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Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 16, 2004, 79 – 89

Abdur Rashid
THE PSEUDOSPECTRAL METHOD FOR THERMOTROPIC PRIMITIVE EQUATION AND ITS ERROR ESTIMATION
(submitted by A. Lapin)

ABSTRACT. In this paper, a pseudospectral method is proposed for solving the periodic problem of thermotropic primitive equation. The strict error estimation is proved.

 ________________ 2000 Mathematical Subject Classification. 35Q35, 65M70,65N30.. Key words and phrases. Thermotropic primitive equation, pseudospectral scheme, error estimation. This work is supported by Gomal University, D.I.Khan, Pakistan..

### 1. Introduction

Thermotropic primitive equation is governed by the following differential equations${}^{\left[1\right]}$:

 $\left\{\begin{array}{c}\frac{\partial U}{\partial t}+U\frac{\partial U}{\partial x}+V\frac{\partial U}{\partial y}+\frac{\partial \phi }{\partial x}-\nu \Delta U-FV=0,\hfill \\ \frac{\partial V}{\partial t}+U\frac{\partial V}{\partial x}+V\frac{\partial V}{\partial y}+\frac{\partial \phi }{\partial y}-\nu \Delta V+FU=0,\hfill \\ \frac{\partial \phi }{\partial t}+U\frac{\partial \phi }{\partial x}+V\frac{\partial \phi }{\partial y}+\phi \left(\frac{\partial U}{\partial x}+\frac{\partial V}{\partial x}\right)=0,\hfill \\ \hfill \end{array}\right\$ (1.1)

where U,V are the components of the speed in x, y directions respectively, g is the acceleration of gravity, H is the height of the geopotential surface, $\phi =gH$, F is coriolis parameter and $\nu$ is the coefficient of friction.
There has been a rapid development in the spectral methods for the last two decades. They have become important tools for numerical solutions of partial differential equations, and have been widely applied to numerical simulations in various fields [2-5]. Although the pseudospectral methods are easier to implement for nonlinear partial differential equations, they are not stable as the spectral ones due to ’aliasing’. Therefore some author proposed the filtering technique [10-11] to remedy the deficiency of instability. Some papers have also been devoted to theoretical study and numerical solutions of (1.1) [6-9].
The aim of this paper is to consider the periodic initial boundary-value problem for thermotropic primitive equation. A pseudospectral scheme with restraint operator in combination with first order time differencing technique is considered for thermotropic primitive equation. The stability and rate of convergence for the approximate problem are proved.

### 2. The Pseudospectral Scheme

Let $\Omega =\left\{\left(x,y\right)\mid -\pi and all functions have the period 2$\pi$ for the variable x and y. The norm of the space ${L}^{q}\left(\Omega \right)$ is denoted by ${∥\cdot ∥}_{{L}^{q}\left(\Omega \right)}$. In particular, the scaler product and the norm of ${L}^{2}\left(\Omega \right)$ are denoted by $\left(\cdot ,\cdot \right)$ and ${∥\cdot ∥}_{{L}^{2}\left(\Omega \right)}$ respectively. Let ${m}_{1},{m}_{2}$ and N be integers and $m=\sqrt{{m}_{1}^{2}+{m}_{2}^{2}}$. Define

${V}_{N}=Span\left\{{e}^{i\left({m}_{1}x+{m}_{2}y\right)}\mid \phantom{\rule{3.26288pt}{0ex}}\mid m\mid \le N\right\},\phantom{\rule{1em}{0ex}}N>0.$

Let ${P}_{N}$ be the orthogonal projection operator, i.e.

$\left({P}_{N}\eta ,\psi \right)=\left(\eta ,\psi \right),\phantom{\rule{1em}{0ex}}\forall \psi \in {V}_{N}.$

For the pseudospectral approximation, we put the nodes

$\left({x}_{{j}_{1}},{y}_{{j}_{2}}\right)=\left(\frac{2\pi {j}_{1}}{2N+1},\frac{2\pi {j}_{2}}{2N+1}\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-N\le {j}_{1},{j}_{2}\le N,$

and let $\stackrel{˜}{{P}_{c}}$ be the interpolation operator, i.e. for $\eta \left(x,y\right)\in C\left(\Omega \right)$

$\stackrel{˜}{{P}_{c}}\eta \left({x}_{{j}_{1}},{y}_{{j}_{2}}\right)=\eta \left({x}_{{j}_{1}},{y}_{{j}_{2}}\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-N\le {j}_{1},{j}_{2}\le N.$

Define ${P}_{c}={P}_{N}\stackrel{˜}{{P}_{c}}$. To weaken the nonlinear instability of computation, we follow the work of  to adopt the filtering operator ${R}_{\gamma }$ with $\gamma >1$, i.e. if

$\eta \left(x,y\right)=\sum _{\mid m\mid \le N}{\eta }_{{m}_{1},{m}_{2}}{e}^{i\left({m}_{1}x+{m}_{2}y\right)},$

then

${R}_{\gamma }\eta \left(x,y\right)=\sum _{\mid m\mid \le N}\left(1-{\left(\frac{\mid m\mid }{N}\right)}^{\gamma }\right){\eta }_{{m}_{1},{m}_{2}}{e}^{i\left({m}_{1}x+{m}_{2}y\right)}.$

Let $\tau$ be the mesh spacing of the variable t and define

${S}_{\tau }=\left\{t=k\tau \mid k=0,1,2,\cdot \cdot \cdot \right\}.$

${\eta }_{t}\left(t\right)=\frac{\eta \left(t+\tau \right)-\eta \left(t\right)}{\tau }.$

