let us introduce the following analogy of Gårding’s inequality.

######  Lemma 7 (Gårding’s inequality).

For a given $\phi\in W^{1}_{\infty}(\Omega)$. Then there is a positive
constant, $\beta<\infty$, such that

(19)
$D_{h}(\phi;v_{h},v_{h})+\beta\|v_{0}\|^{2}\geq\gamma({|||}v_{h}{|||}^{2}+\|v_{0}\|^{2}),\quad\forall
v_{h}=\\{v_{0},v_{b}\\}\in V_{h}^{0},$

where $\gamma=\min\\{\frac{\alpha_{0}}{2},1\\}$.

###### Proof.

Using the boundness of $a(u)$, we have

$\displaystyle
D_{h}(\phi;v_{h},v_{h})+\beta\|v_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}$
$\displaystyle=A_{h}(\phi;v_{h},v_{h})+(a_{u}(\phi)\nabla\phi
v_{0},\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}+\beta\|v_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}$
$\displaystyle\geq\alpha_{0}\|\nabla_{w}v_{h}\|_{L^{2}({\mathcal{T}}_{h})}^{2}+s_{h}(v_{h},v_{h})+(a_{u}(\phi)\nabla\phi
v_{0},\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}+\beta\|v_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}.$

By Hölder’s inequality,

$\displaystyle|(a_{u}(\phi)\nabla\phi
v_{0},\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}|\leq
M_{a}|\phi|_{W^{1}_{\infty}(\Omega)}\|\nabla_{w}v_{h}\|_{L^{2}({\mathcal{T}}_{h})}\|v_{0}\|_{L^{2}({\mathcal{T}}_{h})}.$

Therefore,

$\displaystyle
D_{h}(\phi;v_{h},v_{h})+\beta\|v_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}$
$\displaystyle\geq\alpha_{0}\|\nabla_{w}v_{h}\|_{L^{2}({\mathcal{T}}_{h})}^{2}+s_{h}(v_{h},v_{h})+\beta\|v_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}$
$\displaystyle\quad-
M_{a}|\phi|_{W^{1}_{\infty}(\Omega)}\|\nabla_{w}v\|_{L^{2}({\mathcal{T}}_{h})}\|v_{0}\|_{L^{2}({\mathcal{T}}_{h})}.$

Let $\gamma=\min\\{\frac{\alpha_{0}}{2},1\\}$. Provided

$\displaystyle\beta\geq\gamma+\frac{M_{a}^{2}|\phi|_{W^{1}_{\infty}(\Omega)}^{2}}{2\alpha_{0}},$

from the Young’s inequality, we conclude that

$\displaystyle
D_{h}(\phi;v_{h},v_{h})+\beta\|v_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}$
$\displaystyle\geq\frac{\alpha_{0}}{2}\|\nabla_{w}v_{h}\|_{L^{2}({\mathcal{T}}_{h})}^{2}+s_{h}(v_{h},v_{h})+(\beta-\frac{M_{a}^{2}|\phi|_{W^{1}_{\infty}(\Omega)}^{2}}{2\alpha_{0}})\|v_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}$
$\displaystyle\geq\gamma({|||}v_{h}{|||}^{2}+\|v_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}).$

The proof is completed. ∎

###  3.2 Error equation

Now for the exact solution $u$ of (1a)-(1b), we define the error between the
WG solution $u_{h}=\\{u_{0},u_{b}\\}$ and the projection
$\mathcal{Q}_{h}u=\\{\mathcal{Q}_{0}u,\mathcal{Q}_{b}u\\}$ of $u$ as

$e_{h}=\mathcal{Q}_{h}u-u_{h}:=\\{e_{0},e_{b}\\},$

with

$e_{0}=\mathcal{Q}_{0}u-u_{0},\quad e_{b}=\mathcal{Q}_{b}u-u_{b}.$

The aim of this subsection is to obtain an error equation for $e_{h}$ by the
use of bilinear form $D_{h}(u;\cdot,\cdot)$.

######  Lemma 8 (Error equation).

Let $u$ and $u_{h}$ be the solutions of the problem (1a)-(1b) and the WG
scheme (5), respectively. For any $v=\\{v_{0},v_{b}\\}\in V_{h}^{0}$, we have

(20) $\displaystyle
D_{h}(u;\mathcal{Q}_{h}u-u_{h},v)=E_{h}(u,v)+R_{h}(u;u_{h},v),$

where

(21) $\displaystyle E_{h}(u,v)$
$\displaystyle=\ell(u,v)+s_{h}(\mathcal{Q}_{h}u,v),$ $\displaystyle
R_{h}(u;u_{h},v)$ $\displaystyle=((u-u_{0})\widetilde{a}_{u}(u_{0})(\nabla
u-\nabla_{w}u_{h}),\nabla_{w}v)_{{\mathcal{T}}_{h}}$ (22)
$\displaystyle\qquad+((u-u_{0})^{2}\widetilde{a}_{uu}(u_{0}),\nabla_{w}v)_{{\mathcal{T}}_{h}},$

with $\ell(u,v):=\sum\nolimits_{i=1}^{3}\ell_{i}(u,v)$, and

(23a) $\displaystyle\ell_{1}(u,v)$ $\displaystyle:=(a(u)(\Pi_{h}(\nabla
u)-\nabla u),\nabla_{w}v)_{{\mathcal{T}}_{h}},$ (23b)
$\displaystyle\ell_{2}(u,v)$ $\displaystyle:=\langle(a(u)\nabla
u-\Pi_{h}(a(u)\nabla
u))\cdot\bm{n},v_{0}-v_{b}\rangle_{\partial{\mathcal{T}}_{h}},$ (23c)
$\displaystyle\ell_{3}(u,v)$
$\displaystyle:=((\mathcal{Q}_{0}u-u)a_{u}(u)\nabla{u},\nabla_{w}v)_{{\mathcal{T}}_{h}}.$

###### Proof.

For notational convenience, we denote $a(u)\nabla u$ by $\sigma$. For any
$v=\\{v_{0},v_{b}\\}\in V_{h}^{0}$, testing (1a) by $v_{0}$ and using the fact
that

$\sum_{K\in{\mathcal{T}}_{h}}\langle\sigma\cdot\bm{n},v_{b}\rangle_{\partial
K}=0$

and integration by parts, we arrive at

(24)
$\displaystyle(f,v_{0})=-(\nabla\cdot\sigma,v_{0})_{{\mathcal{T}}_{h}}=(\sigma,\nabla
v_{0})_{{\mathcal{T}}_{h}}-\langle\sigma\cdot\bm{n},v_{0}-v_{b}\rangle_{\partial{\mathcal{T}}_{h}}.$

Next we investigate the term $(\sigma,\nabla v_{0})_{{\mathcal{T}}_{h}}$ in
the above equation. Using the definition of $\Pi_{h}$, integration by parts
and the definition of weak gradient, we have

$\displaystyle(\sigma,\nabla v_{0})_{{\mathcal{T}}_{h}}$ $\displaystyle=$
$\displaystyle(\Pi_{h}\sigma,\nabla v_{0})_{{\mathcal{T}}_{h}}$
$\displaystyle=$
$\displaystyle-(v_{0},\nabla\cdot\Pi_{h}\sigma)_{{\mathcal{T}}_{h}}+\langle
v_{0},\Pi_{h}\sigma\cdot\bm{n}\rangle_{\partial{\mathcal{T}}_{h}}$
$\displaystyle=$
$\displaystyle(\nabla_{w}v,\Pi_{h}\sigma)_{{\mathcal{T}}_{h}}+\langle
v_{0}-v_{b},\Pi_{h}\sigma\cdot\bm{n}\rangle_{\partial{\mathcal{T}}_{h}}$
$\displaystyle=$
$\displaystyle(\sigma,\nabla_{w}v)_{{\mathcal{T}}_{h}}+\langle\Pi_{h}\sigma\cdot\bm{n},v_{0}-v_{b}\rangle_{\partial{\mathcal{T}}_{h}},$

which together with (24) yields

(25)
$\displaystyle(\sigma,\nabla_{w}v)_{{\mathcal{T}}_{h}}=(f,v_{0})+\langle(\sigma-\Pi_{h}\sigma)\cdot\bm{n},v_{0}-v_{b}\rangle_{\partial{\mathcal{T}}_{h}}.$

Recalling Lemma 5 and adding $s_{h}(\mathcal{Q}_{h}u,v)$ on both sides of
(25), we arrive at

$\displaystyle(a(u)\nabla_{w}(\mathcal{Q}_{h}u),\nabla_{w}v)_{{\mathcal{T}}_{h}}+s_{h}(\mathcal{Q}_{h}u,v)=(f,v_{0})+\sum_{i=1}^{2}\ell_{i}(u,v)+s_{h}(\mathcal{Q}_{h}u,v).$

Subtracting the WG scheme (5) from the above equation yields

$\displaystyle
A_{h}(u;\mathcal{Q}_{h}u-u_{h},v)=\sum_{i=1}^{2}\ell_{i}(u,v)+s_{h}(\mathcal{Q}_{h}u,v)+((a(u_{0})-a(u))\nabla_{w}u_{h},\nabla_{w}v)_{{\mathcal{T}}_{h}},$

then adding the term
$((\mathcal{Q}_{0}u-u_{0})a_{u}(u)\nabla{u},\nabla_{w}v)_{{\mathcal{T}}_{h}}$
on both sides gives

(26) $\displaystyle D_{h}(u;\mathcal{Q}_{h}u-u_{h},v)=E_{h}(u,v)+H,$

with $E_{h}(u,v)$ is defined by (21), and

$\displaystyle H$
$\displaystyle=((a(u_{0})-a(u))\nabla_{w}u_{h},\nabla_{w}v)_{{\mathcal{T}}_{h}}+((u-u_{0})a_{u}(u)\nabla{u},\nabla_{w}v)_{{\mathcal{T}}_{h}}$
$\displaystyle=((a(u_{0})-a(u))(\nabla_{w}u_{h}-\nabla
u),\nabla_{w}v)_{{\mathcal{T}}_{h}}$ (27)
$\displaystyle\quad+((a(u_{0})-a(u)-(u_{0}-u)a_{u}(u))\nabla{u},\nabla_{w}v)_{{\mathcal{T}}_{h}}.$

