西南大学学报 (自然科学版)  2018, Vol. 40 Issue (9): 84-90.  DOI: 10.13718/j.cnki.xdzk.2018.09.013
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  • 非定常不可压Navier-Stokes方程基于欧拉格式的两水平变分多尺度方法    [PDF全文]
    薛菊峰, 尚月强     
    西南大学 数学与统计学院, 重庆 400715
    摘要:主要研究了基于两个高斯积分的两水平全离散有限元变分多尺度方法.该方法对每个时间步长首先在粗网格上求解稳定的非线性Navier-Stokes系统,然后在细网格上求解稳定的线性问题去校正粗网格上的解.基于向后欧拉格式的时间离散推导的速度的误差估计关于时间是一阶收敛的.数值实验验证了理论的正确性和方法的有效性.
    关键词Navier-Stokes方程    两水平法    向后欧拉格式    误差估计    

    Navier-Stokes方程为描述不可压缩的牛顿黏性流提供了一种数学模型,而且广泛用于天气、海流等生活实际方面.最近几十年,许多作者研究了解Navier-Stokes方程的有限元方法,如:文献[1]给出了有限元Galerkin方法,但是有限元Galerkin方法对大雷诺数的流体不再适用;为了研究大雷诺数的流体,文献[2]介绍了人工粘性法;文献[3]得到了defect-correct方法;文献[4]得到了亚格子稳定方法;文献[5]得到了变分多尺度方法;文献[6]得到了羌分方法.文献[5]中的基于高斯积分的变分多尺度方法虽然适用于大雷诺数流体,但是需要花费大量的计算时间.本文在文献[5]的向后欧拉格式基础上给出Navier-Stokes方程的两水平变分多尺度方法并推导了速度的误差估计.在解精确度几乎一样的前提下,我们的方法相比文献[5]格式1的方法不仅适用于大雷诺数流体而且可以节约大约一半的计算时间.

    1 预备知识

    定义1  设Ω是在${{\mathbb{R}}^{2}} $上具有利普希茨连续边界的有界区域,那么有下面的Navier-Stokes方程:

    $ {\mathit{\boldsymbol{u}}_t} - \nu \Delta \mathit{\boldsymbol{u}} + \left( {\mathit{\boldsymbol{u}} \cdot \nabla } \right)\mathit{\boldsymbol{u}} + \nabla \mathit{\boldsymbol{p}} = \mathit{\boldsymbol{f}},{\rm{in}}\;\mathit{\Omega } \times \left( {0,T} \right] $ (1)
    $ \nabla \cdot \mathit{\boldsymbol{u}} = 0,{\rm{in}}\;\mathit{\Omega } \times \left( {0,T} \right] $ (2)
    $ \mathit{\boldsymbol{u}} = 0,{\rm{on}}\partial \mathit{\Omega } \times \left( {0,T} \right] $ (3)
    $ \mathit{\boldsymbol{u}}\left| {_{t = 0}} \right. = {\mathit{\boldsymbol{u}}_0},{\rm{in}}\;\mathit{\Omega } $ (4)

    其中u$ \mathit{\Omega} \to {{\mathbb{R}}^{2}}$表示速度矢量,p$ \mathit{\Omega} \to {{\mathbb{R}}}$是压力,f$ \mathit{\Omega} \to {{\mathbb{R}}^{2}}$是流体驱动的体积力,ν>0为流体粘性系数,u0是使得$\nabla \cdot {\mathit{\boldsymbol{u}}_0} = 0 $的初始速度,并且${\mathit{\boldsymbol{u}}_t} = \frac{{\partial \mathit{\boldsymbol{u}}}}{{\partial t}} $.

    定义2  对于定义1的数学问题,我们引进下面的希尔伯特空间:

    $ \mathit{X = H}_0^1{\left( \mathit{\Omega } \right)^2},Y = {L^2}{\left( \mathit{\Omega } \right)^2},M = L_0^2\left( \mathit{\Omega } \right) = \left\{ {q \in {L^2}\left( \mathit{\Omega } \right):\int_\mathit{\Omega } {q{\rm{d}}x} = 0} \right\} $

    其中:(·,·)表示空间L2(Ω)2L2(Ω)的标准内积,($ \nabla \mathit{\boldsymbol{u}}$$ \nabla \mathit{\boldsymbol{v}}$)和‖$ \nabla \mathit{\boldsymbol{u}}$0为空间X上的一般标量积和范数.用字母c表示一个与时间步长和网格参数无关的正数而且可能在每个式子中代表的数值都不相同.

    定义3[5]   三线性项b(·,·,·)的定义为

    $ \begin{array}{l} b\left( {\mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}},\mathit{\boldsymbol{w}}} \right) = \left( {\left( {\mathit{\boldsymbol{u}} \cdot \nabla } \right)\mathit{\boldsymbol{v}},\mathit{\boldsymbol{w}}} \right) + \frac{1}{2}\left( {\left( {\nabla \cdot \mathit{\boldsymbol{u}}} \right)\mathit{\boldsymbol{v}},\mathit{\boldsymbol{w}}} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\left( {\left( {\mathit{\boldsymbol{u}} \cdot \nabla } \right)\mathit{\boldsymbol{v}},\mathit{\boldsymbol{w}}} \right) - \frac{1}{2}\left( {\left( {\mathit{\boldsymbol{u}} \cdot \nabla } \right)\mathit{\boldsymbol{w}},\mathit{\boldsymbol{v}}} \right),\forall \mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}},\mathit{\boldsymbol{w}} \in X \end{array} $

    它有如下的性质:

    $ b\left( {\mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}},\mathit{\boldsymbol{w}}} \right) = - b\left( {\mathit{\boldsymbol{u}},\mathit{\boldsymbol{w}},\mathit{\boldsymbol{v}}} \right),\forall \mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}},\mathit{\boldsymbol{w}} \in X $ (5)
    $ \left| {b\left( {\mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}},\mathit{\boldsymbol{w}}} \right)} \right| \le c{\left\| {\nabla \mathit{\boldsymbol{u}}} \right\|_0}{\left\| {\nabla \mathit{\boldsymbol{v}}} \right\|_0}{\left\| {\nabla \mathit{\boldsymbol{w}}} \right\|_0},\forall \mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}},\mathit{\boldsymbol{w}} \in X $ (6)
    $ \left| {b\left( {\mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}},\mathit{\boldsymbol{w}}} \right)} \right| \le c{\left\| {\nabla \mathit{\boldsymbol{u}}} \right\|_0}{\left\| {\nabla \mathit{\boldsymbol{v}}} \right\|_0}\left\| \mathit{\boldsymbol{w}} \right\|_0^{\frac{1}{2}}\left\| {\nabla \mathit{\boldsymbol{w}}} \right\|_0^{\frac{1}{2}},\forall \mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}},\mathit{\boldsymbol{w}} \in X $ (7)

    定义4   方程(1)-(4)的变分形式为:对于任意的t∈(0,T],存在(up)∈X×M,使得

    $ \left( {{\mathit{\boldsymbol{u}}_t},\mathit{\boldsymbol{v}}} \right) + \nu \left( {\nabla \mathit{\boldsymbol{u}},\nabla \mathit{\boldsymbol{v}}} \right) + b\left( {\mathit{\boldsymbol{u}},\mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}}} \right) - \left( {\nabla \cdot \mathit{\boldsymbol{v}},\mathit{\boldsymbol{p}}} \right) + \left( {\nabla \cdot \mathit{\boldsymbol{u}},\mathit{\boldsymbol{q}}} \right) = \left( {\mathit{\boldsymbol{f}},\mathit{\boldsymbol{v}}} \right),\forall \left( {\mathit{\boldsymbol{v}},\mathit{\boldsymbol{q}}} \right) \in X \times M $ (8)

    定义5[7]   对于方程(8)的有限元离散,我们假设${T^\mu }\left( \mathit{\Omega} \right) = \left\{ K \right\}\left( {\mu = H, {\rm{ }}h, H > h} \right) $是准均匀的三角形网格剖分并且网格尺寸0<μ<1.细网格Th(Ω)可以被认为是由粗网格加密而产生的.协调有限元(XμMμ)满足下面的inf-sup条件:存在常数β>0使得

    $ \mathop {\inf }\limits_{{\mathit{\boldsymbol{q}}_\mu } \in {M_\mu }} \mathop {\sup }\limits_{{\mathit{\boldsymbol{v}}_\mu } \in {X_\mu }} \frac{{\left( {\nabla \cdot {\mathit{\boldsymbol{v}}_\mu },{\mathit{\boldsymbol{q}}_\mu }} \right)}}{{{{\left\| {\nabla {\mathit{\boldsymbol{v}}_\mu }} \right\|}_0}{{\left\| {{\mathit{\boldsymbol{q}}_\mu }} \right\|}_0}}} \ge \beta > 0 $ (9)

    定义6[8-11]   设速度空间Xμ满足$ \forall K \in {T^\mu }\left( \mathit{\Omega} \right), {\rm{ }}{v_\mu } \in {X_\mu }{\rm{, }}(\nabla {v_\mu }){|_K}$是线性的,那么在本文中提到的方法仅适用于速度限制在(P2)2上的有限元对.如:Taylor-Hood元,P2-P0元和Scott-Vogelius(P2-P1disc)元等.

    定义7[12]  我们定义$ V = \left\{ {v \in X:{\rm{ }}\left( {\nabla \cdot{\rm{ }}\mathit{\boldsymbol{v}}, {\rm{ }}\mathit{\boldsymbol{q}}} \right) = 0, {\rm{ }}\forall \mathit{\boldsymbol{q}} \in \mathit{\boldsymbol{M}}} \right\}{\rm{ }}{V_\mu } = \{ {\mathit{\boldsymbol{v}}_\mu } \in {X_\mu }:(\nabla \cdot{\rm{ }}{\mathit{\boldsymbol{v}}_\mu }, {\mathit{\boldsymbol{q}}_\mu }) = 0, \forall {\mathit{\boldsymbol{q}}_\mu } \in {M_\mu }\} $,则有如下的估计:

    $ \mathop {\inf }\limits_{{\mathit{\boldsymbol{v}}_\mu } \in {V_\mu }} {\left\| {\nabla \left( {\mathit{\boldsymbol{v}} - {\mathit{\boldsymbol{v}}_\mu }} \right)} \right\|_0} \le C\left( {1 + \frac{1}{\beta }} \right)\mathop {\inf }\limits_{{\mathit{\boldsymbol{v}}_\mu } \in {X_\mu }} {\left\| {\nabla \left( {\mathit{\boldsymbol{v}} - {\mathit{\boldsymbol{v}}_\mu }} \right)} \right\|_0},\forall \mathit{\boldsymbol{v}} \in V $ (10)

    定义8[12]   设$ {P_{{V_\mu }}}:Y \to V$L2Vμ上的正交投影,则满足$ (\mathit{\boldsymbol{\xi }} - {P_{{V_\mu }}}\mathit{\boldsymbol{\xi }}, {\mathit{\boldsymbol{v}}_\mu }) = 0, {\rm{ }}\forall \mathit{\boldsymbol{\xi }} \in Y, {\mathit{\boldsymbol{v}}_\mu } \in {V_\mu }$.

    引理1[13]  离散Gronwall引理:对于任意整数n≥0,令ΔtHanbncndn是非负数,满足

    $ {a_l} + \Delta t\sum\limits_{n = 0}^l {{b_n}} \le \Delta t\sum\limits_{n = 0}^l {{d_n}{a_n}} + \Delta t\sum\limits_{n = 0}^l {{c_n}} + H,l \ge 0 $

    和Δtdn$1\forall n $.有

    $ {a_l} + \Delta t\sum\limits_{n = 0}^l {{b_n}} \le \exp \left( {\Delta t\sum\limits_{n = 0}^l {\frac{{{d_n}}}{{1 - \Delta t{d_n}}}} } \right)\left( {\Delta t\sum\limits_{n = 0}^l {{c_n}} + H} \right),l \ge 0 $
    2 向后欧拉格式的有限元变分多尺度方法