To approximate the nonlinear terms, we define

${d}_{\alpha }\left(\eta ,u,v\right)=\alpha {d}^{\left(1\right)}\left(\eta ,u,v\right)+\left(1-\alpha \right){d}^{\left(2\right)}\left(\eta ,u,v\right),\phantom{\rule{2em}{0ex}}0\le \alpha \le 1,$

${d}^{\left(1\right)}\left(\eta ,u,v\right)={P}_{c}\left(u\frac{\partial \eta }{\partial x}+v\frac{\partial \eta }{\partial y}\right),$

${d}^{\left(2\right)}\left(\eta ,u,v\right)=\frac{\partial }{\partial x}{P}_{c}\left(u\eta \right)+\frac{\partial }{\partial y}{P}_{c}\left(v\eta \right).$

Let ${u}^{N},{v}^{N},{\varphi }^{N}$ be the approximations to U, V and $\phi$ respectively, where for all $\left(x,y\right)\in \Omega$ and $t\in {S}_{\tau },$

${\eta }^{N}\left(x,y,t\right)=\sum _{\mid m\mid ,\mid n\mid \le N}{\eta }_{m,n}^{N}\left(t\right){e}^{i\left(mx+ny\right)},\phantom{\rule{2em}{0ex}}\eta =u,v,\varphi .$

The pseudospectral scheme for solving (1.1) is

 $\left\{\begin{array}{c}{u}_{t}^{N}+{R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }\left({u}^{N}+\delta \tau {u}_{t}^{N}\right),{u}^{N},{v}^{N}\right)+\frac{\partial }{\partial x}{\varphi }^{N}-\nu \Delta \left({u}^{N}+\sigma \tau {u}_{t}^{N}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{3.26288pt}{0ex}}\phantom{\rule{2.6108pt}{0ex}}-F\left({v}^{N}+\delta \tau {v}_{t}^{N}\right)=0,\hfill \\ {v}_{t}^{N}+{R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }\left({v}^{N}+\delta \tau {v}_{t}^{N}\right),{u}^{N},{v}^{N}\right)+\frac{\partial }{\partial y}{\varphi }^{N}-\nu \Delta \left({v}^{N}+\sigma \tau {v}_{t}^{N}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.6108pt}{0ex}}\phantom{\rule{3.26288pt}{0ex}}+F\left({u}^{N}+\delta \tau {u}_{t}^{N}\right)=0,\hfill \\ {\varphi }_{t}^{N}+{R}_{\gamma }{d}_{0}\left({R}_{\gamma }\left({\varphi }^{N}+\delta \tau {\varphi }_{t}^{N}\right),{u}^{N},{v}^{N}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.6108pt}{0ex}}\phantom{\rule{3.26288pt}{0ex}}+A\left({\varphi }^{N}+\delta \tau {\varphi }_{t}^{N},{u}^{N}+\delta \tau {u}_{t}^{N},{v}^{N}+\delta \tau {v}_{t}^{N}\right)=0,\hfill \\ \hfill \end{array}\right\$ (2.1)

where $0\le \delta \le 1,\phantom{\rule{1em}{0ex}}0\le \sigma \le 1$ and $A\left(\eta ,\xi ,{\eta }^{\ast }\right)={P}_{c}\left[\eta \left(\frac{\partial \xi }{\partial x}+\frac{\partial {\eta }^{\ast }}{\partial x}\right)\right].$

### 3. Some Lemmas

Lemma 1. $\left[1\right].$ For all $\eta \left(x,y,t\right)$

$2\left(\eta \left(t\right),{\eta }_{t}\left(t\right)\right)={\left({∥\eta \left(t\right)∥}^{2}\right)}_{t}-\tau {∥{\eta }_{t}\left(t\right)∥}^{2}.$

Lemma 2. $\left[5\right].$ For all $\eta \left(x,y,t\right)\in {V}_{N}$, then

${∥\frac{\partial \eta }{\partial x}∥}^{2}\le {N}^{2}{∥\eta \left(t\right)∥}^{2},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{∥\frac{\partial \eta }{\partial y}∥}^{2}\le {N}^{2}{∥\eta \left(t\right)∥}^{2}.$

Lemma 3. $\left[10\right].$ For all $\eta \left(x,y,t\right)\in {V}_{N}$ and $\xi \left(x,y,t\right)\in {V}_{N}$, then

${∥\eta \left(t\right)\xi \left(t\right)∥}^{2}\le {\left(2N+1\right)}^{2}{∥\eta \left(t\right)∥}^{2}{∥\xi \left(t\right)∥}^{2}.$

Lemma 4. $\left[15\right].$ For all $\eta \left(x,y,t\right)\in {H}^{\beta }\left(\Omega \right)$ and $\xi \left(x,y,t\right)\in {V}_{N}$, then

$\begin{array}{llll}\hfill {∥{P}_{N}\eta \left(t\right)-\eta \left(t\right)∥}_{{H}^{S}\left(\Omega \right)}& \le {C}_{1}{N}^{S-\beta }{∥\eta \left(t\right)∥}_{{H}^{\beta }\left(\Omega \right)},\phantom{\rule{2.6664pt}{0ex}}0\le s\le \beta ,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {∥{P}_{C}\eta \left(t\right)-\eta \left(t\right)∥}_{{H}^{S}\left(\Omega \right)}& \le {C}_{2}{N}^{S-\beta }{∥\eta \left(t\right)∥}_{{H}^{\beta }\left(\Omega \right)},\phantom{\rule{2.6664pt}{0ex}}0\le s\le \beta ,\phantom{\rule{2.6664pt}{0ex}}\beta >1,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {∥{R}_{\gamma }\xi \left(t\right)-\xi \left(t\right)∥}_{{H}^{S}\left(\Omega \right)}& \le {C}_{3}{N}^{S-\beta }{∥\xi \left(t\right)∥}_{{H}^{\beta }\left(\Omega \right)},\phantom{\rule{2.6664pt}{0ex}}0\le s\le \beta ,\phantom{\rule{2.6664pt}{0ex}}\gamma >\beta -s,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {∥{R}_{\gamma }{P}_{N}\eta \left(t\right)-\eta \left(t\right)∥}_{{H}^{S}\left(\Omega \right)}& \le {C}_{4}{N}^{S-\beta }{∥\eta \left(t\right)∥}_{{H}^{\beta }\left(\Omega \right)},\phantom{\rule{2.6664pt}{0ex}}0\le s\le \beta ,\phantom{\rule{2.6664pt}{0ex}}\gamma >\beta -s,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

where ${C}_{1}-{C}_{4}$ are positive constants.