Using the Taylor expansions (16) and (17), we get

$\displaystyle H=((u-u_{0})\widetilde{a}_{u}(u_{0},u)(\nabla
u-\nabla_{w}u_{h}),\nabla_{w}v)_{{\mathcal{T}}_{h}}+((u-u_{0})^{2}\widetilde{a}_{uu}(u_{0},u),\nabla_{w}v)_{{\mathcal{T}}_{h}},$

which together with (26) completes the proof. ∎

######  Lemma 9.

Assume $u\in H^{k+1}(\Omega)\cap W_{\infty}^{1}(\Omega)$. There exists a
constant $C$ such that the following estimates hold true:

(28) $\displaystyle|\ell_{1}(u,v)|$ $\displaystyle\leq
Ch^{k}|u|_{H^{k+1}(\Omega)}\|\nabla_{w}v\|_{L^{2}({\mathcal{T}}_{h})},$ (29)
$\displaystyle|\ell_{2}(u,v)|$ $\displaystyle\leq
Ch^{k}|u|_{H^{k+1}(\Omega)}s_{h}^{1/2}(v,v),$ (30)
$\displaystyle|\ell_{3}(u,v)|$ $\displaystyle\leq
Ch^{k+1}|u|_{H^{k+1}(\Omega)}\|\nabla_{w}v\|_{L^{2}({\mathcal{T}}_{h})},$ (31)
$\displaystyle|s_{h}(\mathcal{Q}_{h}u,v)|$ $\displaystyle\leq
Ch^{k}|u|_{H^{k+1}(\Omega)}s_{h}^{1/2}(v,v).$

###### Proof.

By the triangle inequality, the Hölder’s inequality and the approximation
property of $\Pi_{h}$, we have

$\displaystyle|\ell_{1}(u,v)|$
$\displaystyle\leq\sum_{K\in{\mathcal{T}}_{h}}|(\Pi_{h}(\nabla u)-\nabla
u,a(u)\nabla_{w}v)_{K}|$
$\displaystyle\leq\sum_{K\in{\mathcal{T}}_{h}}M_{a}\|\Pi_{h}(\nabla u)-\nabla
u\|_{L^{2}(K)}\|\nabla_{w}{v}\|_{L^{2}(K)}$ (32) $\displaystyle\leq
Ch^{k}|u|_{H^{k+1}(\Omega)}\|\nabla_{w}v\|_{L^{2}({\mathcal{T}}_{h})}.$

For the estimate (29), from the Cauchy-Schwarz inequality, (15) and the trace
inequality (9), it follows that

$\displaystyle|\ell_{2}(u,v)|$
$\displaystyle\leq\sum_{K\in{\mathcal{T}}_{h}}\|a(u)\nabla
u-\Pi_{h}(a(u)\nabla u)\|_{L^{2}(\partial K)}\|v_{0}-v_{b}\|_{L^{2}(\partial
K)}$ $\displaystyle\leq(\sum_{K\in{\mathcal{T}}_{h}}h_{K}\|a(u)\nabla
u-\Pi_{h}(a(u)\nabla u)\|_{L^{2}(\partial K)}^{2})^{1/2}$
$\displaystyle\quad\times(\sum_{K\in{\mathcal{T}}_{h}}h_{K}^{-1}\|v_{0}-v_{b}\|_{L^{2}(\partial
K)}^{2})^{1/2}$ $\displaystyle\leq
Ch^{k}|u|_{H^{k+1}(\Omega)}s_{h}^{1/2}(v,v).$

By the Hölder’s inequality and (14), we can deduce that

$\displaystyle|\ell_{3}(u,v)|$
$\displaystyle\leq\sum_{K\in{\mathcal{T}}_{h}}M_{a}\|\mathcal{Q}_{0}u-u\|_{L^{2}(K)}\|\nabla
u\|_{L^{\infty}(K)}\|\nabla_{w}v\|_{L^{2}(K)}$ $\displaystyle\leq
C\sum_{K\in{\mathcal{T}}_{h}}h_{K}^{k+1}|u|_{H^{k+1}(K)}|u|_{W_{\infty}^{1}(K)}\cdot\|\nabla_{w}v\|_{L^{2}(K)}$
$\displaystyle\leq
Ch^{k+1}|u|_{H^{k+1}(\Omega)}\|\nabla_{w}v\|_{L^{2}(\Omega)}.$

Now we consider the estimate (31). By the trace inequality (9) and (14), we
have

$\displaystyle\|\mathcal{Q}_{0}u-u\|_{L^{2}(\partial K)}\leq
Ch_{K}^{k+1/2}|u|_{H^{k+1}(K)}.$

Then, it follows from the Cauchy-Schwarz inequality and the definition of
$\mathcal{Q}_{b}$ that

$\displaystyle|s_{h}(\mathcal{Q}_{h}u,v)|$
$\displaystyle\leq\sum_{K\in{\mathcal{T}}_{h}}h_{K}^{-1}|\langle\mathcal{Q}_{0}u-\mathcal{Q}_{b}u,v_{0}-v_{b}\rangle_{\partial
K}|$
$\displaystyle=\sum_{K\in{\mathcal{T}}_{h}}h_{K}^{-1}|\langle\mathcal{Q}_{0}u-u,v_{0}-v_{b}\rangle_{\partial
K}|$
$\displaystyle\leq(\sum_{K\in{\mathcal{T}}_{h}}h_{K}^{-1}\|\mathcal{Q}_{0}u-u\|_{L^{2}({\partial
K})}^{2})^{1/2}(\sum_{K\in{\mathcal{T}}_{h}}h_{K}^{-1}\|v_{0}-v_{b}\|_{L^{2}({\partial
K})}^{2})^{1/2}$ $\displaystyle\leq
Ch^{k}|u|_{H^{k+1}(\Omega)}s_{h}^{1/2}(v,v).$

We have completed the proof. ∎

###  3.3 Construction of a fixed point mapping

Motivated by the form of the error equation (20), we introduce the fixed point
mapping $\mathcal{F}_{h}$ as follows.

######  Definition 10 (The fixed point mapping $\mathcal{F}_{h}$).

For a given $\xi\in V_{h}$, let $\mathcal{F}_{h}:V_{h}\rightarrow V_{h}$ be a
map $\psi=\mathcal{F}_{h}(\xi)\in V_{h}$ satisfying

(33) $\displaystyle
D_{h}(u;\mathcal{Q}_{h}u-\psi,v_{h})=E_{h}(u,v_{h})+R_{h}(u;\xi,v_{h}),$

for any $v_{h}\in V_{h}^{0}$.

######  Remark 3.1.

Equivalently, we can restate (33) as follows: find $\psi\in V_{h}$ such that

(34) $\displaystyle D_{h}(u;\psi,v_{h})=\Psi_{\xi}(v_{h}),$

for any $v_{h}\in V_{h}^{0}$, with $\Psi_{\xi}(v)$ is defined by

$\Psi_{\xi}(v):=D_{h}(u;\mathcal{Q}_{h}u,v)-E_{h}(u,v)-R_{h}(u;\xi,v).$

Obviously, $\Psi_{\xi}(v)$ is a continuous linear functional of $v$. By Riesz
representation theorem, there exists a function $\widehat{f}$ such that
$(\widehat{f},v)=\Psi_{\xi}(v)$. The solution $\psi\in V_{h}$ of (34) can be
viewed as a WG finite element solution of the linear second order elliptic
equation $L^{\prime}[u]\phi=\widehat{f}$. By the use of Gårding’s inequality
(19), it is easy to show that there exists a unique solution of (34) as the
proof in [29]. Then the map $\psi=\mathcal{F}_{h}(\xi)$ given by (33) is well
defined.

Now consider the ball

$\mathbb{B}_{h}(\mathcal{Q}_{h}u)=\\{\omega\in
V_{h}:{|||}\mathcal{Q}_{h}u-\omega{|||}\leq Ch^{k}|u|_{H^{k+1}(\Omega)}\\},$

where $C$ is a general positive constant. Since
$\mathcal{Q}_{h}u\in\mathbb{B}_{h}(\mathcal{Q}_{h}u)$, the set
$\mathbb{B}_{h}(\mathcal{Q}_{h}u)$ is nonempty.

In the below, we shall show that $\mathcal{F}_{h}$ has a fixed point in
$\mathbb{B}_{h}(\mathcal{Q}_{h}u)$. Since (20) is equivalent to the WG scheme
(5), there exists a solution $u_{h}\in V_{h}$ of the nonlinear WG finite
element scheme (5).