    定义9[5]   设数值格式中出现的变分多尺度稳定项为

    $ G\left( {{\mathit{\boldsymbol{u}}_\mu },{\mathit{\boldsymbol{v}}_\mu }} \right) = \alpha \sum\limits_{K \in {T^\mu }\left( \mathit{\Omega } \right)} {\left( {\int_{K,m} {\nabla {\mathit{\boldsymbol{u}}_\mu } \cdot \nabla {\mathit{\boldsymbol{v}}_\mu }{\rm{d}}x} - \int_{K,1} {\nabla {\mathit{\boldsymbol{u}}_\mu } \cdot \nabla {\mathit{\boldsymbol{v}}_\mu }{\rm{d}}x} } \right)} ,\forall {\mathit{\boldsymbol{u}}_\mu },{\mathit{\boldsymbol{v}}_\mu } \in {X_\mu } $ (11)

    这里$ {\smallint _{K, {\rm{ }}s}}\left( \cdot \right){\rm{d}}x$表示K上适当的高斯积分,该积分对于次数不超过s(s=m,1,m≥2)的多项式是准确的. α>0是一个自定义的稳定项参数.

    定义

    $ {R_0} = \left\{ {\mathit{\boldsymbol{v}} \in {L^2}\left( \mathit{\Omega } \right):\mathit{\boldsymbol{v}}\left| {_K} \right. \in {P_0},\forall K \in {T^\mu }\left( \mathit{\Omega } \right)} \right\},{L_\mu } = R_0^{2 \times 2},L = {L^2}{\left( \mathit{\Omega } \right)^{2 \times 2}} $

    其中P0是常量元素K的空间.那么标准的L2-正交投影Πμ$ L \to {L_\mu }$有下面的性质:

    $ \left( {{\mathit{\Pi }_\mu }\nabla \mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}}} \right) = \left( {\nabla \mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}}} \right),\forall \mathit{\boldsymbol{u}} \in X,\mathit{\boldsymbol{v}} \in {L_\mu } $ (12)
    $ {\left\| {{\mathit{\Pi }_\mu }\nabla \mathit{\boldsymbol{v}}} \right\|_0} \le c{\left\| {\nabla \mathit{\boldsymbol{v}}} \right\|_0},\forall \mathit{\boldsymbol{v}} \in X $ (13)

    注1  稳定项(11)还可以表示为:

    $ G\left( {{\mathit{\boldsymbol{u}}_\mu },{\mathit{\boldsymbol{v}}_\mu }} \right) = \alpha \left( {\nabla {\mathit{\boldsymbol{u}}_\mu } \cdot \nabla {\mathit{\boldsymbol{v}}_\mu }} \right) - \alpha \left( {{\mathit{\Pi }_\mu }\nabla {\mathit{\boldsymbol{u}}_\mu } \cdot \nabla {\mathit{\boldsymbol{v}}_\mu }} \right) = \alpha \left( {\left( {I - {\mathit{\Pi }_\mu }} \right)\nabla {\mathit{\boldsymbol{u}}_\mu },\left( {I - {\mathit{\Pi }_\mu }} \right)\nabla {\mathit{\boldsymbol{v}}_\mu }} \right) $ (14)

    根据定义9,我们给出Navier-Stokes方程的标准的有限元变分多尺度方法.

    方法1  标准的有限元变分多尺度方法[5].

    给定uμ0,存在${(\mathit{\boldsymbol{u}}_\mu ^{n + 1}, \mathit{\boldsymbol{p}}_\mu ^{n + 1})_{n \ge 0}} \in ({X_\mu }, {\rm{ }}{M_\mu }) $使得:

    $ \begin{array}{l} \frac{1}{{\Delta t}}\left( {\mathit{\boldsymbol{u}}_\mu ^{n + 1} - \mathit{\boldsymbol{u}}_\mu ^n,{\mathit{\boldsymbol{v}}_\mu }} \right) + \nu \left( {\nabla \mathit{\boldsymbol{u}}_\mu ^{n + 1},\nabla {\mathit{\boldsymbol{v}}_\mu }} \right) + b\left( {\mathit{\boldsymbol{u}}_\mu ^{n + 1},\mathit{\boldsymbol{u}}_\mu ^{n + 1},{\mathit{\boldsymbol{v}}_\mu }} \right) - \left( {\nabla \cdot {\mathit{\boldsymbol{v}}_\mu },\mathit{\boldsymbol{p}}_\mu ^{n + 1}} \right) + \\ \left( {\nabla \cdot \mathit{\boldsymbol{u}}_\mu ^{n + 1},{\mathit{\boldsymbol{q}}_\mu }} \right) + G\left( {\mathit{\boldsymbol{u}}_\mu ^{n + 1},{\mathit{\boldsymbol{v}}_\mu }} \right) = \left( {{\mathit{\boldsymbol{f}}^{n + 1}},{\mathit{\boldsymbol{v}}_\mu }} \right),\forall \left( {{\mathit{\boldsymbol{v}}_\mu },{\mathit{\boldsymbol{q}}_\mu }} \right) \in {X_\mu } \times {M_\mu } \end{array} $ (15)

    令时间步长的尺寸Δt满足0<Δt<1,${t_n} = n\Delta t, {\rm{ }}n = 0, {\rm{ }}1, {\rm{ }} \ldots , {\rm{ }}N - 1 $,和$ N = \frac{T}{{\Delta t}}$. φ1表示函数φ在时间t1时的值,并且${\varphi ^{\frac{1}{2}}} = \frac{1}{2}({\varphi ^1} + {\varphi ^0}) $.初始速度$\mathit{\boldsymbol{u}}_\mu ^0 = {P_{{V_\mu }}}{\mathit{\boldsymbol{u}}_0} $.