Lemma 5. $\left[9\right].$ Assume that the following conditions are fulfilled:
(i) $\xi \left(t\right)$ and $\eta \left(t\right)$ are non-negative functions defined on ${S}_{\tau };$
(ii) $\rho ,a,{M}_{1},{M}_{2},$ and ${M}_{3}$ are nonnegative constants;
(iii) A(x) is a function such that, if $x\le {M}_{3}$, then $A\left(x\right)\le 0;$
(iv) $\xi \left(t\right)\le \rho +\tau {\sum }_{{t}^{\prime }=0}^{t-\tau }\left[{M}_{1}\xi \left({t}^{\prime }\right)+{M}_{2}{N}^{a}{\xi }^{2}\left({t}^{\prime }\right)+A\left(\xi \left({t}^{\prime }\right)\right)\eta \left({t}^{\prime }\right)\right];$
(v) $\rho {e}^{\left({M}_{1}+{M}_{2}\right)T}\le min\left({M}_{3},\frac{1}{{N}^{a}}\right)$, $\xi \left(0\right)\le \rho$, $t\le T$.
Then

$\xi \left(t\right)\le \rho {e}^{\left({M}_{1}+{M}_{2}\right)t}.$

In particular, if ${M}_{2}=0$ and $A\left(\xi \left({t}^{\prime }\right)\right)=0$, then for all $\rho$ and n.

$\xi \left(t\right)\le \rho {e}^{{M}_{1}t}.$

### 4. Error Estimation

For simplicity, we take $\delta =0,$ let ${U}^{N}={P}_{N}U,\phantom{\rule{3.26288pt}{0ex}}{V}^{N}={P}_{N}V,$ and ${\phi }^{N}={P}_{N}\phi$, then (1.1) leads to

 $\left\{\begin{array}{c}{U}_{t}^{N}+{R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }{U}^{N},{U}^{N},{V}^{N}\right)+\frac{\partial }{\partial x}{\phi }^{N}-\nu \Delta \left({U}^{N}+\sigma \tau {U}_{t}^{N}\right)-F{V}^{N}=0,\hfill \\ {V}_{t}^{N}+{R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }{V}^{N},{U}^{N},{V}^{N}\right)+\frac{\partial }{\partial y}{\phi }^{N}-\nu \Delta \left({V}^{N}+\sigma \tau {V}_{t}^{N}\right)+F{U}^{N}=0,\hfill \\ {\phi }_{t}^{N}+{R}_{\gamma }{d}_{0}\left({R}_{\gamma }{\phi }^{N},{U}^{N},{V}^{N}\right)+A\left({\phi }^{N},{U}^{N},{V}^{N}\right)=0,\hfill \end{array}\right\$ (4.1)

where

$\begin{array}{llll}\hfill {G}_{1}^{N}& ={U}_{t}^{N}-\frac{\partial {U}^{N}}{\partial t},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {G}_{2}^{N}& ={R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }{U}^{N},{U}^{N},{V}^{N}\right)-{P}_{N}\left[U\frac{\partial U}{\partial x}+V\frac{\partial U}{\partial y}\right],\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {G}_{3}^{N}& =-\nu \sigma \tau \Delta {U}_{t}^{N},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {G}_{4}^{N}& ={V}_{t}^{N}-\frac{\partial {V}^{N}}{\partial t},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {G}_{5}^{N}& ={R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }{V}^{N},{U}^{N},{V}^{N}\right)-{P}_{N}\left[U\frac{\partial V}{\partial x}+V\frac{\partial V}{\partial y}\right],\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {G}_{6}^{N}& =-\nu \sigma \tau \Delta {V}_{t}^{N},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {G}_{7}^{N}& ={\phi }_{t}^{N}-\frac{\partial {\phi }^{N}}{\partial t},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {G}_{8}^{N}& ={R}_{\gamma }{d}_{0}\left({R}_{\gamma }{\phi }^{N},{U}^{N},{V}^{N}\right)-{P}_{N}\left[U\frac{\partial \phi }{\partial x}+V\frac{\partial \phi }{\partial y}\right],\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {G}_{9}^{N}& =A\left({\phi }^{N},{U}^{N},{V}^{N}\right)-{P}_{N}\left[\phi \frac{\partial U}{\partial x}+\phi \frac{\partial V}{\partial x}\right].\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Put

$\stackrel{˜}{u}=u-{U}^{N},\phantom{\rule{1em}{0ex}}\stackrel{˜}{v}=v-{V}^{N},\phantom{\rule{1em}{0ex}}\stackrel{˜}{\varphi }=\varphi -{\phi }^{N}.$