######  Lemma 11.

Assume $u\in H^{k+1}(\Omega)\cap W_{\infty}^{1}(\Omega)$. For any $\xi\in
V_{h}$ and $v_{h}\in V_{h}^{0}$, there exists a constant $C>0$, independent of
$h$, such that

(35) $\displaystyle|R_{h}(u;\xi,v_{h})|\leq
Ch^{-1/2}(h^{k}|u|_{H^{k+1}(\Omega)}+{|||}\mathcal{Q}_{h}u-\xi{|||})^{2}\|\nabla_{w}{v_{h}}\|_{L^{2}({\mathcal{T}}_{h})}.$

###### Proof.

For any $\xi\in V_{h}$, let $\xi=\\{\xi_{0},\xi_{b}\\}$. From (8), we know

$\displaystyle R_{h}(u;\xi,v_{h})$
$\displaystyle=((u-\xi_{0})\widetilde{a}_{u}(\xi_{0},u)(\nabla
u-\nabla_{w}\xi),\nabla_{w}{v_{h}})_{{\mathcal{T}}_{h}}$
$\displaystyle\quad+((u-\xi_{0})^{2}\widetilde{a}_{uu}(\xi_{0},u)\nabla
u,\nabla_{w}{v_{h}})_{{\mathcal{T}}_{h}}.$

By Hölder’s inequality and the inverse inequality (11), we can deduce that

$\displaystyle|((u-\xi_{0})\widetilde{a}_{u}(\xi_{0},u)(\nabla
u-\nabla_{w}\xi),\nabla_{w}{v_{h}})_{{\mathcal{T}}_{h}}|$ $\displaystyle\leq
M_{a}\|u-\xi_{0}\|_{L^{4}({\mathcal{T}}_{h})}\|\nabla
u-\nabla_{w}\xi\|_{L^{2}({\mathcal{T}}_{h})}\|\nabla_{w}{v_{h}}\|_{L^{4}({\mathcal{T}}_{h})}$
(36) $\displaystyle\leq
Ch^{-1/2}\|u-\xi_{0}\|_{L^{4}({\mathcal{T}}_{h})}\|\nabla
u-\nabla_{w}\xi\|_{L^{2}({\mathcal{T}}_{h})}\|\nabla_{w}{v_{h}}\|_{L^{2}({\mathcal{T}}_{h})},$

and

$\displaystyle|((u-\xi_{0})^{2}\widetilde{a}_{uu}(\xi_{0},u)\nabla
u,\nabla_{w}{v_{h}})_{{\mathcal{T}}_{h}}|$ $\displaystyle\leq
M_{a}\|(u-\xi_{0})^{2}\|_{L^{2}({\mathcal{T}}_{h})}\|\nabla
u\|_{L^{\infty}(\Omega)}\|\nabla_{w}{v_{h}}\|_{L^{2}({\mathcal{T}}_{h})}$ (37)
$\displaystyle\leq
C\|u-\xi_{0}\|_{L^{4}({\mathcal{T}}_{h})}^{2}\|\nabla_{w}{v_{h}}\|_{L^{2}({\mathcal{T}}_{h})}.$

Thanks to Lemma 4, we get

$\displaystyle\|u-\xi_{0}\|_{L^{4}({\mathcal{T}}_{h})}\leq
C_{P}\|u-\xi\|_{1,h}\leq
C_{P}(\|u-\mathcal{Q}_{0}u\|_{1,h}+\|\mathcal{Q}_{h}u-\xi\|_{1,h}),$

then using (14), the trace inequality (9) and Lemma 2, we obtain

(38) $\displaystyle\|u-\xi_{0}\|_{L^{4}({\mathcal{T}}_{h})}\leq
C(h^{k}|u|_{H^{k+1}(\Omega)}+{|||}\mathcal{Q}_{h}u-\xi{|||}).$

By the triangle inequality, (15) and Lemma 5, we have

$\displaystyle\|\nabla u-\nabla_{w}\xi\|_{L^{2}({\mathcal{T}}_{h})}$
$\displaystyle\leq\|\nabla u-\Pi_{h}(\nabla
u)\|_{L^{2}({\mathcal{T}}_{h})}+\|\nabla_{w}(\mathcal{Q}_{h}u-\xi)\|_{L^{2}({\mathcal{T}}_{h})}$
(39) $\displaystyle\leq
C(h^{k}|u|_{H^{k+1}(\Omega)}+{|||}\mathcal{Q}_{h}u-\xi{|||}).$

Plugging (38) and (3.3) back into (3.3) and (3.3), respectively, completes the
proof. ∎

######  Lemma 12.

Assume $u\in H^{k+1}(\Omega)\cap W_{\infty}^{1}(\Omega)$. For any
$\xi\in\mathbb{B}_{h}(\mathcal{Q}_{h}u)$, let $\psi=\mathcal{F}_{h}(\xi)$ and
denote by $\zeta_{h}=\mathcal{Q}_{h}u-\psi$. Then there exists a constant
$C>0$, independent of $h$, such that

(40)
$\displaystyle\gamma{|||}\zeta_{h}{|||}\leq\beta\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})}+Ch^{k}|u|_{H^{k+1}(\Omega)},$

for sufficiently small $h$.

###### Proof.

Taking $v_{h}=\zeta_{h}$ in (33) yields

(41) $\displaystyle
D_{h}(u;\zeta_{h},\zeta_{h})=E_{h}(u,\zeta_{h})+R_{h}(u;\xi,\zeta_{h}).$

From Lemma 9, we can see that

(42) $\displaystyle|E_{h}(u,\zeta_{h})|\leq
Ch^{k}|u|_{H^{k+1}(\Omega)}{|||}\zeta_{h}{|||}.$

In view of Lemma 11 and noting that $\xi\in\mathbb{B}_{h}(\mathcal{Q}_{h}u)$,
we arrive at

$\displaystyle|R_{h}(u;\xi,\zeta_{h})|\leq
Ch^{2k-1/2}|u|_{H^{k+1}(\Omega)}^{2}{|||}\zeta_{h}{|||},$

which combining with (41) and (42) leads to

$\displaystyle D_{h}(u;\zeta_{h},\zeta_{h})$ $\displaystyle\leq
C(1+h^{k-1/2}|u|_{H^{k+1}(\Omega)})h^{k}|u|_{H^{k+1}(\Omega)}{|||}\zeta_{h}{|||}$
$\displaystyle\leq Ch^{k}|u|_{H^{k+1}(\Omega)}{|||}\zeta_{h}{|||},$

for sufficiently small $h$.

Appealing to the Gårding’s inequality (19) yields

$\displaystyle\gamma({|||}\zeta_{h}{|||}^{2}+\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2})$
$\displaystyle\leq\beta\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}+D_{h}(u;\zeta_{h},\zeta_{h})$
$\displaystyle\leq\beta\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}+Ch^{k}|u|_{H^{k+1}(\Omega)}{|||}\zeta_{h}{|||},$

which implies

$\displaystyle\gamma{|||}\zeta_{h}{|||}\leq\beta\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})}+Ch^{k}|u|_{H^{k+1}(\Omega)}.$

The proof is completed. ∎

In order to bound $\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})}$ we use a duality
argument. Consider the following dual problem

(45)
$\displaystyle\left\\{\begin{array}[]{rr}-\nabla\cdot(a(u)\nabla\varphi)+a_{u}(u)\nabla
u\cdot\nabla\varphi=\zeta_{0}&\quad\mbox{in}\;\;\Omega,\\\\[3.0pt]
\varphi=0&\quad\mbox{on}\;\partial\Omega.\end{array}\right.$

Assume that the dual problem has the $H^{2}$-regularity in the sense that
there exists a constant $C$ such that

(46) $\|\varphi\|_{H^{2}(\Omega)}\leq
C\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})}.$

######  Lemma 13.

Assume $u\in H^{k+1}(\Omega)\cap W_{\infty}^{1}(\Omega)$ and $\varphi\in
H^{2}(\Omega)$. There exists a constant $C$ such that

$\displaystyle|E_{h}(u,\mathcal{Q}_{h}\varphi)|$ $\displaystyle\leq
Ch^{k+1}|u|_{H^{k+1}(\Omega)}\|\varphi\|_{H^{2}(\Omega)}.$

###### Proof.

By the triangle inequality, the Hölder’s inequality and the approximation
property of $\Pi_{h}$, we have

$\displaystyle|\ell_{1}(u,\mathcal{Q}_{h}\varphi)|$
$\displaystyle\leq\sum_{K\in{\mathcal{T}}_{h}}|(\Pi_{h}(\nabla u)-\nabla
u,a(u)\nabla_{w}(\mathcal{Q}_{h}\varphi))_{K}|$
$\displaystyle\leq\sum_{K\in{\mathcal{T}}_{h}}|(\Pi_{h}(\nabla u)-\nabla
u,a(u)(\Pi_{h}(\nabla\varphi)-\nabla\varphi))_{K}|$
$\displaystyle\quad+\sum_{K\in{\mathcal{T}}_{h}}|(\Pi_{h}(\nabla u)-\nabla
u,a(u)\nabla\varphi-\Pi_{h}(a(u)\nabla\varphi))_{K}|$
$\displaystyle\leq\sum_{K\in{\mathcal{T}}_{h}}\|\Pi_{h}(\nabla u)-\nabla
u\|_{L^{2}(K)}(M_{a}\|\Pi_{h}(\nabla\varphi)-\nabla\varphi\|_{L^{2}(K)}$
$\displaystyle\qquad\qquad+\|a(u)\nabla\varphi-\Pi_{h}(a(u)\nabla\varphi)\|_{L^{2}(K)})$
(47) $\displaystyle\leq
Ch^{k+1}|u|_{H^{k+1}(\Omega)}|\varphi|_{H^{2}(\Omega)}.$

First of all, we get the bound of
$s_{h}(\mathcal{Q}_{h}\varphi,\mathcal{Q}_{h}\varphi)$. By the definition of
$\mathcal{Q}_{b}$, it is easy to see that

$\|\mathcal{Q}_{b}\varphi-\varphi\|_{L^{2}(\partial
K)}\leq\|\mathcal{Q}_{0}\varphi-\varphi\|_{L^{2}(\partial K)},$

then using the triangle inequality, we have

$\|\mathcal{Q}_{0}\varphi-\mathcal{Q}_{b}\varphi\|_{L^{2}(\partial K)}\leq
2\|\mathcal{Q}_{0}\varphi-\varphi\|_{L^{2}(\partial K)}.$

Thus, from the Cauchy-Schwarz inequality, (14) and the trace inequality (9),
it follows that

$\displaystyle s_{h}(\mathcal{Q}_{h}\varphi,\mathcal{Q}_{h}\varphi)$
$\displaystyle=\sum_{K\in{\mathcal{T}}_{h}}h_{K}^{-1}\|\mathcal{Q}_{0}\varphi-\mathcal{Q}_{b}\varphi\|_{L^{2}(\partial
K)}^{2}$ $\displaystyle\leq
4\sum_{K\in{\mathcal{T}}_{h}}h_{K}^{-1}\|\mathcal{Q}_{0}\varphi-\varphi\|_{L^{2}(\partial
K)}^{2}$ (48) $\displaystyle\leq Ch^{2}|\varphi|_{H^{2}(\Omega)}^{2}.$

