有限元方法作为求解偏微分方程的重要数值方法之一, 经过70多年的发展已经形成了多个分支和研究方向, 其中, 非协调元的研究、超逼近性和超收敛性分析研究一直是有限元研究的热点问题, 研究成果非常丰富, 其基本理论和方法也都比较完整和成熟^[1-5], 但这些研究成果和理论分析大部分都基于对单元剖分的正则性条件:即剖分单元的直径与单元内切球的直径之比有界, 也就是说, 在对求解区域进行有限单元剖分时, 不能出现窄边单元, 否则会破坏正则性条件.但随着有限元分析所涉及研究领域的不断扩大和深入, 以及有限元方法的广泛应用, 这些传统有限元理论分析中的基本条件在很大程度上限制了有限元方法的应用范围, 比如, 有些复杂问题的解可能会在某些区域的拐角处或者边界层呈现出各向异性特征, 即偏微分方程的真解仅沿某一方向剧列变化; 有些问题本身的求解区域就是窄边情形, 如, 在喷气式飞机的机翼部位进行有限元的收敛性分析时, 需要对该区域网格剖分进行适当的加密处理, 而此时有限元分析中的正则性条件或拟一致条件往往不满足要求; 又如在处理多边形区域上的对流扩散问题和奇异摄动问题时, 或者在边界层或内部层及区域的拐角处求解偏微分方程的解时, 都可能会呈现出各向异性特征, 即不满足正则性条件.虽然此类问题可以通过在不同方向上用不同尺寸的剖分来解决, 但计算量将会增加很多, 如果在不满足正则性条件的网格单元上, 能得到与正则性条件下类似的收敛性质和结果, 对扩展有限元的应用范围具有重要意义.

最近, 国内外有限元专家对各向异性有限元的研究引起了广泛兴趣, 关于各向异性有限元的研究成为目前有限元的研究热点问题之一. 1992年, Apel等人在文献[6-7]中主要对各向异性Lagrange元进行了研究, 给出了各向异性元的验证条件, 但该条件使用起来非常不方便; 2004年, 陈绍春教授等人对Apel的结果进行了改进^[8], 给出了验证单元具有各向异性特征的更方便易行的方法, 并将研究的范围扩展到Hermite单元以及非协调元^[9-12]; 在此基础上, 石东洋教授又开展了各向异性超收敛分析的研究工作, 并且有一系列研究成果.在石东洋教授关于各向异性有限元超收敛分析的结果^[12]中, 所采取的核心技术是林群院士和严宁宁研究员提出的积分恒等式技巧, 但此方法非常复杂, 没有充分利用单元构造本身的特征, 同时对于双三次Hermite元的各向异性点态超收敛分析还未见报道.基于此, 本文采用一种新的、简便的分析方法-双线性引理和Bramble-Hilbert引理, 将双线性元、双二次元的各向异性超收敛性和点态超收敛性推广到双三次Hermite元.

考虑四阶问题:

$\left\{ {{\begin{array}{ll} {\Delta^{2}u=f} {\hbox{在}}{\;\;\Omega }, \\ {\left. u \right|_{\Gamma } =\left. {\dfrac{\partial u}{\partial n}} \right|_{\Gamma } =0} {\hbox{在}}{ \Gamma =\partial \Omega }, \end{array} }} \right.$

(2.1)

其中$\Omega \subset R^{2}$为有界矩形区域, 则相应变分问题为:求$u\in H_{0}^{2} (\Omega )$, 使得

$a(u, v) = f(v), \forall v \in H_0^2(\Omega ), $

(2.2)

其中$a(u, v)=\int_\Omega {(u_{xx} v_{xx} +2u_{xy} v_{xy} +u_{yy} v_{yy} )} dxdy$, $f(v)=\int_\Omega {fvdxdy} $.