Then from (1.1) and (2.1), we obtain

 $\left\{\begin{array}{c}{\stackrel{˜}{u}}_{t}^{N}+{\xi }_{1}^{N}+{\xi }_{2}^{N}+\frac{\partial }{\partial x}{\varphi }^{N}-\nu \Delta \left({\stackrel{˜}{u}}^{N}+\sigma \tau {\stackrel{˜}{u}}_{t}^{N}\right)-F{\stackrel{˜}{v}}^{N}=\sum _{l=1}^{3}{G}_{l}^{N},\hfill \\ {\stackrel{˜}{v}}_{t}^{N}+{\xi }_{3}^{N}+{\xi }_{4}^{N}+\frac{\partial }{\partial y}{\varphi }^{N}-\nu \Delta \left({\stackrel{˜}{v}}^{N}+\sigma \tau {\stackrel{˜}{v}}_{t}^{N}\right)+F{\stackrel{˜}{u}}^{N}=\sum _{l=4}^{6}{G}_{l}^{N},\hfill \\ {\stackrel{˜}{\varphi }}_{t}^{N}+\sum _{k=5}^{9}{\xi }_{k}^{N}=\sum _{l=7}^{9}{G}_{l}^{N},\hfill \end{array}\right\$ (4.2)

where

$\begin{array}{llll}\hfill {\xi }_{1}^{N}& ={R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }{\stackrel{˜}{u}}^{N},{U}^{N},{v}^{N}\right)+{R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }{U}^{N},{\stackrel{˜}{u}}^{N},{\stackrel{˜}{v}}^{N}\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\xi }_{2}^{N}& ={R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }{\stackrel{˜}{u}}^{N},{\stackrel{˜}{u}}^{N},{\stackrel{˜}{v}}^{N}\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\xi }_{3}^{N}& ={R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }{\stackrel{˜}{v}}^{N},{U}^{N},{V}^{N}\right)+{R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }{V}^{N},{\stackrel{˜}{u}}^{N},{\stackrel{˜}{v}}^{N}\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\xi }_{4}^{N}& ={R}_{\gamma }{d}_{1∕2}\left({R}_{\gamma }{\stackrel{˜}{v}}^{N},{\stackrel{˜}{u}}^{N},{\stackrel{˜}{v}}^{N}\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\xi }_{5}^{N}& ={R}_{\gamma }{d}_{0}\left({R}_{\gamma }{\stackrel{˜}{\varphi }}^{N},{U}^{N},{V}^{N}\right)+{R}_{\gamma }{d}_{0}\left({R}_{\gamma }{\phi }^{N},{\stackrel{˜}{u}}^{N},{\stackrel{˜}{v}}^{N}\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\xi }_{6}^{N}& ={R}_{\gamma }{d}_{0}\left({R}_{\gamma }{\stackrel{˜}{\varphi }}^{N},{\stackrel{˜}{u}}^{N},{\stackrel{˜}{v}}^{N}\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\xi }_{7}^{N}& =-A\left({\stackrel{˜}{\varphi }}^{N},{\stackrel{˜}{u}}^{N},{\stackrel{˜}{v}}^{N}\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\xi }_{8}^{N}& =-A\left({\stackrel{˜}{\varphi }}^{N},{U}^{N},{V}^{N}\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\xi }_{9}^{N}& =-A\left({\phi }^{N},{\stackrel{˜}{u}}^{N},{\stackrel{˜}{v}}^{N}\right).\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

We shall use the following notations

$E=\left(U,V,\phi \right),\phantom{\rule{1em}{0ex}}{E}^{N}=\left({U}^{N},{V}^{N},{\phi }^{N}\right),\phantom{\rule{1em}{0ex}}{\stackrel{˜}{E}}^{N}=\left({\stackrel{˜}{u}}^{N},{\stackrel{˜}{v}}^{N},{\stackrel{˜}{\varphi }}^{N}\right),$

${∥{\stackrel{˜}{E}}^{N}\left(t\right)∥}^{2}={∥{\stackrel{˜}{u}}^{N}\left(t\right)∥}^{2}+{∥{\stackrel{˜}{v}}^{N}\left(t\right)∥}^{2}+{∥{\stackrel{˜}{\varphi }}^{N}\left(t\right)∥}^{2},$
${∥{\stackrel{˜}{E}}_{t}^{N}\left(t\right)∥}^{2}={∥{\stackrel{˜}{u}}_{t}^{N}\left(t\right)∥}^{2}+{∥{\stackrel{˜}{v}}_{t}^{N}\left(t\right)∥}^{2}+{∥{\stackrel{˜}{\varphi }}_{t}^{N}\left(t\right)∥}^{2},$
${\left|{\stackrel{˜}{E}}^{N}\left(t\right)\right|}_{1}^{2}={\left|{\stackrel{˜}{u}}^{N}\left(t\right)\right|}_{1}^{2}+{\left|{\stackrel{˜}{v}}^{N}\left(t\right)\right|}_{1}^{2}+{\left|{\stackrel{˜}{\varphi }}^{N}\left(t\right)\right|}_{1}^{2}.$

Let ${H}^{\beta }\left(\Omega \right)$ be the Sobolev space equipped with the norm ${∥\cdot ∥}_{{H}^{\beta }\left(\Omega \right)}$. In particular ${L}^{2}\left(\Omega \right)={H}^{0}\left(\Omega \right)$, we define ${E}_{1}=\left(U,V,\phi \right)$

${∥{E}_{1}∥}_{{H}^{\beta }\left(\Omega \right)}^{2}={∥U\left(t\right)∥}_{{H}^{\beta }\left(\Omega \right)}^{2}+{∥V\left(t\right)∥}_{{H}^{\beta }\left(\Omega \right)}^{2}+{∥\phi \left(t\right)∥}_{{H}^{\beta }\left(\Omega \right)}^{2},$

$∥\mid {E}_{1}∥{\mid }_{\beta }={max}_{0\le t\le \tau }{∥{E}_{1}\left(t\right)∥}_{{H}^{\beta }\left(\Omega \right)}.$