Then, using (29) of Lemma 9 and (3.3), we obtain

(49) $\displaystyle|\ell_{2}(u,\mathcal{Q}_{h}\varphi)|\leq
Ch^{k}|u|_{H^{k+1}(\Omega)}s_{h}^{1/2}(\mathcal{Q}_{h}\varphi,\mathcal{Q}_{h}\varphi)\leq
Ch^{k+1}|u|_{H^{k+1}(\Omega)}|\varphi|_{H^{2}(\Omega)}.$

Similarly, we have

(50) $\displaystyle|s_{h}(\mathcal{Q}_{h}u,\mathcal{Q}_{h}\varphi)|\leq
Ch^{k+1}|u|_{H^{k+1}(\Omega)}|\varphi|_{H^{2}(\Omega)}.$

By the Hölder’s inequality and (14), we can deduce that

$\displaystyle|\ell_{3}(u,\mathcal{Q}_{h}\varphi)|$
$\displaystyle\leq\sum_{K\in{\mathcal{T}}_{h}}M_{a}\|\mathcal{Q}_{0}u-u\|_{L^{2}(K)}\|\nabla
u\|_{L^{\infty}(K)}\|\nabla_{w}(\mathcal{Q}_{h}\varphi)\|_{L^{2}(K)}$
$\displaystyle\leq
C\sum_{K\in{\mathcal{T}}_{h}}h_{K}^{k+1}|u|_{H^{k+1}(K)}|u|_{W_{\infty}^{1}(K)}\cdot\|\Pi_{h}(\nabla\varphi)\|_{L^{2}(K)}$
(51) $\displaystyle\leq
Ch^{k+1}|u|_{H^{k+1}(\Omega)}|\varphi|_{H^{1}(\Omega)}.$

With the help of the estimates (3.3), (49), (3.3) and (50), we completes the
proof. ∎

######  Lemma 14.

Assume $u\in H^{k+1}(\Omega)\cap W_{\infty}^{1}(\Omega)$ and $\varphi\in
H^{2}(\Omega)$. For any $\xi\in\mathbb{B}_{h}(\mathcal{Q}_{h}u)$, there exists
a constant $C>0$, independent of $h$, such that

$\displaystyle|R_{h}(u;\xi,\mathcal{Q}_{h}\varphi)|\leq
Ch^{2k}|u|_{H^{k+1}(\Omega)}^{2}\|\varphi\|_{H^{2}(\Omega)}.$

###### Proof.

By Lemma 5, $\nabla_{w}(\mathcal{Q}_{h}\varphi)=\Pi_{h}(\nabla\varphi)$. Then
from (8), we know

$\displaystyle R_{h}(u;\xi,\mathcal{Q}_{h}\varphi)$
$\displaystyle=((u-\xi_{0})\widetilde{a}_{u}(\xi_{0},u)(\nabla
u-\nabla_{w}\xi),\Pi_{h}(\nabla\varphi))_{{\mathcal{T}}_{h}}$
$\displaystyle\quad+((u-\xi_{0})^{2}\widetilde{a}_{uu}(\xi_{0},u)\nabla
u,\Pi_{h}(\nabla\varphi))_{{\mathcal{T}}_{h}}.$

By Hölder’s inequality and the inverse inequality (11), we can deduce that

$\displaystyle|((u-\xi_{0})\widetilde{a}_{u}(\xi_{0},u)(\nabla
u-\nabla_{w}\xi),\Pi_{h}(\nabla\varphi))_{{\mathcal{T}}_{h}}|$ (52)
$\displaystyle\leq M_{a}\|u-\xi_{0}\|_{L^{4}({\mathcal{T}}_{h})}\|\nabla
u-\nabla_{w}\xi\|_{L^{2}({\mathcal{T}}_{h})}\|\Pi_{h}(\nabla\varphi)\|_{L^{4}({\mathcal{T}}_{h})},$

and

$\displaystyle|((u-\xi_{0})^{2}\widetilde{a}_{uu}(\xi_{0},u)\nabla
u,\Pi_{h}(\nabla\varphi))_{{\mathcal{T}}_{h}}|$ (53) $\displaystyle\leq
M_{a}\|(u-\xi_{0})^{2}\|_{L^{2}({\mathcal{T}}_{h})}\|\nabla
u\|_{L^{\infty}(\Omega)}\|\Pi_{h}(\nabla\varphi)\|_{L^{2}({\mathcal{T}}_{h})}.$

From the triangle inequality and (15), it follows that

$\displaystyle\|\Pi_{h}(\nabla\varphi)\|_{L^{2}({\mathcal{T}}_{h})}$
$\displaystyle\leq\|\Pi_{h}(\nabla\varphi)-\nabla\varphi\|_{L^{2}({\mathcal{T}}_{h})}+\|\nabla\varphi\|_{L^{2}({\mathcal{T}}_{h})}$
$\displaystyle\leq Ch|\varphi|_{H^{2}(\Omega)}+|\varphi|_{H^{1}(\Omega)}$ (54)
$\displaystyle\leq C\|\varphi\|_{H^{2}(\Omega)}.$

Plugging (38), (3.3) and (3.3) back into (3.3) and (3.3), respectively,
completes the proof. ∎

######  Lemma 15.

Assume $u\in H^{k+1}(\Omega)\cap W_{\infty}^{1}(\Omega)$. For any
$\xi\in\mathbb{B}_{h}(\mathcal{Q}_{h}u)$, let $\psi=\mathcal{F}_{h}(\xi)$ and
denote by $\zeta_{h}=\mathcal{Q}_{h}u-\psi$. Then there exists a constant
$C>0$, independent of $h$, such that

(55) $\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})}\leq
C[h{|||}\zeta_{h}{|||}+h^{k+1}|u|_{H^{k+1}(\Omega)}+h^{2k}|u|_{H^{k+1}(\Omega)}^{2}],$

for sufficiently small $h$.

###### Proof.

Testing the first equation of the dual problem (45) by $\zeta_{0}$ yields

$\displaystyle\|\zeta_{0}\|_{L^{2}(\Omega)}^{2}=-(\nabla\cdot(a(u)\nabla\varphi),\zeta_{0})_{{\mathcal{T}}_{h}}+(\zeta_{0}a_{u}(u)\nabla
u,\nabla\varphi)_{{\mathcal{T}}_{h}}.$

Then using a similar procedure as in the proof of the equation (25), we obtain

$\displaystyle-(\nabla\cdot(a(u)\nabla\varphi),\zeta_{0})_{{\mathcal{T}}_{h}}$
$\displaystyle=(a(u)\nabla_{w}\zeta_{h},\nabla\varphi)_{{\mathcal{T}}_{h}}+\langle\zeta_{0}-\zeta_{b},(\Pi_{h}(a(u)\nabla\varphi)-a(u)\nabla\varphi)\cdot\bm{n}\rangle_{\partial{\mathcal{T}}_{h}}.$

Therefore,

$\displaystyle\|\zeta_{0}\|_{L^{2}(\Omega)}^{2}$
$\displaystyle=(a(u)\nabla_{w}\zeta_{h},\nabla\varphi)_{{\mathcal{T}}_{h}}+(\zeta_{0}a_{u}(u)\nabla
u,\nabla\varphi)_{{\mathcal{T}}_{h}}$
$\displaystyle\quad+\langle\zeta_{0}-\zeta_{b},(\Pi_{h}(a(u)\nabla\varphi)-a(u)\nabla\varphi)\cdot\bm{n}\rangle_{\partial{\mathcal{T}}_{h}}.$

With the following notations

(56) $\displaystyle J_{1}$
$\displaystyle=\langle\zeta_{0}-\zeta_{b},(\Pi_{h}(a(u)\nabla\varphi)-a(u)\nabla\varphi)\cdot\bm{n}\rangle_{\partial{\mathcal{T}}_{h}},$
(57) $\displaystyle J_{2}$
$\displaystyle=(a(u)(\nabla\varphi-\Pi_{h}(\nabla\varphi)),\nabla_{w}\zeta_{h})_{{\mathcal{T}}_{h}},$
(58) $\displaystyle J_{3}$ $\displaystyle=(a_{u}(u)\nabla
u\zeta_{0},\nabla\varphi-\Pi_{h}(\nabla\varphi))_{{\mathcal{T}}_{h}},$

we could rewrite the term $\|\zeta_{0}\|_{L^{2}(\Omega)}^{2}$ as follows:

$\displaystyle\|\zeta_{0}\|_{L^{2}(\Omega)}^{2}$
$\displaystyle=(a(u)\nabla_{w}\zeta_{h},\nabla_{w}(\mathcal{Q}_{h}\varphi))_{{\mathcal{T}}_{h}}+(\zeta_{0}a_{u}(u)\nabla
u,\nabla_{w}(\mathcal{Q}_{h}\varphi))_{{\mathcal{T}}_{h}}$
$\displaystyle\quad+J_{1}+J_{2}+J_{3}$ (59)
$\displaystyle=D_{h}(u;\zeta_{h},\mathcal{Q}_{h}\varphi)-s_{h}(\zeta_{h},\mathcal{Q}_{h}\varphi)+J_{1}+J_{2}+J_{3}.$

Taking $v=\mathcal{Q}_{h}\varphi$ in (33) yields

$\displaystyle
D_{h}(u;\zeta_{h},\mathcal{Q}_{h}\varphi)=E_{h}(u,\mathcal{Q}_{h}\varphi)+R_{h}(u;\xi,\mathcal{Q}_{h}\varphi),$

which together with (3.3) leads to

$\displaystyle\|\zeta_{0}\|_{L^{2}(\Omega)}^{2}$
$\displaystyle=E_{h}(u,\mathcal{Q}_{h}\varphi)+R_{h}(u;\xi,\mathcal{Q}_{h}\varphi)$
(60) $\displaystyle\quad-
s_{h}(\zeta_{h},\mathcal{Q}_{h}\varphi)+J_{1}+J_{2}+J_{3}.$

By Cauchy-Schwarz inequality and (3.3, we get

(61) $\displaystyle|s_{h}(\zeta_{h},\mathcal{Q}_{h}\varphi)|\leq
s_{h}^{1/2}(\zeta_{h},\zeta_{h})s_{h}^{1/2}(\mathcal{Q}_{h}\varphi,\mathcal{Q}_{h}\varphi)\leq
Ch{|||}\zeta_{h}{|||}|\varphi|_{H^{2}(\Omega)}.$