设$J_{h} $是$\Omega $的一族各向异性矩形网格部分, 对$\forall T\in J_{h} $有diam$(T)\leq h$, 其顶点$a_{i} $的坐标为$(x_{i}, y_{i} )$, $i=1, 2, 3, 4$; 中心点坐标为$(x_{T}, y_{T} )$, 在X轴, Y轴方向边长分别为$2h_{x}, 2h_{y} $, 令$h_{T} =\max \{h_{x}, h_{y} \}$, $h=\max \limits_{T\in J_{h} } h_{T} $, 设$\widehat T = [ - 1,1] \times [ - 1,1]$为其参考单元, 顶点为$\hat{{a}}_{1} (-1, -1), \hat{{a}}_{2} (1, -1), $ $\hat{{a}}_{3} (1, 1), \hat{{a}}_{4} (-1, 1)$.在$\hat{{T}}$上定义有限元$(\hat{{T}}, \hat{{P}}, \hat{{\sum }})$, 这里

$\hat{{\sum }}=\left\{ {\hat{{v}}_{i}, \hat{{v}}_{i\varepsilon }, \hat{{v}}_{i\eta }, \hat{{v}}_{i\varepsilon \eta }, i=1, 2, 3, 4} \right\}, \hat{{P}}={\hbox{span}}\{p_{1}, p_{2}, \cdots, p_{16} \}, $

其中

$p_{1} =\frac{1}{16}(1-\varepsilon )^{2}(1-\eta )^{2}(2+\varepsilon )(2+\eta ), p_{2} =\frac{1}{16}(1+\varepsilon )^{2}(1-\eta )^{2}(2-\varepsilon )(2+\eta ), \\ p_{3} =\frac{1}{16}(1+\varepsilon )^{2}(1+\eta )^{2}(2-\varepsilon )(2-\eta ), p_{4} =\frac{1}{16}(1-\varepsilon )^{2}(1+\eta )^{2}(2+\varepsilon )(2-\eta ), \\ p_{5} =-\frac{1}{16}(1-\varepsilon )(\varepsilon^{2}-1)(1-\eta )^{2}(2+\eta ), p_{6} =\frac{1}{16}(1+\varepsilon )(\varepsilon^{2}-1)(1-\eta )^{2}(2+\eta ), \\ p_{7} =\frac{1}{16}(1+\varepsilon )(\varepsilon^{2}-1)(1+\eta )^{2}(2-\eta ), p_{8} =-\frac{1}{16}(1-\varepsilon )(\varepsilon^{2}-1)(1+\eta )^{2}(2-\eta ), \\ p_{9} =-\frac{1}{16}(1-\eta )(\eta^{2}-1)(1-\varepsilon )^{2}(2+\varepsilon ), p_{10} =-\frac{1}{16}(1-\eta )(\eta^{2}-1)(1+\varepsilon )^{2}(2-\varepsilon ), \\ p_{11} =\frac{1}{16}(1+\eta )(\eta^{2}-1)(1+\varepsilon )^{2}(2-\varepsilon ), p_{12} =\frac{1}{16}(1+\eta )(\eta^{2}-1)(1-\varepsilon )^{2}(2+\varepsilon ), \\ p_{13} =\frac{1}{16}(1-\varepsilon^{2})(1-\eta^{2})(1-\varepsilon )(1-\eta ), p_{14} =-\frac{1}{16}(1-\varepsilon^{2})(1-\eta^{2})(1+\varepsilon )(1-\eta ), \\ p_{15} =\frac{1}{16}(1-\varepsilon^{2})(1-\eta^{2})(1+\varepsilon )(1+\eta ), p_{16} =-\frac{1}{16}(1-\varepsilon^{2})(1-\eta^{2})(1-\varepsilon )(1+\eta ), $

则$\forall \hat{{v}}\in \hat{{P}}$, $\prod_{\hat{{T}}} \hat{{v}}=\sum\limits_{i=1}^{16} {p_{i} (\varepsilon, \eta )a_{i} } $, 其中$a_{i} (\hat{{v}})=\hat{{v}}_{i} $, $a_{i+4} (\hat{{v}})=\hat{{v}}_{i\varepsilon } $, $a_{i+8} (\hat{{v}})=\hat{{v}}_{i\eta } $, $a_{i+12} (\hat{{v}})=\hat{{v}}_{i\varepsilon \eta } (i=1, 2, 3, 4)$