    引理2[5]  令fL2(0,TH-1(Ω)2)和u0L2(Ω)2.则格式(15)的解是稳定的且满足任意的0<lN

    $ \left\| {\mathit{\boldsymbol{u}}_\mu ^l} \right\|_0^2 + \sum\limits_{n = 0}^{l - 1} {\left\| {\mathit{\boldsymbol{u}}_\mu ^{n + 1} - \mathit{\boldsymbol{u}}_\mu ^n} \right\|_0^2} + \Delta t\sum\limits_{n = 0}^{l - 1} {\nu \left\| {\nabla \mathit{\boldsymbol{u}}_\mu ^{n + 1}} \right\|_0^2} \le {\nu ^{ - 1}}\left\| \mathit{\boldsymbol{f}} \right\|_{{L^2}\left( {0,T;{H^{ - 1}}{{\left( \mathit{\Omega } \right)}^2}} \right)}^2 + \left\| {{\mathit{\boldsymbol{u}}_0}} \right\|_0^2 $ (16)
    $ \Delta t\sum\limits_{n = 0}^{l - 1} {{{\left\| {\mathit{\boldsymbol{p}}_\mu ^{n + 1}} \right\|}_0}} \le C\left( {\nu ,\alpha ,{\mathit{\boldsymbol{u}}_0},\mathit{\boldsymbol{f}},T,\mathit{\Omega }} \right) $ (17)

    引理3[5]   Navier-Stokes方程的精确解(u p)满足uL(0,TH1(Ω)2),uttL(0,TH1(Ω)2),和utttL(0,TL2(Ω)2).那么由格式(15)计算的全离散解有如下估计:

    $ \begin{array}{*{20}{c}} {\left\| {\mathit{\boldsymbol{u}}\left( T \right) - \mathit{\boldsymbol{u}}_\mu ^N} \right\|_0^2 + \nu \Delta t\sum\limits_{n = 0}^{N - 1} {\left\| {\nabla \left( {\mathit{\boldsymbol{u}}\left( {{t_{n + 1}}} \right) - \mathit{\boldsymbol{u}}_\mu ^{n + 1}} \right)} \right\|_0^2} \le c\left( {{\alpha ^2} + \Delta {t^2}} \right) + \mathop {\inf }\limits_{{\mathit{\boldsymbol{w}}_\mu } \in {X_\mu }} \left\| {\mathit{\boldsymbol{u}}\left( T \right) - {\mathit{\boldsymbol{w}}_\mu }} \right\|_0^2 + }\\ {C\Delta t\sum\limits_{n = 0}^{N - 1} {\left( {\mathop {\inf }\limits_{{\mathit{\boldsymbol{w}}_\mu } \in {X_\mu }} \left\| {\nabla \left( {\mathit{\boldsymbol{u}}\left( {{t_{n + 1}}} \right) - {\mathit{\boldsymbol{w}}_\mu }} \right)} \right\|_0^2 + \mathop {\inf }\limits_{{\mathit{\boldsymbol{\lambda }}_\mu } \in {M_\mu }} \left\| {\mathit{\boldsymbol{p}}\left( {{t_{n + 1}}} \right) - {\mathit{\boldsymbol{\lambda }}_\mu }} \right\|_0^2} \right)} } \end{array} $ (18)

    两水平有限元变分多尺度方法如下:

    方法2  两水平有限元变分多尺度方法.

    给定uH0uh0,存在(uh1ph1)n≥0Xh×Mh.

    1) 寻找粗网格上的一个解(uH1pH1)n≥0∈(XHMH)使得

    $ \begin{array}{*{20}{c}} {\frac{1}{{\Delta t}}\left( {\mathit{\boldsymbol{u}}_H^{n + 1} - \mathit{\boldsymbol{u}}_H^n,{\mathit{\boldsymbol{v}}_H}} \right) + \nu \left( {\nabla \mathit{\boldsymbol{u}}_H^{n + 1},\nabla {\mathit{\boldsymbol{v}}_H}} \right) + b\left( {\mathit{\boldsymbol{u}}_H^{n + 1},\mathit{\boldsymbol{u}}_H^{n + 1},{\mathit{\boldsymbol{v}}_H}} \right) - \left( {\nabla \cdot {\mathit{\boldsymbol{v}}_H},\mathit{\boldsymbol{p}}_H^{n + 1}} \right) + }\\ {\left( {\nabla \cdot {\mathit{\boldsymbol{u}}_H}^1,{\mathit{\boldsymbol{q}}_H}} \right) + G\left( {\mathit{\boldsymbol{u}}_H^{n + 1},{\mathit{\boldsymbol{v}}_H}} \right) = \left( {{\mathit{\boldsymbol{f}}^{n + 1}},{\mathit{\boldsymbol{v}}_H}} \right),\forall \left( {{\mathit{\boldsymbol{v}}_H},{\mathit{\boldsymbol{q}}_H}} \right) \in {X_H} \times {M_H}} \end{array} $ (19)

    2) 寻找细网格上的一个解(uh1ph1)n≥0∈(XhMh)使得

    $ \begin{array}{*{20}{c}} {\frac{1}{{\Delta t}}\left( {\mathit{\boldsymbol{u}}_h^{n + 1} - \mathit{\boldsymbol{u}}_h^n,{\mathit{\boldsymbol{v}}_h}} \right) + \nu \left( {\nabla \mathit{\boldsymbol{u}}_h^{n + 1},\nabla {\mathit{\boldsymbol{v}}_h}} \right) + b\left( {\mathit{\boldsymbol{u}}_H^{n + 1},\mathit{\boldsymbol{u}}_h^{n + 1},{\mathit{\boldsymbol{v}}_h}} \right) - \left( {\nabla \cdot {\mathit{\boldsymbol{v}}_h},\mathit{\boldsymbol{p}}_h^{n + 1}} \right) + }\\ {\left( {\nabla \cdot \mathit{\boldsymbol{u}}_h^{n + 1},{\mathit{\boldsymbol{q}}_h}} \right) + {G^ * }\left( {\mathit{\boldsymbol{u}}_h^{n + 1},{\mathit{\boldsymbol{v}}_h}} \right) = \left( {{\mathit{\boldsymbol{f}}^{n + 1}},{\mathit{\boldsymbol{v}}_h}} \right),\forall \left( {{\mathit{\boldsymbol{v}}_h},{\mathit{\boldsymbol{q}}_h}} \right) \in {X_h} \times {M_h}} \end{array} $ (20)