It follows from Cauchy-Schwarz inequality and (15) that

$\displaystyle|J_{1}|$
$\displaystyle\leq(\sum_{K\in{\mathcal{T}}_{h}}h_{K}^{-1}\|\zeta_{0}-\zeta_{b}\|_{L^{2}(\partial
K)}^{2})^{1/2}$
$\displaystyle\quad\times(\sum_{K\in{\mathcal{T}}_{h}}h_{K}\|\Pi_{h}(a(u)\nabla\varphi)-a(u)\nabla\varphi\|_{L^{2}(\partial
K)}^{2})^{1/2}$ (62) $\displaystyle\leq
Ch{|||}\zeta_{h}{|||}|\varphi|_{H^{2}(\Omega)},$

and

$\displaystyle|J_{2}|$ $\displaystyle\leq
M_{a}\|\nabla\varphi-\Pi_{h}(\nabla\varphi)\|_{L^{2}({\mathcal{T}}_{h})}\|\nabla_{w}\zeta_{h}\|_{L^{2}({\mathcal{T}}_{h})}$
(63) $\displaystyle\leq Ch{|||}\zeta_{h}{|||}|\varphi|_{H^{2}(\Omega)}.$

From Hölder’s inequality and (15), it follows that

$\displaystyle|J_{3}|$ $\displaystyle\leq
M_{a}\|\nabla\varphi-\Pi_{h}(\nabla\varphi)\|_{L^{2}({\mathcal{T}}_{h})}\|\nabla
u\|_{L^{\infty}(\Omega)}\|\zeta_{0}\|_{L^{2}(\Omega)}$ (64) $\displaystyle\leq
Ch\|\zeta_{0}\|_{L^{2}(\Omega)}|\varphi|_{H^{2}(\Omega)}.$

Finally, appealing to Lemma 13 and 14, and the estimates of (61), (3.3), (3.3)
and (3.3), we have

$\displaystyle\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}$ $\displaystyle\leq
C[h^{k+1}|u|_{H^{k+1}(\Omega)}+h({|||}\zeta_{h}{|||}+\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})})$
$\displaystyle\qquad+h^{2k}|u|_{H^{k+1}(\Omega)}^{2}]\|\varphi\|_{H^{2}(\Omega)},$

which together with (46) leads to

$\displaystyle\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})}$ $\displaystyle\leq
C[h^{k+1}|u|_{H^{k+1}(\Omega)}+h({|||}\zeta_{h}{|||}+\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})})+h^{2k}|u|_{H^{k+1}(\Omega)}^{2}],$

which implies (55) for sufficiently small $h$. The proof is completed. ∎

######  Theorem 16.

Assume $u\in H^{k+1}(\Omega)\cap W_{\infty}^{1}(\Omega)$. For any
$\xi\in\mathbb{B}_{h}(\mathcal{Q}_{h}u)$, let $\psi=\mathcal{F}_{h}(\xi)$ and
denote by $\zeta_{h}=\mathcal{Q}_{h}u-\psi$. Then there exists a constant
$C>0$, independent of $h$, such that

(65) ${|||}\zeta_{h}{|||}\leq Ch^{k}|u|_{H^{k+1}(\Omega)},$

and

(66) $\|\zeta_{0}\|_{L^{2}({\mathcal{T}}_{h})}\leq
C(h^{k+1}+h^{2k}|u|_{H^{k+1}(\Omega)})|u|_{H^{k+1}(\Omega)},$

for sufficiently small $h$.

###### Proof.

Plugging (55) of Lemma 15 back into (40) of Lemma 12 yields

${|||}\zeta_{h}{|||}\leq
C[h{|||}\zeta_{h}{|||}+h^{k}|u|_{H^{k+1}(\Omega)}+h^{2k}|u|_{H^{k+1}(\Omega)}^{2}],$

which implies

${|||}\zeta_{h}{|||}\leq
Ch^{k}|u|_{H^{k+1}(\Omega)}(1+h^{k}|u|_{H^{k+1}(\Omega)})\leq
Ch^{k}|u|_{H^{k+1}(\Omega)},$

for sufficiently small $h$.

Combining (65) with (55) of Lemma 15 leads to the $L^{2}$ norm estimate (66).
The proof is completed. ∎

According to (65) of Theorem 16, we can easily conclude the following
statements.

######  Theorem 17.

For sufficiently small $h$, the map $\mathcal{F}_{h}$ maps
$\mathbb{B}_{h}(\mathcal{Q}_{h}u)$ into itself.

###### Proof.

For any $\xi\in\mathbb{B}_{h}(\mathcal{Q}_{h}u)$, from (65) of Theorem 16 it
follows that

${|||}\mathcal{Q}_{h}u-\mathcal{F}_{h}(\xi){|||}\leq
Ch^{k}|u|_{H^{k+1}(\Omega)},$

which implies $\mathcal{F}_{h}(\xi)\in\mathbb{B}_{h}(\mathcal{Q}_{h}u)$.
Therefore,
$\mathcal{F}_{h}(\mathbb{B}_{h}(\mathcal{Q}_{h}u))\subseteq\mathbb{B}_{h}(\mathcal{Q}_{h}u)$.
The proof is completed. ∎

The following theorem shows that the mapping $\mathcal{F}_{h}$ is a
contraction in the norm ${|||}\cdot{|||}$ of the ball
$\mathbb{B}_{h}(\mathcal{Q}_{h}u)$.

######  Theorem 18.

For any given $\xi,\theta\in\mathbb{B}_{h}(\mathcal{Q}_{h}u)$, there holds

(67)
$\displaystyle{|||}\mathcal{F}_{h}(\xi)-\mathcal{F}_{h}(\theta){|||}\leq\rho(h){|||}\xi-\theta{|||},$

with $\rho(h)\in(0,1)$ for sufficiently small $h$.

###### Proof.

For the sake of simplicity, let $\psi_{h}=\mathcal{F}_{h}(\xi)$ and
$\phi_{h}=\mathcal{F}_{h}(\theta)$. From the definition (33) of
$\mathcal{F}_{h}$, it is easy to see that

(68) $\displaystyle
D_{h}(u;\phi_{h}-\psi_{h},v_{h})=R_{h}(u;\xi,v_{h})-R_{h}(u;\theta,v_{h}),\quad\forall\;v_{h}\in
V_{h}^{0}.$

From (3.2), we find

$\displaystyle
R_{h}(u;\xi,v_{h})=(T_{1}(\xi),\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}+(T_{2}(\xi)\nabla
u,\nabla_{w}v_{h})_{{\mathcal{T}}_{h}},$

with

$\displaystyle T_{1}(\xi)$
$\displaystyle=(a(\xi_{0})-a(u))(\nabla_{w}\xi-\nabla u),$ $\displaystyle
T_{2}(\xi)$ $\displaystyle=a(\xi_{0})-a(u)-(\xi_{0}-u)a_{u}(u).$

Similarly, we have

$\displaystyle
R_{h}(u;\theta,v_{h})=(T_{1}(\theta),\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}+(T_{2}(\theta)\nabla
u,\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}.$

Thus,

$\displaystyle R_{h}(u;\xi,v_{h})-R_{h}(u;\theta,v_{h})$
$\displaystyle=(T_{1}(\xi)-T_{1}(\theta),\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}+((T_{2}(\xi)-T_{2}(\theta))\nabla
u,\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}$ (69) $\displaystyle=I_{1}+I_{2}.$

By the Taylor series expansions (16) and (17), we get

$\displaystyle T_{1}(\xi)-T_{1}(\theta)$
$\displaystyle=(a(\xi_{0})-a(u))(\nabla_{w}\xi-\nabla
u)-(a(\theta_{0})-a(u))(\nabla_{w}\theta-\nabla u)$
$\displaystyle=(a(\xi_{0})-a(\theta_{0}))(\nabla_{w}\theta-\nabla
u)+(a(\xi_{0})-a(u))\nabla_{w}(\xi-\theta)$
$\displaystyle=(\xi_{0}-\theta_{0})\widetilde{a}_{u}(\xi_{0},\theta_{0})(\nabla_{w}\theta-\nabla
u)+(\xi_{0}-u)\widetilde{a}_{u}(\xi_{0},u)\nabla_{w}(\xi-\theta),$

and

$\displaystyle T_{2}(\xi)-T_{2}(\theta)$
$\displaystyle=a(\xi_{0})-a(\theta_{0})-(\xi_{0}-\theta_{0})a_{u}(u)$
$\displaystyle=[a(\xi_{0})-a(\theta_{0})-(\xi_{0}-\theta_{0})a_{u}(\theta_{0})]$
$\displaystyle\qquad+(a_{u}(\theta_{0})-a_{u}(u))(\xi_{0}-\theta_{0})$
$\displaystyle=(\xi_{0}-\theta_{0})^{2}\widetilde{a}_{uu}(\xi_{0},\theta_{0})+(\xi_{0}-\theta_{0})(\theta_{0}-u)\widetilde{a}_{uu}(\theta_{0},u).$

Then, by Hölder’s inequality and the inverse inequality (11), we have

$\displaystyle|I_{1}|$
$\displaystyle\leq|((\xi_{0}-\theta_{0})\widetilde{a}_{u}(\xi_{0},\theta_{0})(\nabla_{w}\theta-\nabla
u),\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}|$
$\displaystyle\quad+|((\xi_{0}-u)\widetilde{a}_{u}(\xi_{0},u)\nabla_{w}(\xi-\theta),\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}|$
$\displaystyle\leq
M_{a}(\|\xi_{0}-\theta_{0}\|_{L^{4}({\mathcal{T}}_{h})}\|\nabla_{w}\theta-\nabla
u\|_{L^{2}({\mathcal{T}}_{h})}$
$\displaystyle\quad+\|\xi_{0}-u\|_{L^{4}({\mathcal{T}}_{h})}\|\nabla_{w}(\xi-\theta)\|_{L^{2}({\mathcal{T}}_{h})})\|\nabla_{w}v_{h}\|_{L^{4}({\mathcal{T}}_{h})}$
$\displaystyle\leq
Ch^{-1/2}(\|\xi_{0}-\theta_{0}\|_{L^{4}({\mathcal{T}}_{h})}\|\nabla_{w}\theta-\nabla
u\|_{L^{2}({\mathcal{T}}_{h})}$ (70)
$\displaystyle\quad+\|\xi_{0}-u\|_{L^{4}({\mathcal{T}}_{h})}{|||}\xi-\theta{|||})\|\nabla_{w}v_{h}\|_{L^{2}({\mathcal{T}}_{h})}.$