设$F_{T} $为从$\hat{{T}}\to T$的仿射变换: $F_{T} :\left\{ {\begin{array}{l} x=x_{T} +h_{x} \varepsilon, \\ y=y_{T} +h_{y} \eta, \end{array}} \right.$则有限元空间$V_{h} $可定义为

$\begin{array}{l} {V_h} = \{ {v_h}:{\widehat v_h} = {\left. {{v_h}} \right|_T} \cdot {F_T} \in \widehat P, \forall T \in {J_h}\}, \\ V_0^h = \{ v \in {V_h};{v_h}{|_{\partial \Omega }} = \frac{{\partial v}}{{\partial n}}{|_{\partial \Omega }} = 0\} . \end{array}$

对于坐标为$(x, y)$的一般矩形单元$T$, 插值算子可定义为$\prod_{T} :H^{5}(T)\to \hat{{P}}\cdot F_{T}^{-1} $, $\prod_{T} v:H^{5}(T)\to (\hat{{\prod }}\hat{{v}})\cdot F_{T}^{-1} $, $\prod_{h} :H^{5}(\Omega )\to V_{h} $, $\left. {\prod_{h} } \right|_{T} =\prod_{T} $, 则问题(2.2) 的逼近形式为:求$u_{h} \in V_{h} $, 使得

$a({u_h}, {v_h}) = f({v_h}), \forall {v_h} \in {V_h}.$

(2.3)

容易验证$\left\| {\cdot } \right\|_{h} =(a(\cdot, \cdot ))^{\frac{1}{2}}=(\sum\limits_{T\in J_{h} } {\left| {\cdot } \right|_{2, T}^{2} } )^{\frac{1}{2}}$是$V_{h} $上的范数.

引理2.1  假设$u\in H^{5}(\Omega )\cap H_{0}^{2} (\Omega )$是问题(2.1) 的解, $\Pi_{h} u$为$u$的分片双三次插值, 设$w=u-\Pi_{h} u$, 则$\forall v\in V_{h} $有

$\begin{equation} \left\{ {\begin{array}{l} \displaystyle \int_\Omega {w_{xx} v_{xx} dxdy} \leq ch^{3}\left| u \right|_{5, \Omega } \left| v \right|_{2, \Omega }, \\\\[-8pt] \displaystyle \int_\Omega {w_{yy} v_{yy} dxdy} \leq ch^{3}\left| u \right|_{5, \Omega } \left| v \right|_{2, \Omega }, \\\\[-8pt] \displaystyle \int_\Omega {w_{xy} v_{xy} dxdy} \leq ch^{3}\left| u \right|_{5, \Omega } \left| v \right|_{2, \Omega }, \end{array}} \right. \end{equation}$

(2.4)

其中$c$与$\dfrac{h_{T} }{\rho_{T} }, \dfrac{h}{h_{T} }$无关.

证 设$Q(\hat{{u}}, \hat{{v}})=\displaystyle\int_{\hat{{T}}} {\frac{\partial ^{2}(\hat{{u}}-\hat{{\Pi }}_{\hat{{T}}} \hat{{u}})}{\partial \varepsilon ^{2}}\frac{\partial^{2}\hat{{v}}}{\partial \varepsilon^{2}}d\varepsilon d\eta } $, 则$Q(\hat{{u}}, \hat{{v}})$为$H^{5}(\Omega )\times H_{0}^{2} (\Omega )$上的双线性型.下面证明$\displaystyle\frac{\partial^{2}\hat{{u}}}{\partial \varepsilon^{2}}\in P_{2} (\hat{{T}})$时, 有$Q(\hat{{u}}, \hat{{v}})=0$.