    注2

    $ {G^ * }\left( {{\mathit{\boldsymbol{u}}_h},{\mathit{\boldsymbol{v}}_h}} \right) = \alpha \sum\limits_{K \in {T^h}\left( \mathit{\Omega } \right)} {\left( {\int_{K,m} {\nabla {\mathit{\boldsymbol{u}}_h} \cdot \nabla {\mathit{\boldsymbol{v}}_h}{\rm{d}}x} - \int_{K,1} {\nabla {\mathit{\boldsymbol{u}}_H} \cdot \nabla {\mathit{\boldsymbol{v}}_h}{\rm{d}}x} } \right)} ,\forall {\mathit{\boldsymbol{v}}_h} \in {X_h} $
    3 主要结果

    定理1   Navier-Stokes方程的精确解(up)满足uL(0,TH1(Ω)2),uttL(0,TL2(Ω)2),那么由式(19)-(20)得到的全离散解有下面的估计式:

    $ \begin{array}{*{20}{c}} {\left\| {\mathit{\boldsymbol{u}}\left( T \right) - {\mathit{\boldsymbol{u}}_h}^N} \right\|_0^2 + \nu \Delta t\sum\limits_{n = 0}^{N - 1} {\left\| {\nabla \left( {\mathit{\boldsymbol{u}}\left( {{t_{n + 1}}} \right) - {\mathit{\boldsymbol{u}}_h}^{n + 1}} \right)} \right\|_0^2} \le }\\ {\mathop {\inf }\limits_{{\mathit{\boldsymbol{w}}_h} \in {X_h}} \left\| {\mathit{\boldsymbol{u}}\left( T \right) - {\mathit{\boldsymbol{w}}_h}} \right\|_0^2 + c\Delta t\sum\limits_{n = 0}^{N - 1} {\left\| {\nabla \left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1}} \right)} \right\|_0^2} + }\\ {c\Delta t\sum\limits_{n = 0}^{N - 1} {\left( {\mathop {\inf }\limits_{{\mathit{\boldsymbol{w}}_h} \in {X_h}} \left\| {\nabla \left( {\mathit{\boldsymbol{u}}\left( {{t_{n + 1}}} \right) - {\mathit{\boldsymbol{w}}_h}} \right)} \right\|_0^2 + \mathop {\inf }\limits_{{\mathit{\boldsymbol{\lambda }}_h} \in {M_h}} \left\| {\mathit{\boldsymbol{p}}\left( {{t_{n + 1}}} \right) - {\mathit{\boldsymbol{\lambda }}_h}} \right\|_0^2} \right)} + c\left( {{\alpha ^2} + \Delta {t^2}} \right)} \end{array} $ (21)

      令$ (\mathit{\boldsymbol{u}}({t_{n + 1}}), \mathit{\boldsymbol{p}}({t_{n + 1}})) = ({\mathit{\boldsymbol{u}}^{n + 1}}, {\mathit{\boldsymbol{p}}^{n + 1}}), {\rm{ }}({\mathit{\boldsymbol{e}}^{n + 1}}, {\mathit{\boldsymbol{\eta }}^{n + 1}}) = ({\mathit{\boldsymbol{u}}^{n + 1}} - {\mathit{\boldsymbol{u}}_h}^{n + 1}, {\mathit{\boldsymbol{p}}^{n + 1}} - {\mathit{\boldsymbol{p}}_h}^{n + 1}), n = 1, 2, \ldots , N - 1$.当时间t=tn+1时,式(8)减去式(20)得:$\forall ({\mathit{\boldsymbol{v}}_h}, {\rm{ }}{\mathit{\boldsymbol{q}}_h}) \in ({X_h}, {\rm{ }}{M_h}) $

    $ \begin{array}{*{20}{c}} {\frac{1}{{\Delta t}}\left( {{\mathit{\boldsymbol{e}}^{n + 1}} - {\mathit{\boldsymbol{e}}^n},{\mathit{\boldsymbol{v}}_h}} \right) + \left( {\nu + \alpha } \right)\left( {\nabla {\mathit{\boldsymbol{e}}^{n + 1}},\nabla {\mathit{\boldsymbol{v}}_h}} \right) + b\left( {{\mathit{\boldsymbol{u}}^{n + 1}},{\mathit{\boldsymbol{e}}^{n + 1}},{\mathit{\boldsymbol{v}}_h}} \right) - }\\ {b\left( {{u^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1},{\mathit{\boldsymbol{e}}^{n + 1}},{\mathit{\boldsymbol{v}}_h}} \right) + b\left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1},{\mathit{\boldsymbol{u}}^{n + 1}},{\mathit{\boldsymbol{v}}_h}} \right) - \left( {\nabla \cdot {\mathit{\boldsymbol{v}}_h},{\mathit{\boldsymbol{\eta }}^{n + 1}}} \right) = }\\ {\alpha \left( {\nabla {\mathit{\boldsymbol{u}}^{n + 1}},\nabla {\mathit{\boldsymbol{v}}_h}} \right) - \alpha \left( {{\mathit{\boldsymbol{ \boldsymbol{\varPi} }}_h}\nabla \mathit{\boldsymbol{u}}_H^{n + 1},\nabla {\mathit{\boldsymbol{v}}_h}} \right) - \left( {{\mathit{\boldsymbol{u}}_t}\left( {{t_{n + 1}}} \right) - \frac{{{\mathit{\boldsymbol{u}}^{n + 1}} - {\mathit{\boldsymbol{u}}^n}}}{{\Delta t}},{\mathit{\boldsymbol{v}}_h}} \right)} \end{array} $ (22)

    由于

    $ {V_h} \subset {X_h} $

    所以

    $ {\mathit{\boldsymbol{v}}_h} \in {V_h} \subset {X_h} $

    从而令

    $ {\mathit{\boldsymbol{e}}^{n + 1}} = {\mathit{\boldsymbol{\chi }}^{n + 1}} - {\mathit{\boldsymbol{\varphi }}_h}^{n + 1} $

    $ {\mathit{\boldsymbol{\chi }}^{n + 1}} = {\mathit{\boldsymbol{u}}^{n + 1}} - {P_{{v_h}}}{\mathit{\boldsymbol{u}}^{n + 1}},{\mathit{\boldsymbol{\varphi }}_h}^{n + 1} = {\mathit{\boldsymbol{u}}_h}^{n + 1} - {P_{{v_h}}}{\mathit{\boldsymbol{u}}^{n + 1}} $

    vh=φhn+1代入式(22)利用式(5)和$ 2\left( {a - b, {\rm{ }}a} \right) = {a^2} - {b^2} + {\left( {a - b} \right)^2}$得:

    $ \begin{array}{*{20}{c}} {\frac{1}{{2\Delta t}}\left( {\left\| {{\varphi _h}^{n + 1}} \right\|_0^2 - \left\| {\varphi _h^n} \right\|_0^2 + \left\| {\varphi _h^{n - 1} - \varphi _h^n} \right\|_0^2} \right) + \left( {\nu + \alpha } \right)\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2 = }\\ {\left( {\frac{{{\chi ^{n + 1}} - {\chi ^n}}}{{\Delta t}},{\varphi _h}^{n + 1}} \right) + \left( {\nu + \alpha } \right)\left( {\nabla {\chi ^{n + 1}},\nabla {\varphi _h}^{n + 1}} \right) + b\left( {{\mathit{\boldsymbol{u}}^{n + 1}},{\mathit{\boldsymbol{\chi }}^{n + 1}},{\varphi _h}^{n + 1}} \right) -\\ b\left( {{\mathit{\boldsymbol{u}}^{n + 1}},\mathit{\boldsymbol{u}}_H^{n + 1},{\chi ^{n + 1}},{\varphi _h}^{n + 1}} \right) + }\\ {b\left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1},{\mathit{\boldsymbol{u}}^{n + 1}},{\varphi _h}^{n + 1}} \right) - \left( {\nabla \cdot {\varphi _h}^{n + 1},{\mathit{\boldsymbol{p}}^{n + 1}} - {\lambda _h}} \right) + \left( {\nabla \cdot {\varphi _h}^{n + 1},{\mathit{\boldsymbol{p}}_h}^{n + 1} - {\lambda _h}} \right) -\\ \alpha \left( {\nabla {\mathit{\boldsymbol{u}}^{n + 1}},\nabla {\varphi _h}^{n + 1}} \right) + }\\ {\alpha \left( {{\mathit{\Pi }_h}\nabla \mathit{\boldsymbol{u}}_H^{n + 1},\nabla {\varphi _h}^{n + 1}} \right) + \left( {{\mathit{\boldsymbol{u}}_t}\left( {{t_{n + 1}}} \right) - \frac{{{\mathit{\boldsymbol{u}}^{n + 1}} - {\mathit{\boldsymbol{u}}^n}}}{{\Delta t}},{\varphi _h}^{n + 1}} \right)} \end{array} $ (23)

    其中:λhMhpn+1的近似值.

    根据定义7中Vh的定义,有

    $ \left( {\nabla \cdot {\varphi _h}^{n + 1},{\mathit{\boldsymbol{p}}_h}^{n + 1} - {\lambda _h}} \right) = 0 $

    由于χn+1-χnVhφhn+1Vh,利用投影算子我们有

    $ \frac{1}{{\Delta t}}\left( {{\chi ^{n + 1}} - {\chi ^n},{\varphi _h}^{n + 1}} \right) = 0 $

    所以由式(23)可得:

    $ \begin{array}{*{20}{c}} {\frac{1}{{2\Delta t}}\left( {\left\| {{\varphi _h}^{n + 1}} \right\|_0^2 - \left\| {\varphi _h^n} \right\|_0^2 + \left\| {\varphi _h^n - \varphi _h^{n - 1}} \right\|_0^2} \right) + \left( {\nu + \alpha } \right)\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2 = }\\ {\left( {\nu + \alpha } \right)\left( {\nabla {\chi ^{n + 1}},\nabla {\varphi _h}^{n + 1}} \right) + b\left( {{\mathit{\boldsymbol{u}}^{n + 1}},{\mathit{\boldsymbol{\chi }}^{n + 1}},{\varphi _h}^{n + 1}} \right) - b\left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1},{\chi ^{n + 1}},{\varphi _h}^{n + 1}} \right) + }\\ {b\left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1},{\mathit{\boldsymbol{u}}^{n + 1}},{\varphi _h}^{n + 1}} \right) - \left( {\nabla \cdot {\varphi _h}^{n + 1},{\mathit{\boldsymbol{p}}^{n + 1}} - {\lambda _h}} \right) - \alpha \left( {\nabla {\mathit{\boldsymbol{u}}^{n + 1}},\nabla {\varphi _h}^{n + 1}} \right) + }\\ {\alpha \left( {{\mathit{\Pi }_h}\nabla \mathit{\boldsymbol{u}}_H^{n + 1},\nabla {\varphi _h}^{n + 1}} \right) + \left( {{\mathit{\boldsymbol{u}}_t}\left( {{t_{n + 1}}} \right) - \frac{{{\mathit{\boldsymbol{u}}^{n + 1}} - {\mathit{\boldsymbol{u}}^n}}}{{\Delta t}},{\varphi _h}^{n + 1}} \right)} \end{array} $ (24)

    现在利用施瓦兹不等式,Young不等式和式(13)对式(24)的右边进行估计:

    $ \nu \left( {\nabla {\chi ^{n + 1}},\nabla {\varphi _h}^{n + 1}} \right) \le \frac{7}{2}\nu \left\| {\nabla {\chi ^{n + 1}}} \right\|_0^2 + \frac{\nu }{{14}}\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2 $ (25)
    $ \alpha \left( {\nabla {\chi ^{n + 1}},\nabla {\varphi _h}^{n + 1}} \right) \le \alpha \left\| {\nabla {\chi ^{n + 1}}} \right\|_0^2 + \frac{\alpha }{4}\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2 $ (26)
    $ - \left( {\nabla \cdot {\varphi _h}^{n + 1},{\mathit{\boldsymbol{p}}^{n + 1}} - {\lambda _h}} \right) \le c{\nu ^{ - 1}}\left\| {{\mathit{\boldsymbol{p}}^{n + 1}} - {\lambda _h}} \right\|_0^2 + \frac{\nu }{{14}}\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2 $ (27)
    $ - \alpha \left( {\nabla {\mathit{\boldsymbol{u}}^{n + 1}},\nabla {\varphi _h}^{n + 1}} \right) \le c{\alpha ^2}{\nu ^{ - 1}}\left\| {\nabla {\mathit{\boldsymbol{u}}^{n + 1}}} \right\|_0^2 + \frac{\nu }{{14}}\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2 $ (28)
    $ \alpha \left( {{\mathit{\Pi }_h}\nabla \mathit{\boldsymbol{u}}_H^{n + 1},\nabla {\varphi _h}^{n + 1}} \right) \le c{\alpha ^2}{\nu ^{ - 1}}\left\| {\nabla \mathit{\boldsymbol{u}}_H^{n + 1}} \right\|_0^2 + \frac{\nu }{{14}}\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2 $ (29)