Now we suppose

$\rho \left(t\right)=\parallel {\stackrel{˜}{E}}^{N}\left(0\right){\parallel }^{2}\right)+\nu \tau \left(\sigma +\frac{q}{2}\right)\mid {\stackrel{˜}{E}}^{N}\left(0\right){\mid }_{1}^{2}+\tau \sum _{{t}^{\prime }=0}^{t-\tau }\parallel {G}_{l}^{N}\left({t}^{\prime }\right){\parallel }^{2},\phantom{\rule{1em}{0ex}}l=1,...,9$

$\begin{array}{cc}\begin{array}{rl}\stackrel{˜}{E}\left(t\right)& =\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}+\nu \tau \left(\sigma +\frac{q}{2}\right)\mid {\stackrel{˜}{E}}^{N}\left(0\right){\mid }_{1}^{2}\\ & +\tau \sum _{{t}^{\prime }=0}^{t-\tau }{r}_{1}\tau \left[\parallel {\stackrel{˜}{E}}_{t}^{N}\left({t}^{\prime }\right){\parallel }^{2}+\nu \left(2-7\varepsilon \right)\mid {\stackrel{˜}{E}}^{N}\left({t}^{\prime }\right){\mid }_{1}^{2}\right].\end{array}& \end{array}$

Theorem 1. Suppose the following conditions are fulfilled
(i) $\delta =0,\phantom{\rule{1em}{0ex}}\tau {N}^{2}<\infty$,
(ii) $\sigma >1∕2\phantom{\rule{2.6664pt}{0ex}}\phantom{\rule{3.33237pt}{0ex}}or\phantom{\rule{2.6664pt}{0ex}}\phantom{\rule{3.33237pt}{0ex}}\tau {N}^{2}<\frac{2}{v\left(1-2\sigma \right)},$
(iii) for suitably small positive constant ${M}_{1}$ and all $t\le T$ , such that $\rho \left(T\right)\le \frac{{M}_{1}}{{N}^{2}}.$
Then there exists a positive constant ${M}_{2}$ such that for all $t\in {S}_{\tau }$, $t\le T$, we have

$\stackrel{˜}{E}\left(t\right)\le \rho \left(t\right){e}^{{M}_{2}t}.$

Theorem 2. Assume that the conditions (i), (ii) of Theorem 1 are satisfied. In addition $E\in {C}^{2}\left(0,T;{m}_{0}^{0}\left(\Omega \right)\right),{E}_{1}\in C\left(0,T;{H}_{\frac{5}{2}+r}\left(\Omega \right)\bigcap {m}_{0}^{B+1}\left(\Omega \right)\right),r>0,\phantom{\rule{1em}{0ex}}\beta \ge 1,$ then

$\stackrel{˜}{E}\left(t\right)\le {M}_{3}\left({\tau }^{2}+{N}^{-2\beta }\right){e}^{{M}_{4}t}.$

${M}_{\ell }$ being positive constants depending only on $\parallel \mid {E}_{1}\parallel {\mid }_{\frac{5}{2}+r}$ and $\nu$.

Now we define

$\parallel \eta \left(t\right){\parallel }_{{m}_{r}^{\beta }}={max}_{0\le s\le q}{\left(\frac{1}{4{\pi }^{2}}\sum _{a+b=0}^{\beta }\underset{\Omega }{\iint }{\left(\frac{{\partial }^{a+b}\eta \left(x,y,t\right)}{\partial {x}^{a}\partial {y}^{b}}\right)}^{2}dxdy\right)}^{1∕2},$

$\parallel \eta {\parallel }_{{C}^{q}\left(0,T;{m}_{r}^{\beta }\left(\Omega \right)\right)}={max}_{0\le s\le q}{max}_{0\le t\le T}{∥\frac{{\partial }^{s}\eta \left(t\right)}{\partial {t}^{s}}∥}_{{m}_{r}^{\beta }\left(\Omega \right)},$

${C}^{q}\left(0,T;{m}_{r}^{\beta }\left(\Omega \right)\right)=\left(\eta \left(x,y,t\right)\mid \phantom{\rule{1em}{0ex}}\parallel \eta {\parallel }_{{C}^{q}\left(0,T;{m}_{r}^{\beta }\left(\Omega \right)\right)}<\infty \right).$

### 5. The Proof of Theorem 1

Let $c$ be a positive constant which may be different in different cases, $q$ denote an undetermined positive constant and $\varepsilon >0$. Taking the scaler product (4.2) with $2{\stackrel{˜}{u}}^{N}+q\tau {\stackrel{˜}{u}}_{t}^{N}$, we have

 $\begin{array}{cc}\begin{array}{rl}{\left(\parallel {\stackrel{˜}{u}}^{N}\left(t\right){\parallel }^{2}\right)}_{t}& +\tau \left(q-1-\varepsilon \right)\parallel {\stackrel{˜}{u}}_{t}^{N}\left(t\right){\parallel }^{2}\\ & +\left(2{\stackrel{˜}{u}}^{N}\left(t\right)+q\tau {\stackrel{˜}{u}}_{t}^{N}\left(t\right),{\stackrel{˜}{\xi }}_{1}^{N}\left(t\right)+{\stackrel{˜}{\xi }}_{2}^{N}\left(t\right)\\ & +\frac{\partial {\stackrel{˜}{\phi }}^{N}}{\partial x}\left(t\right)+F{\stackrel{˜}{v}}^{N}\left(t\right)\right)+2\nu \mid {\stackrel{˜}{u}}^{N}\left(t\right){\mid }_{1}^{2}\\ & +\nu \tau \left(\sigma +\frac{q}{2}\right){\left(\mid \stackrel{˜}{u}\left(t\right){\mid }_{1}^{2}\right)}_{t}+\nu {\tau }^{2}\left(\sigma q-\sigma -\frac{q}{2}\right)\mid {\stackrel{˜}{u}}_{t}^{N}\left(t\right){\mid }_{1}^{2}\\ & \le c\parallel {\stackrel{˜}{u}}^{N}\left(t\right){\parallel }^{2}+c\left(1+\frac{{q}^{2}\tau }{4\varepsilon }\right)\sum _{l=1}^{3}\parallel {G}_{l}^{N}\left(t\right){\parallel }^{2}.\end{array}& \end{array}$ (5.1)