事实上, 当$\hat{{u}}\in P_{3} (\hat{{T}})$, 由插值函数的定义$\hat{{\Pi }}_{\hat{{T}}} \hat{{u}}=\hat{{u}}$, 从而$\displaystyle\int_{\hat{{T}}} {\frac{\partial ^{2}(\hat{{u}}-\hat{{\Pi }}_{\hat{{T}}} \hat{{u}})}{\partial \varepsilon ^{2}}\frac{\partial^{2}\hat{{v}}}{\partial \varepsilon^{2}}d\varepsilon d\eta } =0$, 故只需验证$\hat{{u}}=\varepsilon^{4}, \hat{{u}}=\eta ^{4}$时, $Q(\hat{{u}}, \hat{{v}})=0$即可.当$\hat{{u}}=\varepsilon ^{4}$时, 由插值函数知$\hat{{\Pi }}_{\hat{{T}}} \hat{{u}}=2\varepsilon ^{2}-1$, 由此得到

$ \int_{\hat{{T}}} {\frac{\partial^{2}(\hat{{u}}-\hat{{\Pi }}_{\hat{{T}}} \hat{{u}})}{\partial \varepsilon^{2}}\frac{\partial^{2}\hat{{v}}}{\partial \varepsilon^{2}}d\varepsilon d\eta } =\int_{\hat{{T}}} {4(3\varepsilon ^{2}-1)\frac{\partial^{2}\hat{{v}}}{\partial \varepsilon^{2}}d\varepsilon d\eta } =0. $

当$\displaystyle\frac{\partial^{2}\hat{{v}}}{\partial \varepsilon^{2}}\in P_{0} (\hat{{T}})$时, 由插值条件和Green公式可知

$ \int_{\hat{{T}}} {\frac{\partial^{2}(\hat{{u}}-\hat{{\Pi }}_{\hat{{T}}} \hat{{u}})}{\partial \varepsilon^{2}}\frac{\partial^{2}\hat{{v}}}{\partial \varepsilon^{2}}d\varepsilon d\eta } =\frac{\partial ^{2}\hat{{v}}}{\partial \varepsilon^{2}}\int_{\partial \hat{{T}}} {\frac{\partial (\hat{{u}}-\hat{{\Pi }}_{\hat{{T}}} \hat{{u}})}{\partial \varepsilon }n_{\varepsilon } d\varepsilon =0}. $

故由双线性引理

$\begin{array}{l} \left| {Q(\widehat u, \widehat v)} \right| = \left| {\int_{\widehat T} {\frac{{{\partial ^2}(\widehat u - {{\widehat \Pi }_{\widehat T}}\widehat u)}}{{\partial {\varepsilon ^2}}}\frac{{{\partial ^2}\widehat v}}{{\partial {\varepsilon ^2}}}d\varepsilon d\eta } } \right|\\ \le c{\left( {\sum\limits_{\left| \alpha \right| = 3} {\left\| {{D^\alpha }\frac{{{\partial ^2}\widehat u}}{{\partial {\varepsilon ^2}}}} \right\|_{0, \widehat T}^2} } \right)^{\frac{1}{2}}}{\left( {\sum\limits_{\left| \alpha \right| = 1} {\left\| {{D^\alpha }\frac{{{\partial ^2}\widehat v}}{{\partial {\varepsilon ^2}}}} \right\|_{0, \widehat T}^2} } \right)^{\frac{1}{2}}}. \end{array}$

(2.5)