    利用式(5)-(7)和Young不等式估计下面的三线性项

    $ b\left( {{\mathit{\boldsymbol{u}}^{n + 1}},{\chi ^{n + 1}},\varphi _h^{n + 1}} \right) \le c{\nu ^{ - 1}}\left\| {\nabla \mathit{\boldsymbol{u}}_H^{n + 1}} \right\|_0^2\left\| {\nabla {\chi ^{n + 1}}} \right\|_0^2 + \frac{\nu }{{14}}\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2 $ (30)
    $ - b\left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1},{\chi ^{n + 1}},\varphi _h^{n + 1}} \right) \le c{\nu ^{ - 1}}\left\| {\nabla \left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1}} \right)} \right\|_0^2\left\| {\nabla {\chi ^{n + 1}}} \right\|_0^2 + \frac{\nu }{{14}}\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2 $ (31)
    $ b\left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1},{\mathit{\boldsymbol{u}}^{n + 1}},\varphi _h^{n + 1}} \right) \le c{\nu ^{ - 1}}\left\| {\nabla \left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1}} \right)} \right\|_0^2\left\| {\nabla {\mathit{\boldsymbol{u}}^{n + 1}}} \right\|_0^2 + \frac{\nu }{{14}}\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2 $ (32)

    对式(24)最后一项$ \left( {{\mathit{\boldsymbol{u}}_t}({t_{n + 1}}) - \frac{{{\mathit{\boldsymbol{u}}^{n + 1}} - {\mathit{\boldsymbol{u}}^n}}}{{\Delta t}}, {\varphi _h}^{n + 1}} \right)$利用泰勒展式,施瓦兹不等式和Young不等式,得:

    $ \left( {{\mathit{\boldsymbol{u}}_t}\left( {{t_{n + 1}}} \right) - \frac{{{\mathit{\boldsymbol{u}}^{n + 1}} - {\mathit{\boldsymbol{u}}^n}}}{{\Delta t}},\varphi _h^{n + 1}} \right) \le c{\left( {\Delta t} \right)^2}\left\| {{\mathit{\boldsymbol{u}}_{tt}}} \right\|_{{L^\infty }\left( {{t_n},{t_{n + 1}};{L^2}{{\left( \mathit{\Omega } \right)}^2}} \right)}^2 + \left\| {{\varphi _h}^{n + 1}} \right\|_0^2 $ (33)

    将式(25)-(33)代入式(24),有

    $ \begin{array}{*{20}{c}} {\frac{1}{{2\Delta t}}\left( {\left\| {{\varphi _h}^{n + 1}} \right\|_0^2 - \left. {\varphi _h^n} \right\|_0^2 + \left\| {2\varphi _h^{n - 1} - \varphi _h^n} \right\|_0^2} \right) + \frac{1}{2}\nu \left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2 \le }\\ {c\left( {\nu + \alpha + {\nu ^{ - 1}}\left( {\left\| {\nabla {\mathit{\boldsymbol{u}}^{n + 1}}} \right\|_0^2 + \left\| {\nabla \left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1}} \right)} \right\|_0^2} \right)} \right)\left\| {\nabla {\chi ^{n + 1}}} \right\|_0^2 + }\\ {c\left\| {{\varphi _h}^{n + 1}} \right\|_0^2 + c{\nu ^{ - 1}}\left\| {{\mathit{\boldsymbol{p}}^{n + 1}} - {\lambda _h}} \right\|_0^2 + c{\alpha ^2}{\nu ^{ - 1}}\left( {\left\| {\nabla {\mathit{\boldsymbol{u}}^{n + 1}}} \right\|_0^2 + \left\| {\nabla \mathit{\boldsymbol{u}}_H^{n + 1}} \right\|_0^2} \right) + }\\ {c{\nu ^{ - 1}}\left\| {\nabla {\mathit{\boldsymbol{u}}^{n + 1}}} \right\|_0^2\left\| {\nabla \left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1}} \right.} \right\|_0^2 + c\Delta {t^2}\left\| {{\mathit{\boldsymbol{u}}_{tt}}} \right\|_{{L^\infty }\left( {{t_n},{t_{n + 1}};{L^2}{{\left( \mathit{\Omega } \right)}^2}} \right)}^2} \end{array} $ (34)

    将式(34)乘以2Δt,从n=1加到N-1且φh0=0.得:

    $ \begin{array}{*{20}{c}} {\left\| {\varphi _h^N} \right\|_0^2 + \sum\limits_{n = 0}^{N - 1} {\left\| {{\varphi _h}^{n + 1} - \varphi _h^n} \right\|_0^2} + \nu \Delta t\sum\limits_{n = 0}^{N - 1} {\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2} \le }\\ {c\Delta t\sum\limits_{n = 0}^{N - 1} {\left( {\nu + \alpha + {\nu ^{ - 1}}\left( {\left\| {\nabla {\mathit{\boldsymbol{u}}^{n + 1}}} \right\|_0^2 + \left\| {\nabla \left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1}} \right)} \right\|_0^2} \right)} \right)} \left\| {\nabla {\chi ^{n + 1}}} \right\|_0^2 + }\\ {c\Delta t\sum\limits_{n = 0}^{N - 1} {\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2} + c\Delta t\sum\limits_{n = 0}^{N - 1} {{\nu ^{ - 1}}\left\| {{\mathit{\boldsymbol{p}}^{n + 1}} - {\lambda _h}} \right\|_0^2} + }\\ {c{\alpha ^2}{\nu ^{ - 1}}\left( {\left\| {\nabla \mathit{\boldsymbol{u}}} \right\|_{{L^2}\left( {0,T;{L^2}{{\left( \mathit{\Omega } \right)}^2}} \right)}^2 + \Delta t\sum\limits_{n = 0}^{N - 1} {\left\| {\nabla \mathit{\boldsymbol{u}}_H^{n + 1}} \right\|_0^2} } \right) + }\\ {c\Delta {t^2}\left\| {{\mathit{\boldsymbol{u}}_{tt}}} \right\|_{{L^\infty }\left( {0,T;{L^2}{{\left( \mathit{\Omega } \right)}^2}} \right)}^2 + c\Delta t\sum\limits_{n = 0}^{N - 1} {{\nu ^{ - 1}}\left\| {\nabla {\mathit{\boldsymbol{u}}^{n + 1}}} \right\|_0^2\left\| {\nabla \left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1}} \right)} \right\|_0^2} } \end{array} $