Similarly from the second and third formulas of (4.2), we have

 $\begin{array}{cc}\begin{array}{rl}{\left(\parallel {\stackrel{˜}{v}}^{N}\left(t\right){\parallel }^{2}\right)}_{t}& +\tau \left(q-1-\varepsilon \right)\parallel {\stackrel{˜}{v}}_{t}^{N}\left(t\right){\parallel }^{2}\\ & +\left(2{\stackrel{˜}{v}}^{N}\left(t\right)+q\tau {\stackrel{˜}{v}}_{t}^{N}\left(t\right),{\stackrel{˜}{\xi }}_{3}^{N}\left(t\right)+{\stackrel{˜}{\xi }}_{4}^{N}\left(t\right)\\ & +\frac{\partial {\stackrel{˜}{\phi }}^{N}}{\partial y}\left(t\right)+F{\stackrel{˜}{u}}^{N}\left(t\right)\right)+2\nu \mid {\stackrel{˜}{v}}^{N}\left(t\right){\mid }_{1}^{2}\\ & +\nu \tau \left(\sigma +\frac{q}{2}\right){\left(\mid \stackrel{˜}{v}\left(t\right){\mid }_{1}^{2}\right)}_{t}+\nu {\tau }^{2}\left(\sigma q-\sigma -\frac{q}{2}\right)\mid {\stackrel{˜}{v}}_{t}^{N}\left(t\right){\mid }_{1}^{2}\\ & \le c\parallel {\stackrel{˜}{v}}^{N}\left(t\right){\parallel }^{2}+c\left(1+\frac{{q}^{2}\tau }{4\varepsilon }\right)\sum _{l=4}^{6}\parallel {G}_{l}^{N}\left(t\right){\parallel }^{2}\end{array}& \end{array}$ (5.2)
 $\begin{array}{cc}\begin{array}{rl}{\left(\parallel {\stackrel{˜}{\varphi }}^{N}\left(t\right){\parallel }^{2}\right)}_{t}& +\tau \left(q-1-\varepsilon \right)\parallel {\stackrel{˜}{\varphi }}_{t}^{N}\left(t\right){\parallel }^{2}\\ & +\left(2{\stackrel{˜}{\varphi }}^{N}\left(t\right)+q\tau {\stackrel{˜}{\varphi }}_{t}^{N}\left(t\right),\sum _{l=5}^{9}{\stackrel{˜}{\xi }}_{l}^{N}\left(t\right)\right)\\ & \le c\parallel {\stackrel{˜}{\varphi }}^{N}\left(t\right){\parallel }^{2}+c\left(1+\frac{{q}^{2}\tau }{4\varepsilon }\right)\sum _{l=7}^{9}\parallel {G}_{l}^{N}\left(t\right){\parallel }^{2}.\end{array}& \end{array}$ (5.3)

Putting (5.1)-(5.3) together, we get

 $\begin{array}{cc}\begin{array}{rl}{\left(\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}\right)}_{t}& +\tau \left(q-1-\varepsilon \right)\parallel {\stackrel{˜}{E}}_{t}^{N}\left(t\right){\parallel }^{2}\\ & +2\nu \mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}+\nu \tau \left(\sigma +\frac{q}{2}\right){\left(\mid \stackrel{˜}{E}\left(t\right){\mid }_{1}^{2}\right)}_{t}\\ & +\nu {\tau }^{2}\left(\sigma q-\sigma -\frac{q}{2}\right)\mid {\stackrel{˜}{E}}_{t}^{N}\left(t\right){\mid }_{1}^{2}+\sum _{l=1}^{6}{M}_{l}^{N}\left(t\right)\\ & \le c\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}+c\left(1+\frac{{q}^{2}\tau }{4\varepsilon }\right)\sum _{l=1}^{9}\parallel {G}_{l}^{N}\left(t\right){\parallel }^{2},\end{array}& \end{array}$ (5.4)