Using Lemma 4 and 2, one has

$\displaystyle\|\xi_{0}-\theta_{0}\|_{L^{4}({\mathcal{T}}_{h})}\leq
C_{P}\|\xi-\theta\|_{1,h}\leq C{|||}\xi-\theta{|||},$

and

$\displaystyle\|\xi_{0}-u\|_{L^{4}({\mathcal{T}}_{h})}$ $\displaystyle\leq
C_{P}\|\xi-u\|_{1,h}$ $\displaystyle\leq
C_{P}(\|\xi-\mathcal{Q}_{h}u\|_{1,h}+\|\mathcal{Q}_{h}u-u\|_{1,h})$
$\displaystyle\leq
C({|||}\xi-\mathcal{Q}_{h}u{|||}+h^{k}|u|_{H^{k+1}(\Omega)})$
$\displaystyle\leq Ch^{k}|u|_{H^{k+1}(\Omega)},$

where we have used $\xi\in\mathbb{B}_{h}(\mathcal{Q}_{h}u)$ in the last
inequality.

By Lemma 5, the triangle inequality and
$\theta\in\mathbb{B}_{h}(\mathcal{Q}_{h}u)$, we obtain

$\displaystyle\|\nabla_{w}\theta-\nabla
u\|_{L^{2}({\mathcal{T}}_{h})}\leq{|||}\theta-\mathcal{Q}_{h}u{|||}+\|\Pi_{h}(\nabla
u)-\nabla u\|_{L^{2}({\mathcal{T}}_{h})}\leq Ch^{k}|u|_{H^{k+1}(\Omega)}.$

Plugging those estimates back into (3.3) yields

(71) $\displaystyle|I_{1}|\leq
Ch^{k-1/2}|u|_{H^{k+1}(\Omega)}{|||}\xi-\theta{|||}{|||}v_{h}{|||}.$

Similarly, we have the following estimate for $I_{2}$:

$\displaystyle|I_{2}|$
$\displaystyle\leq|((\xi_{0}-\theta_{0})^{2}\widetilde{a}_{uu}(\xi_{0},\theta_{0})\nabla
u,\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}|$
$\displaystyle\quad+|((\xi_{0}-\theta_{0})(\theta_{0}-u)\widetilde{a}_{uu}(\theta_{0},u)\nabla
u,\nabla_{w}v_{h})_{{\mathcal{T}}_{h}}|$ $\displaystyle\leq
M_{a}|u|_{W_{\infty}^{1}(\Omega)}\|\xi_{0}-\theta_{0}\|_{L^{4}({\mathcal{T}}_{h})}(\|\xi_{0}-\theta_{0}\|_{L^{4}({\mathcal{T}}_{h})}+\|\theta_{0}-u\|_{L^{4}({\mathcal{T}}_{h})})\|\nabla_{w}v_{h}\|_{L^{2}({\mathcal{T}}_{h})}$
$\displaystyle\leq
Ch^{k}|u|_{H^{k+1}(\Omega)}{|||}\xi-\theta{|||}{|||}v_{h}{|||},$

which together with (71) leads to

(72) $\displaystyle|R_{h}(u;\xi,v_{h})-R_{h}(u;\theta,v_{h})|\leq
Ch^{k-1/2}|u|_{H^{k+1}(\Omega)}{|||}\xi-\theta{|||}{|||}v_{h}{|||}.$

Denote by $\chi_{h}=\phi_{h}-\psi_{h}$. Taking $v_{h}=\chi_{h}$ in (68) and
then using the Gårding’s inequality (19), we find

$\displaystyle\gamma({|||}\chi_{h}{|||}^{2}+\|\chi_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2})$
$\displaystyle\leq\beta\|\chi_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}+D_{h}(u;\chi_{h},\chi_{h})$
$\displaystyle\leq\beta\|\chi_{0}\|_{L^{2}({\mathcal{T}}_{h})}^{2}+Ch^{k-1/2}|u|_{H^{k+1}(\Omega)}{|||}\xi-\theta{|||}{|||}\chi_{h}{|||},$

which implies

(73) $\displaystyle\gamma{|||}\chi_{h}{|||}$
$\displaystyle\leq\beta\|\chi_{0}\|_{L^{2}({\mathcal{T}}_{h})}+Ch^{k-1/2}|u|_{H^{k+1}(\Omega)}{|||}\xi-\theta{|||}.$

Then an application of duality argument as in Lemma 15 yields

$\displaystyle\|\chi_{0}\|_{L^{2}({\mathcal{T}}_{h})}\leq
C(h{|||}\chi_{h}{|||}+h^{k}|u|_{H^{k+1}(\Omega)}{|||}\xi-\theta{|||}),$

which combining with (73) implies

$\displaystyle{|||}\chi_{h}{|||}\leq
Ch^{k-1/2}|u|_{H^{k+1}(\Omega)}{|||}\xi-\theta{|||}.$

Let $\rho(h)=Ch^{k-1/2}|u|_{H^{k+1}(\Omega)}$. Since $k\geq 1$, then we have
$\rho(h)<1$ for sufficiently small $h$. The proof is completed. ∎

###  3.4 Error estimates

According to Theorem 17 and 18 and Brouwer’s fixed point theorem, we can
deduce that $\mathcal{F}_{h}$ has a unique fixed point $u_{h}$ in
$\mathbb{B}_{h}(\mathcal{Q}_{h}u)$, which is also the solution of the
nonlinear WG finite element scheme (5). By the definition of the ball
$\mathbb{B}_{h}(\mathcal{Q}_{h}u)$, the finite element approximation $u_{h}$
of $u$ satisfies the bound

(74) $\displaystyle{|||}\mathcal{Q}_{h}u-u_{h}{|||}\leq
Ch^{k}|u|_{H^{k+1}(\Omega)},$

for sufficiently small $h$. Furthermore, Theorem 16 gives

(75) $\|\mathcal{Q}_{0}u-u_{0}\|_{L^{2}({\mathcal{T}}_{h})}\leq
C(h^{k+1}+h^{2k}|u|_{H^{k+1}(\Omega)})|u|_{H^{k+1}(\Omega)},$

for sufficiently small $h$.

By the triangle inequality, Lemma 2, (14), (74) and (75), we immediately have
the following results.

######  Theorem 19.

Suppose that $u\in H^{k+1}(\Omega)\cap W_{\infty}^{1}(\Omega)$. Then there
exists a constant $C>0$, independent of $h$, such that

$\|u-u_{h}\|_{1,h}\leq Ch^{k}|u|_{H^{k+1}(\Omega)},$

and

$\|u-u_{0}\|_{L^{2}({\mathcal{T}}_{h})}\leq
C(h^{k+1}+h^{2k}|u|_{H^{k+1}(\Omega)})|u|_{H^{k+1}(\Omega)},$

for sufficiently small $h$.

##  4 Two-grid algorithm of the WG method

In this section, we propose a two-grid algorithm of the WG method for the
quasilinear elliptic problem (1a)-(1b). Let ${\mathcal{T}}_{H}$ and
${\mathcal{T}}_{h}$ be two shape-regular partitions of the domain $\Omega$,
respectively, with different mesh sizes $H$ and $h$ ($H>h$). And the
corresponding WG finite element spaces $V_{H}$ and $V_{h}$ will be called
coarse and fine space, respectively. In the applications given below, we shall
always assume that

$H=O(h^{\lambda})\quad\mbox{for some}\;0<\lambda<1.$

With the discrete variational form $A_{h}(\cdot;\cdot,\cdot)$ defined in (6),
let us introduce a two-grid algorithm of the WG method (5) as follows:

######  Algorithm 2 (Two-grid WG method).

  * Step 1.

On the coarse mesh ${\mathcal{T}}_{H}$, seek $u_{H}\in V_{H}$ such that

$A_{H}(u_{H};u_{H},v)=(f,v_{0}),\quad\forall\;v=\\{v_{0},v_{b}\\}\in
V_{H}^{0}.$

  * Step 2.

On the fine mesh ${\mathcal{T}}_{h}$, seek $u^{h}\in V_{h}$ such that

$A_{h}(u_{H};u^{h},v)=(f,v_{0}),\quad\forall\;v=\\{v_{0},v_{b}\\}\in
V_{h}^{0}.$

In this algorithm, we firstly use the WG metod solve the quasilinear elliptic
problem on a coarse space $V_{H}$, and obtain a rough approximation $u_{H}\in
V_{h}$; and then use $u_{H}$ to linearize the nonlinear scheme on the fine
space $V_{h}$, and solve the resulting linearized problem to get $u^{h}\in
V_{h}$.

In order to prove the convergence of Algorithm 2, we introduce the following
two lemmas.

######  Lemma 20.

Let $u\in H^{2}(\Omega)\cap W_{\infty}^{1}(\Omega)$ and $u_{h}\in V_{h}$ be
the solutions of problem (1a)-(1b) and the discrete problem (5), respectively.
Then, there holds

$\displaystyle\|\nabla_{w}u_{h}\|_{L^{\infty}({\mathcal{T}}_{h})}\leq C(u),$

where $C(u)$ is a generic positive constant, which is independent of $h$ but
dependent on $|u|_{H^{2}(\Omega)}$ and $|u|_{W_{\infty}^{1}(\Omega)}$.