    最后,当Δt足够小时,应用离散Gronwall引理和式(10)有

    $ \begin{array}{*{20}{c}} {\left\| {\varphi _h^N} \right\|_0^2 + \nu \Delta t\sum\limits_{n = 0}^{N - 1} {\left\| {\nabla {\varphi _h}^{n + 1}} \right\|_0^2} \le c\Delta t\sum\limits_{n = 0}^{N - 1} {\left( {\mathop {\inf }\limits_{{w_h} \in {X_h}} \left\| {\nabla \left( {\mathit{\boldsymbol{u}}\left( {{t_{n + 1}}} \right) - {w_h}} \right)} \right\|_0^2 + \mathop {\inf }\limits_{{\lambda _h} \in {M_h}} \left\| {\mathit{\boldsymbol{p}}\left( {{t_{n + 1}}} \right) - {\lambda _h}} \right\|_0^2} \right)} + }\\ {c\Delta t\sum\limits_{n = 0}^{N - 1} {\left\| {\nabla \left( {{\mathit{\boldsymbol{u}}^{n + 1}} - \mathit{\boldsymbol{u}}_H^{n + 1}} \right)} \right\|_0^2} + c\left( {{\alpha ^2} + \Delta {t^2}} \right)} \end{array} $ (35)

    利用三角不等式,可得式(21).

    4 数值实验

    在本节中,我们利用FreeFem++软件[14]进行一些实验验证理论预测的正确性. Taylor-Hood元用于空间离散化.粗网格上的非线性迭代的迭代限差为10-6,并且非线性系统由牛顿迭代法求解.值得注意的是,对于非线性迭代,标准的变分多尺度方法和两水平变分多尺度方法的稳定项可以近似为

    $ \begin{array}{*{20}{c}} {G\left( {{\mathit{\boldsymbol{u}}_h}^j,{\mathit{\boldsymbol{v}}_h}} \right) = \alpha \sum\limits_{K \in {T^h}\left( \mathit{\Omega } \right)} {\left( {\int_{K,m} {\nabla {\mathit{\boldsymbol{u}}_h}^j \cdot \nabla {\mathit{\boldsymbol{v}}_h}{\rm{d}}x} - \int_{K,1} {\nabla \mathit{\boldsymbol{u}}_h^{j - 1} \cdot \nabla {\mathit{\boldsymbol{v}}_h}{\rm{d}}x} } \right)} }&{\forall {\mathit{\boldsymbol{u}}_h}^j,\mathit{\boldsymbol{u}}_h^{j - 1},{\mathit{\boldsymbol{v}}_h} \in {X_h}} \end{array} $

    其中j表示非线性迭代的次数.

    选择Navier-Stokes方程的精确解为:

    $ {u_1} = 10{x^2}{\left( {x - 1} \right)^2}y\left( {y - 1} \right)\left( {2y - 1} \right){{\rm{e}}^{ - t}} $
    $ {u_2} = - 10{y^2}{\left( {y - 1} \right)^2}x\left( {x - 1} \right)\left( {2x - 1} \right){{\rm{e}}^{ - t}} $
    $ p = 10\left( {2x - 1} \right)\left( {2y - 1} \right){{\rm{e}}^{ - t}} $

    其中解的区域Ω=[0, 1]×[0, 1] $\subset {{\mathbb{R}}^{2}} $,且ν=1.0×10-7T=0.01.

    定理1的误差估计在理论上预测了能量范数对于$\mathcal{O} $ (h2)的空间收敛速度.由此可设α=0.1 h2$ H={{h}^{\frac{1}{2}}}$.表 1给出了数值结果.从表 1可知:本文的方法对空间和时间离散是一阶收敛的,同时也表明我们的理论预测是正确的.

    表 1 本文的两水平变分多尺度方法近似解的误差

    为了对比本文的方法和文献[5]中的方法,令网格尺寸$ H=\frac{1}{8}, h=\frac{1}{64}$,时间步长$\Delta t=\frac{1}{800} $,但是粘性系数分别为ν=0.01,0.001,0.000 1,0.000 01和0.000 001的情况下求解.数值结果在表 2中给出.由表 2可知:本文的两水平变分多尺度方法得到的解精确度和标准的网格变分多尺度方法大体一致,但是本文的方法可以节约一半以上的计算时间.

    表 2 近似解的对比
    6 结束语

    本文给出了全离散速度的误差估计.对比标准的变分多尺度方法,本文的方法可以节约很多计算时间.

    参考文献
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    A Finite Element Variational Multiscale Method Based on the Backward Euler Scheme for the Time-Dependent Navier-Stokes Equations
    XUE Ju-feng, SHANG Yue-qiang     
    School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
    Abstract: In this paper, we mainly study a fully discrete finite element variational multiscale method based on two local Gaussian integrations for the time-dependent Navier-Stokes equations. A feature of the method is that a stabilized nonlinear Navier-Stokes system is first solved on a coarse grid, and then a stabilized linear problem is solved on a fine grid to correct the coarse grid solution at each time step. Based on the backward Euler scheme for temporal discretization, we derive error bound of the approximate velocity which is first-order in time. Numerical experiments verify the correctness of the theory and the effectiveness of the method.
    Key words: Navier-Stokes equation    two-grid method    backward Euler scheme    error estimate    
    X