where

$\begin{array}{llll}\hfill {M}_{1}^{N}\left(t\right)& =\left(2{\stackrel{˜}{u}}^{N}\left(t\right)+q\tau {\stackrel{˜}{u}}_{t}^{N}\left(t\right),{\xi }_{1}^{N}\left(t\right)\right)+\left(2{\stackrel{˜}{v}}^{N}\left(t\right)+q\tau {\stackrel{˜}{v}}_{t}^{N}\left(t\right),{\xi }_{3}^{N}\left(t\right)\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & +\left(2{\stackrel{˜}{\varphi }}^{N}\left(t\right)+q\tau {\stackrel{˜}{\varphi }}_{t}^{N}\left(t\right),{\xi }_{5}^{N}\left(t\right)\right)-q\tau \left({\stackrel{˜}{u}}_{t}^{N}\left(t\right),F{\stackrel{˜}{v}}^{N}\left(t\right)\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & +q\tau \left({\stackrel{˜}{v}}_{t}^{N}\left(t\right),F{\stackrel{˜}{u}}^{N}\left(t\right)\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {M}_{2}^{N}\left(t\right)& =\left(2{\stackrel{˜}{u}}^{N}\left(t\right)+q\tau {\stackrel{˜}{u}}_{t}^{N}\left(t\right),{\xi }_{2}^{N}\left(t\right)\right)+\left(2{\stackrel{˜}{v}}^{N}\left(t\right)+q\tau {\stackrel{˜}{v}}_{t}^{N}\left(t\right),{\xi }_{4}^{N}\left(t\right)\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & +\left(2{\stackrel{˜}{\varphi }}^{N}\left(t\right)+q\tau {\stackrel{˜}{\varphi }}_{t}^{N}\left(t\right),{\xi }_{6}^{N}\left(t\right)\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {M}_{3}^{N}\left(t\right)& =\left(2{\stackrel{˜}{u}}^{N}\left(t\right)+q\tau {\stackrel{˜}{u}}_{t}^{N}\left(t\right),\frac{\partial }{\partial x}{\varphi }^{N}\left(t\right)\right)+\left(2{\stackrel{˜}{v}}^{N}\left(t\right)+q\tau {\stackrel{˜}{v}}_{t}^{N}\left(t\right),\frac{\partial }{\partial y}{\stackrel{˜}{\varphi }}^{N}\left(t\right)\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {M}_{4}^{N}\left(t\right)& =\left(2{\stackrel{˜}{\varphi }}^{N}\left(t\right)+q\tau {\stackrel{˜}{\varphi }}_{t}^{N}\left(t\right),{\xi }_{7}^{N}\left(t\right)\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {M}_{5}^{N}\left(t\right)& =\left(2{\stackrel{˜}{\varphi }}^{N}\left(t\right)+q\tau {\stackrel{˜}{\varphi }}_{t}^{N}\left(t\right),{\xi }_{8}^{N}\left(t\right)\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {M}_{4}^{N}\left(t\right)& =\left(2{\stackrel{˜}{\varphi }}^{N}\left(t\right)+q\tau {\stackrel{˜}{\varphi }}_{t}^{N}\left(t\right),{\xi }_{9}^{N}\left(t\right)\right).\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

We now estimate $\mid {M}_{l}^{N}\left(t\right)\mid$. Because of the Schwarz inequality and embedding theorem, we have

$\begin{array}{llll}\hfill \mid {M}_{1}^{N}\left(t\right)\mid & \le \varepsilon \nu \mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}+\varepsilon \tau \parallel {\stackrel{˜}{E}}_{t}^{N}\left(t\right){\parallel }^{2}+\frac{c}{\varepsilon \nu }\left(1+\tau {q}^{2}{N}^{2}\right)\mid \parallel {E}_{1}^{N}\parallel {\mid }_{\frac{5}{2}+r}\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \mid {M}_{2}^{N}\left(t\right)\mid & \le \varepsilon \nu \mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}+\varepsilon \tau \parallel {\stackrel{˜}{E}}_{t}^{N}\left(t\right){\parallel }^{2}+\frac{c{N}^{2}}{\varepsilon \nu }\left(\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & +\mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}+\mid \parallel {E}_{1}^{N}\parallel {\mid }_{\frac{5}{2}+r}\right)\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \mid {M}_{3}^{N}\left(t\right)\mid & \le \varepsilon \tau \mid {\stackrel{˜}{E}}_{t}^{N}\left(t\right){\mid }_{1}^{2}+\varepsilon \nu \mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}+\frac{c}{\varepsilon \nu }\left(1+\tau {q}^{2}{N}^{2}\right)\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \mid {M}_{4}^{N}\left(t\right)\mid & \le \varepsilon \tau \parallel {\stackrel{˜}{E}}_{t}^{N}\left(t\right){\parallel }_{1}^{2}+\varepsilon \nu \mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}+\frac{c}{\varepsilon \nu }\left(1+\tau {q}^{2}{N}^{2}\right)\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \mid {M}_{5}^{N}\left(t\right)\mid & \le \varepsilon \tau \parallel {\stackrel{˜}{E}}_{t}^{N}\left(t\right){\parallel }_{1}^{2}+\varepsilon \nu \mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}+\frac{c{N}^{2}}{\varepsilon \nu }\left(1+\tau {q}^{2}{N}^{2}\right)\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{4},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \mid {M}_{6}^{N}\left(t\right)\mid & \le \varepsilon \tau \parallel {\stackrel{˜}{E}}_{t}^{N}\left(t\right){\parallel }_{1}^{2}+\varepsilon \nu \mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}+\frac{c}{\varepsilon \nu }\mid \parallel {E}_{1}\parallel {\mid }_{\frac{5}{2}+r}\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ By substituting the above estimates into (5.4), we get
 $\begin{array}{cc}\begin{array}{rl}{\left(\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}\right)}_{t}& +\tau \left(q-1-7\varepsilon \right)\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}+\nu \left(2-6\varepsilon \right)\mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}\\ & +\nu \tau \left(\sigma +\frac{q}{2}\right){\left(\mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}\right)}_{t}+\nu {\tau }^{2}\left(\sigma q-\sigma -\frac{q}{2}\right)\mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}\\ & \le {H}^{N}\left(t\right)+{A}_{1}\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}+{B}_{1}\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{4}+{B}_{2}\mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2},\end{array}& \end{array}$ (5.5)

where

${A}_{1}=c\left(1+\frac{c}{\varepsilon \nu }+\frac{c}{\varepsilon \nu }\left(1+\tau {q}^{2}{N}^{2}\right)\right)\mid \parallel {E}_{1}\parallel {\mid }_{\frac{5}{2}+r},$

${B}_{1}=\frac{c{N}^{2}}{\varepsilon \nu }\left(1+\tau {q}^{2}{N}^{2}\right)+\frac{c{N}^{2}}{\varepsilon \nu },$

${B}_{2}=\frac{c{N}^{2}}{\varepsilon \nu }\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2},$

${H}^{N}\left(t\right)=c\left(1+\frac{{q}^{2}\tau }{4\varepsilon }\right)\sum _{l=1}^{9}\parallel {G}_{l}^{N}{\parallel }^{2}.$