###### Proof.

By the triangle inequality and Lemma 5, we have

(76)
$\displaystyle\|\nabla_{w}u_{h}\|_{L^{\infty}({\mathcal{T}}_{h})}\leq\|\nabla_{w}(u_{h}-\mathcal{Q}_{h}u)\|_{L^{\infty}({\mathcal{T}}_{h})}+\|\Pi_{h}(\nabla
u)\|_{L^{\infty}({\mathcal{T}}_{h})}.$

Using the inverse inequality (11) and (74), we get

(77)
$\displaystyle\|\nabla_{w}(u_{h}-\mathcal{Q}_{h}u)\|_{L^{\infty}({\mathcal{T}}_{h})}\leq
Ch^{-1}\|\nabla_{w}(u_{h}-\mathcal{Q}_{h}u)\|_{L^{2}({\mathcal{T}}_{h})}\leq
C|u|_{H^{2}(\Omega)}.$

In view of the approximation property (15), we have

$\displaystyle\|\Pi_{h}(\nabla u)\|_{L^{\infty}({\mathcal{T}}_{h})}$
$\displaystyle\leq\|\Pi_{h}(\nabla u)-\nabla
u\|_{L^{\infty}({\mathcal{T}}_{h})}+\|\nabla u\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq C|u|_{H^{2}(\Omega)}+|u|_{W_{\infty}^{1}(\Omega)},$

which combining with (76) and (77) completes the proof. ∎

######  Lemma 21.

For a given $\phi\in H^{1}({\mathcal{T}}_{h})$, there holds

$\displaystyle
A_{h}(\phi;v_{h},v_{h})\geq\min\\{\alpha_{0},1\\}{|||}v_{h}{|||},\quad\forall\;v_{h}\in
V_{h}^{0}.$

###### Proof.

The proof is trivial, we omit it here. ∎

Now, we are ready to prove convergence estimates in the $H^{1}$-like norm
$\|\cdot\|_{1,h}$ for the two-grid algorithm of the WG method under the
assumption $u\in H^{k+1}(\Omega)\cap W_{\infty}^{1}(\Omega)$.

######  Theorem 22.

Assume that $u\in H^{k+1}(\Omega)\cap W_{\infty}^{1}(\Omega)$. Let $u_{h}\in
V_{h}$ and $u^{h}\in V_{h}$ be the solutions obtained by Algorithm 1 and 2,
respectively. For $H\ll 1$, we have

(78) $\displaystyle\|u-u^{h}\|_{1,h}$ $\displaystyle\leq C(u)(h^{k}+H^{k+1}),$

where $C(u)$ is a positive constant, which is independent of $h$ but dependent
on $|u|_{H^{k+1}(\Omega)}$ and $|u|_{W_{\infty}^{1}(\Omega)}$.

###### Proof.

From (5) and the Step 2 of Algorithm 2, we know that

$\displaystyle
A_{h}(u_{H};u^{h},v)=(f,v_{0})=A_{h}(u_{h};u_{h},v),\quad\forall\;v\in
V_{h}^{0},$

and thus using the definition of $A_{h}(\cdot;\cdot,\cdot)$, we have

$\displaystyle A_{h}(u_{H};u^{h}-u_{h},v)$
$\displaystyle=A_{h}(u_{H};u^{h},v)-A_{h}(u_{H};u_{h},v)$
$\displaystyle=A_{h}(u_{h};u_{h},v)-A_{h}(u_{H};u_{h},v)$
$\displaystyle=((a(u_{h,0})-a(u_{H,0}))\nabla_{w}u_{h},\nabla_{w}v)_{{\mathcal{T}}_{h}}$
(79) $\displaystyle=S_{1}+S_{2},$

with

$\displaystyle S_{1}$
$\displaystyle=((a(u_{h,0})-a(u))\nabla_{w}u_{h},\nabla_{w}v)_{{\mathcal{T}}_{h}},$
$\displaystyle S_{2}$
$\displaystyle=((a(u)-a(u_{H,0}))\nabla_{w}u_{h},\nabla_{w}v)_{{\mathcal{T}}_{h}}.$

By the Taylor expansion (16) and Hölder’s inequality, we obtain

$\displaystyle|S_{1}|$
$\displaystyle=|(\widetilde{a}_{u}(u_{h,0},u)(u_{h,0}-u)\nabla_{w}u_{h},\nabla_{w}v)_{{\mathcal{T}}_{h}}|$
$\displaystyle\leq
M_{a}\|u_{h,0}-u\|_{L^{2}({\mathcal{T}}_{h})}\|\nabla_{w}u_{h}\|_{L^{\infty}({\mathcal{T}}_{h})}\|\nabla_{w}v\|_{L^{2}({\mathcal{T}}_{h})},$

then using Lemma 20 and Theorem 19 yields

(80) $\displaystyle|S_{1}|\leq C(u)h^{k+1}{|||}v{|||}.$

In a similar way, we obtain

$\displaystyle|S_{2}|\leq C(u)H^{k+1}{|||}v{|||},$

which together with (80) and (4) leads to

$\displaystyle A_{h}(u_{H};u^{h}-u_{h},v)\leq C(u)H^{k+1}{|||}v{|||}.$

Taking $v=u^{h}-u_{h}$ in the above equation and using Lemma 21 and 2 yields

$\displaystyle\|u_{h}-u^{h}\|_{1,h}$ $\displaystyle\leq C(u)H^{k+1},$

which is combining with Theorem 19 completes the proof. ∎

######  Remark 4.1.

Theorem 22 suggests that optimal order of convergence in the two-grid WG
method with $k=1$ can be achieved by employing $H=O(h^{1/2})$.

##  5 Numerical Experiments

In this section, we conduct some numerical experiments to verify the
theoretical predication on the WG method (5) and to demonstrate the efficiency
of the two-grid WG method described in Algorithm 2.

The WG scheme (5) yields a nonlinear algebraic system, which will be solved by
Newton’s method. In our numerical experiments, the initial guess for Newton’s
iteration is taken to be the zero solution, i.e., $u_{h}^{(0)}=0$. The Newton
iteration is continued until a tolerance of
${|||}u_{h}^{(k+1)}-u_{h}^{(k)}{|||}<10^{-12}$ is reached, where $u_{h}^{(k)}$
and $u_{h}^{(k+1)}$ are two successive iterative solutions, respectively.

The following two examples are used for our numerical experiments.

######  Example 1.

Consider the model problem (1a)-(1b) with $a(u)=1+u$ in the domain
$\Omega=(0,1)^{2}$. The source data $f$ and Dirichlet data $g$ are chosen so
that the exact solution is

$u=\sin(\pi x)\sin(\pi y).$

######  Example 2.

Consider the model problem (1a)-(1b) with $a(u)=1+\sin(u)/2$ in the domain
$\Omega=(0,1)^{2}$. The source data $f$ and Dirichlet data $g$ are chosen so
that the exact solution is

$u=\varphi(x)\varphi(y),$

with $\varphi(t)=t(1-t)e^{2t}$.

###  5.1 Accuracy test for the WG method

In this subsection, we conduct some numerical tests for Example 1 and 2 to
validate the theoretical results predicated in Theorem 19 for the WG method
(5). We shall consider rectangular grids and polygonal grids. A least-squares
fit is conducted for computing the numerical order of convergence.

(a) The first level grid with $4\times 4$ elements

(b) The second level grid with $8\times 8$ elements

Fig. 1: The demo of polygonal grids.

(a) Rectangular grid with $4\times 4$ elements

(b) Rectangular grid with $8\times 8$ elements

Fig. 2: The demo of Rectangular grids.

Figure 1 shows the first two level of polygonal grids used in our computation,
which are generated by the MATLAB software package PolyMesher [15]. The
rectangular grid shown in Figure 2 is obtained by dividing the domain
$\Omega=(0,1)^{2}$ into $N\times N$ uniform small rectangles.

In Table 1, we report the errors and the orders of convergence for the WG
finite element solutions with $k=1$ and $k=2$ for Example 1. It is expected to
see the order of convergence predicted by Theorem 19, that is, $k$ for
$\|u-u_{h}\|_{1,h}$ and $k+1$ for $\|u-u_{0}\|_{L^{2}}$, provided $k\geq 1$.
We can see in Table 1 that these computed orders of convergence are full in
agreement with the theoretical findings in Theorem 19.

Table 1: Error profiles and convergence rates of the WG solution for Example 1. $k$ | Mesh | Polygonal grid | Rectangular grid
---|---|---|---
$\|u-u_{h}\|_{1,h}$ | $\|u-u_{0}\|_{L^{2}}$ | $\|u-u_{h}\|_{1,h}$ | $\|u-u_{0}\|_{L^{2}}$
1 | $4\times 4$ | 1.38E+00 | 1.56E-01 | 1.63E+00 | 2.05E-01
$8\times 8$ | 7.52E-01 | 4.59E-02 | 8.66E-01 | 5.78E-02
$16\times 16$ | 3.91E-01 | 1.19E-02 | 4.39E-01 | 1.48E-02
$32\times 32$ | 1.97E-01 | 2.96E-03 | 2.20E-01 | 3.74E-03
$64\times 64$ | 9.73E-02 | 7.17E-04 | 1.10E-01 | 9.35E-04
Rate | 0.96 | 1.95 | 0.97 | 1.95
2 | $4\times 4$ | 4.59E-01 | 3.44E-02 | 5.31E-01 | 4.37E-02
$8\times 8$ | 1.13E-01 | 4.30E-03 | 1.39E-01 | 5.44E-03
$16\times 16$ | 2.80E-02 | 5.08E-04 | 3.58E-02 | 6.65E-04
$32\times 32$ | 6.99E-03 | 6.29E-05 | 9.09E-03 | 8.21E-05
$64\times 64$ | 1.76E-03 | 7.90E-06 | 2.29E-03 | 1.02E-05
Rate | 2.01 | 3.03 | 1.97 | 3.02

Table 2 shows the errors and the orders of convergence for the WG finite
element solutions with $k=1$ and $k=2$ for Example 2. It is observed that
these computed orders of convergence match with the theoretical order of
convergence predicated in Theorem 19 again.

Table 2: Error profiles and convergence rates of the WG solution for Example 2. $k$ | Mesh | Polygonal grid | Rectangular grid
---|---|---|---
$\|u-u_{h}\|_{1,h}$ | $\|u-u_{0}\|_{L^{2}}$ | $\|u-u_{h}\|_{1,h}$ | $\|u-u_{0}\|_{L^{2}}$
1 | $4\times 4$ | 1.46E+00 | 1.85E-01 | 1.58E+00 | 2.10E-01
$8\times 8$ | 6.84E-01 | 3.98E-02 | 8.42E-01 | 5.57E-02
$16\times 16$ | 3.18E-01 | 8.38E-03 | 4.30E-01 | 1.42E-02
$32\times 32$ | 1.58E-01 | 2.05E-03 | 2.16E-01 | 3.56E-03
$64\times 64$ | 8.31E-02 | 5.55E-04 | 1.08E-01 | 8.92E-04
Rate | 1.03 | 2.10 | 0.97 | 1.97
2 | $4\times 4$ | 3.26E-01 | 2.16E-02 | 3.59E-01 | 2.64E-02
$8\times 8$ | 8.84E-02 | 3.12E-03 | 1.18E-01 | 3.88E-03
$16\times 16$ | 2.40E-02 | 4.29E-04 | 3.34E-02 | 5.07E-04
$32\times 32$ | 5.85E-03 | 4.81E-05 | 8.81E-03 | 6.35E-05
$64\times 64$ | 1.57E-03 | 6.44E-06 | 2.25E-03 | 7.92E-06
Rate | 1.93 | 2.94 | 1.84 | 2.93

###  5.2 Efficiency test for the two-grid WG method

In this subsection, we shall carry out the two-grid WG method stated in
Algorithm 2 with $k=1$ on rectangular grids. We will take the coarse mesh size
$H=h^{1/2}$, where $h$ is the fine mesh size. In this case, the theoretical
result predicated by Theorem 22 gives

$\displaystyle\|u-u^{h}\|_{1,h}\leq Ch,$

where $u^{h}$ is the numerical solution obtained by employing the two-grid WG
method on the fine mesh ${\mathcal{T}}_{h}$.

The computation is performed on a laptop with an Intel Core i7-7Y75 CPU at
1.60 GHz. In Table 3 and 4, we report the errors in $H^{1}$-like norm
$\|\cdot\|_{1,h}$ of the WG method and the two-grid WG method on the same mesh
level and their computational times (in second) for Example 1 and 2. It can be
observed that (1) the two-grid WG method has the convergence rate of $O(h)$ in
the $H^{1}$-like norm $\|\cdot\|_{1,h}$, which is expected as Theorem 22; (2)
both the WG method and the two-grid WG method have the similar accuracy on the
same mesh level; (3) the cost of time for the two-grid WG method is
significantly less than the one of WG method.

Table 3: Comparisons of the errors and the CPU times of the WG method and the two-grid WG method for Example 1, $H=h^{1/2}$. $h$ | WG method | Two-grid WG method
---|---|---
$\|u-u_{h}\|_{1,h}$ | CPU time (sec.) | $\|u-u^{h}\|_{1,h}$ | CPU time (sec.)
$1/4$ | 1.63E+00 | 0.7031 | 1.66E+00 | 0.4375
$1/16$ | 4.39E-01 | 7.2031 | 4.76E-01 | 2.0469
$1/36$ | 1.96E-01 | 33.4375 | 2.24E-01 | 6.8125
$1/64$ | 1.10E-01 | 108.0156 | 1.28E-01 | 19.2500
$1/100$ | 7.06E-02 | 271.5781 | 8.29E-02 | 53.8750
Rate | 0.98 |  | 0.93 |
Table 4: Comparisons of the errors and the CPU times of the WG method and the two-grid WG method for Example 2, $H=h^{1/2}$. $h$ | WG method | Two-grid WG method
---|---|---
$\|u-u_{h}\|_{1,h}$ | CPU time (sec.) | $\|u-u^{h}\|_{1,h}$ | CPU time (sec.)
$1/4$ | 1.58E+00 | 0.8125 | 1.57E+00 | 0.8594
$1/16$ | 4.30E-01 | 7.8438 | 4.79E-01 | 2.9063
$1/36$ | 1.92E-01 | 44.6563 | 2.25E-01 | 10.0000
$1/64$ | 1.08E-01 | 111.9688 | 1.28E-01 | 23.9063
$1/100$ | 6.93E-02 | 283.5781 | 8.26E-02 | 71.6719
Rate | 0.97 |  | 0.91 |

Declarations

Conflict of interest The authors declare no conflict interests.

## References

  * [1] C. Bi and V. Ginting, Two-grid finite volume element method for linear and nonlinear elliptic problems, Numer. Math., 108(2007), 177-198.
  * [2] C. Bi and V. Ginting, Two-grid discontinuous Galerkin method for quasi-linear elliptic problems, J. Sci. Comput., 49(2011), 311-331.
  * [3] P. Ciarlet, The finite element method for elliptic problems, North-Holland, New York, 1978.
  * [4] J. Douglas and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem, Math. Comput., 29(1975), 689-696.
  * [5] T. Gudi and A. K. Pani, Discontinuous Galerkin methods for quasi-linear elliptic problems of nonmonotone type, SIAM J. Numer. Anal., 45(2007), 163-192.
  * [6] T. Gudi, N. Nataraj, and A. K. Pani, An hp-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type, Math. Comput., 77(2008), 731-756.
  * [7] S. C. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^{1}$ functions, SIAM J. Numer. Anal., 41(2003), 306-324.
  * [8] A. Lasis and E. Süli, Poincaré-Type inequalities for broken Sobolev spaces, Isaac Newton Institute for Mathematical Sciences, Preprint No. NI03067-CPD, (2003).
  * [9] F. A. Milner, Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comput., 44(1985), 303-320.
  * [10] C. Ortner and E. Süli, Discontinuous Galerkin finite element approximation of nonlinear second-order elliptic and hyperbolic systems, SIAM J. Numer. Anal., 45(2007), 1370-1397.
  * [11] J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33(1996), 1759-1777.
  * [12] J. Xu, A novel two-grid method for semi-linear equations, SIAM J. Sci. Comput., 15(1994), 231-237.
  * [13] E. J. Park, Mixed finite element methods for nonlinear second-order elliptic problems, SIAM J. Numer. Anal., 32(1995), 865-885.
  * [14] S. Sun, Z. Huang and C. Wang, Weak Galerkin finite element method for a class of quasilinear elliptic problems, Appl. Math. Letters, 79(2018), 67-72.
  * [15] C. Talischi, G. H. Paulino, A. Pereira, I. F. M. Menezes, PolyMesher: a general-purpose mesh generator for polygonal elements written in Maltab, Struct. Multidiscip. Optim., 45(3)(2012), 309-328.
  * [16] F. Gao and L. Mu, On $L^{2}$ error estimate for weak Galerkin finite element methods for parabolic problems, J. Comput. Math., 32(2)(2014), 195-204.
  * [17] Q. Guan, M. Gunzburger and W. Zhao, Weak-Galerkin finite element methods for a second-order elliptic variational inequlaity, Computer Methods Appl. Mech. Engrg., 337(2018), 677-688.
  * [18] R. Lin, X. Ye, S. Zhang and P. Zhu, A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems, SIAM J. Numer. Anal., 56(2018), 1482-1497.
  * [19] L. Mu, J. Wang, and X. Ye, Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12(2015), 31-53.
  * [20] L. Mu, J. Wang, and X. Ye, A weak Galerkin finite element method with polynomial reduction, J. Comput. Appl. Math., 285(2015), 45-58.
  * [21] L. Mu, J. Wang, and X. Ye, A weak Galerkin finite element method for biharmonic equations on polytopal meshes, Numer. Meth. PDE, 30 (2014), 1003-1029.
  * [22] L. Mu, J. Wang, X. Ye, S. Zhang, A $C^{0}$ Weak Galerkin finite element methods for the biharmonic equation, J. Sci. Comput., 59 (2014), 437-495.
  * [23] L. Mu, X. Ye and S. Zhang, Development of a $P_{2}$ element with optimal $L^{2}$ convergence for biharmonic equation, Numer. Meth. PDE, 21 (2019), 1497-1508.
  * [24] L. Mu, J. Wang and X. Ye, weak Galerkin finite element method for the Helmholtz equation with large wave number on polytopal meshes. IMA J. Numer. Anal., 35 (2015), 1228-1255.
  * [25] L. Mu, J. Wang, X. Ye and S. Zhang, A weak Galerkin finite element method for the Maxwell equations. J. Sci. Comput., 65 (2015), 363-386.
  * [26] L. Mu, J. Wang, X. Ye and S. Zhao, A new weak Galerkin finite element method for elliptic interface problems. J. Comput. Phy., 325 (2016), 157-173.
  * [27] C. Wang and J. Wang, An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes, Comput. Math. Appl., 68(2014), 2314-2330.
  * [28] J. Wang and X. Ye, A Weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp., 83 (2014), 2101-2126.
  * [29] J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comp. Appl. Math. 241 (2013), 103-115.
  * [30] X. Ye, S. Zhang and P. Zhu, A weak Galerkin finite element method for nonlinear conservation laws, Electronic Research Archive, 2020, doi:10.3934/era.2020097.
  * [31] X. Ye and S. Zhang, A stabilizer-free weak Galerkin finite element method on polytopal meshes, J. Comput. Appl. Math., 372 (2020), 112699, arXiv:1906.06634.
  * [32] X. Ye and S. Zhang, A stabilizer free weak Galerkin finite element method for the biharmonic equation on polytopal meshes, SIAM J. Numer. Anal., 58(2020), no. 5, 2572-2588.
  * [33] R. Zhang and Q. Zhai, A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order, J. Sci. Comput., 64 (2015), no. 2, 559-585.
  * [34] X. Ye, S. Zhang and Y. Zhu, Stabilizer-free weak Galerkin methods for monotone quasilinear elliptic PDEs, Results in Applied Mathematics, https://doi.org/10.1016/j.rinam.2020.100097