Now let $\varepsilon$ be suitably small, ${r}_{1}>0$, and

$\begin{array}{llll}\hfill {q}_{1}& =max\left[1+{r}_{1}+7\varepsilon ,\frac{2\sigma }{2\sigma -1}\right],\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {q}_{2}& ={r}_{1}+1+7\varepsilon +\frac{\nu \tau {N}^{2}}{2},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {q}_{3}& =\left(2{r}_{1}+2+14\varepsilon +2\sigma \nu \tau {N}^{2}\right){\left[2-\nu \tau {N}^{2}\left(1-2\sigma \right)\right]}^{-1}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

If $\sigma >1∕2$, we put $q={q}_{1}$, and it follows from (5.5) that

 $\begin{array}{cc}\begin{array}{rl}{\left(\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}\right)}_{t}& +{r}_{1}\tau \parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}+\nu \left(2-6\varepsilon \right)\mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}\\ & +\nu \tau \left(\sigma +\frac{q}{2}\right){\left(\mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}\right)}_{t}\\ & \le {H}^{N}\left(t\right)+{A}_{1}\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}+{B}_{1}\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{4}+{B}_{2}\mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}.\end{array}& \end{array}$ (5.6)

If $\sigma =1∕2$, we put $q={q}_{2}$, and so

$\tau \left(q-1-7\varepsilon \right)\parallel {\stackrel{˜}{E}}_{t}^{N}{\parallel }^{2}+\nu {\tau }^{2}\left(\sigma q-\sigma -\frac{q}{2}\right)\mid {\stackrel{˜}{E}}_{t}^{N}\left(t\right){\mid }_{1}^{2}\ge {r}_{1}\tau \mid {\stackrel{˜}{E}}_{t}^{N}\left(t\right){\mid }^{2}.$

Therefor (5.6) is still holds.
If $\sigma <1∕2$, $\tau {N}^{2}<\frac{2}{\nu \left(1-2\sigma \right)}$, we put $q={q}_{3}$, and thus (5.12) holds.
By summing up (5.6) for $t\in {S}_{\tau }$ , we get

 $\begin{array}{cc}\begin{array}{rl}{\left(\parallel {\stackrel{˜}{E}}^{N}\left(t\right){\parallel }^{2}\right)}_{t}& +\nu \tau \left(\sigma +\frac{q}{2}\right){\left(\mid {\stackrel{˜}{E}}^{N}\left(t\right){\mid }_{1}^{2}\right)}_{t}+\tau \sum _{{t}^{\prime }=0}^{t-\tau }{r}_{1}\tau \parallel {\stackrel{˜}{E}}_{t}^{N}\left({t}^{\prime }\right){\parallel }^{2}\\ & +\nu \left(2-7\varepsilon \right)\mid {\stackrel{˜}{E}}^{N}\left({t}^{\prime }\right){\mid }_{1}^{2}\le \rho \left(t\right)\\ & +\tau \sum _{{t}^{\prime }=0}^{t-\tau }{A}_{1}{\stackrel{˜}{E}}_{1}^{N}\left({t}^{\prime }\right)+{B}_{1}{\stackrel{˜}{E}}_{1}^{2}\left({t}^{\prime }\right)\\ & +{B}_{2}\mid {\stackrel{˜}{E}}_{2}^{N}\left({t}^{\prime }\right){\mid }_{1}^{2},\end{array}& \end{array}$ (5.7)

where

$\rho \left(t\right)=\parallel {\stackrel{˜}{E}}^{N}\left(0\right){\parallel }^{2}\right)+\nu \tau \left(\sigma +\frac{q}{2}\right)\mid {\stackrel{˜}{E}}^{N}\left(0\right){\mid }_{1}^{2}+\tau \sum _{{t}^{\prime }=0}^{t-\tau }{H}^{N}\left({t}^{\prime }\right)$

from which and Lemma 5, the proof is completed.

### 6. The Proof of Theorem 2

We first have

$\parallel {G}_{l}^{N}\left(t\right)\le c\tau \parallel E{\parallel }_{{C}^{2}\left(0,T;{m}_{0}^{0}\left(\Omega \right)\right)},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}l=1,4,7.$

From Lemma 4 and the embedding theorem, we get

$\parallel {G}_{l}^{N}\left(t\right)\parallel \le c\parallel \mid {E}_{1}\parallel {\mid }_{\frac{5}{2}+r}{N}^{-\beta }\parallel {E}_{1}{\parallel }_{{m}_{0}^{\beta +1}\left(\Omega \right)},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}l=2,5,8.$

It is easy to show that

$\parallel {G}_{l}^{N}\left(t\right)\parallel \le c\tau \left({∥\frac{\partial E}{\partial t}\left(t\right)∥}_{{m}_{2}^{0}\left(\Omega \right)}+{∥\frac{\partial E}{\partial t}\left(t\right)∥}_{{m}_{2}^{0}\left(\Omega \right)}\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}l=3,6.$

We have also

$\parallel {G}_{9}^{N}\left(t\right)\parallel \le c{N}^{-\beta }\parallel \mid {E}_{1}\parallel {\mid }_{\frac{5}{2}+r}\parallel E\left(t\right){\parallel }_{{m}_{0}^{\beta +1}\left(\Omega \right)},\phantom{\rule{2.6108pt}{0ex}}\parallel {\stackrel{˜}{E}}^{N}\left(0\right)\parallel \le c{N}^{-\beta }\parallel E\left(0\right){\parallel }_{{m}_{0}^{\beta }\left(\Omega \right)}.$

Therefore if the conditions of Theorem 1 are fulfilled, then

$\rho \left(t\right)\le c\left({\tau }^{2}+{N}^{-2\beta }\right).$

By combining the above estimations with Theorem 1, we complete the proof of Theorem 2.

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DEPARTMENT OF MATHEMATICS, GOMAL UNIVERSITY, D.I.KHAN, PAKISTAN.