由(2.5) 式及有限维空间上的模等价性得

$\begin{array}{l} \left| {\int_\Omega {{w_{xx}}{v_{xx}}dxdy} } \right| \le \sum\limits_{T \in {J_h}} {\left| {\int_T {{w_{xx}}{v_{xx}}dxdy} } \right|} = \sum\limits_{\widehat T \in {J_h}} {h_x^{ - 3}{h_y}\left| {\int_{\widehat T} {{w_{\varepsilon \varepsilon }}{v_{\varepsilon \varepsilon }}d\varepsilon d\eta } } \right|} \\ \le \sum\limits_{\widehat T \in {J_h}} {h_x^{ - 3}{h_y}c{{\left( {\sum\limits_{\left| \alpha \right| = 3} {\left\| {{D^\alpha }\frac{{{\partial ^2}\widehat u}}{{\partial {\varepsilon ^2}}}} \right\|_{0, \widehat T}^2} } \right)}^{\frac{1}{2}}}{{\left( {\sum\limits_{\left| \alpha \right| = 1} {\left\| {{D^\alpha }\frac{{{\partial ^2}\widehat v}}{{\partial {\varepsilon ^2}}}} \right\|_{0, \widehat T}^2} } \right)}^{\frac{1}{2}}}} \\ \le \sum\limits_{\widehat T \in {J_h}} {c{{\left( {\sum\limits_{\left| \alpha \right| = 3} {\left\| {{D^\alpha }\frac{{{\partial ^2}\widehat u}}{{\partial {\varepsilon ^2}}}} \right\|_{0, \widehat T}^2} h_T^{2\alpha }} \right)}^{\frac{1}{2}}}{{\left( {\sum\limits_{\left| \alpha \right| = 1} {\left\| {{D^\alpha }\frac{{{\partial ^2}\widehat v}}{{\partial {\varepsilon ^2}}}} \right\|_{0, \widehat T}^2} } \right)}^{\frac{1}{2}}}} \\ \le c{h^3}{\left| u \right|_{5, \Omega }}{\left| v \right|_{2, \Omega }}, \quad \forall v \in {V_h}. \end{array}$

通过引进双线性型$Q(\hat{{u}}, \hat{{v}})=\displaystyle\int_{\hat{{T}}} {\frac{\partial ^{2}(\hat{{u}}-\hat{{\Pi }}_{\hat{{T}}} \hat{{u}})}{\partial \eta ^{2}}\frac{\partial^{2}\hat{{v}}}{\partial \eta^{2}}d\varepsilon d\eta } $以及$Q(\hat{{u}}, \hat{{v}})=\displaystyle\int_{\hat{{T}}} \frac{\partial ^{2}(\hat{{u}}-\hat{{\Pi }}_{\hat{{T}}} \hat{{u}})}{\partial \varepsilon \partial \eta }$ $\displaystyle\frac{\partial^{2}\hat{{v}}}{\partial \varepsilon \partial \eta }$ $d\varepsilon d\eta $, 同理可证(2.4) 其它二个不等式.

定理2.1  设$u$, $\hat{{u}}$分别是问题(2.1) 和(2.3) 的解, 其中$u\in H^{5}(\Omega )\cap H_{0}^{2} (\Omega )$, $V_{h} $为各向异性矩形网格下的双三次有限元空间, $\Pi_{h} u\in V_{h} $为$u$的双三次Hermite插值, $\left. {\Pi_{h} } \right|_{T} =\Pi_{T} $, 则$\left\| {u_{h} -\Pi_{h} u} \right\|_{h} \leq ch^{3}\left| u \right|_{5, \Omega } $.

证 由引理2.1可知$\forall v\in V_{h} $,

$\begin{array}{l} a({u_h} - {\Pi _h}u, v) = a({u_h} - u + u - {\Pi _h}u, v) = a(u - {\Pi _h}u, v)\\ = \sum\limits_{T \in {J_h}} {\int_T {({w_{xx}}{v_{xx}} + 2{w_{xy}}{v_{xy}} + {w_{yy}}{v_{yy}})} dxdy} \le c{h^3}{\left| u \right|_{5, \Omega }}{\left| v \right|_{2, \Omega }}. \end{array}$

取$v=u_{h} -\Pi_{h} u$, 得$\left\| {u_{h} -\Pi_{h} u} \right\|_{h} =(a(u_{h} -\Pi_{h} u, u_{h} -\Pi_{h} u))^{\frac{1}{2}}\leq ch^{3}\left| u \right|_{5, \Omega } $.

定理3.1  假设$u\in H^{5}(\Omega )\cap H_{0}^{2} (\Omega )$是问题(2.1) 的解, $u_{h} $是(2.3) 式的解, $O_{T} $是单元$T$内的点, 其在参考单元$\hat{{T}}$上对应的点$\hat{{O}}_{\hat{{T}}} $满足$3\varepsilon^{2}-1=0$和$3\eta^{2}-1=0$, 即$O_{T} $是单元$T$上的高斯点, 则

$ \left(\sum\limits_{T\in J_{h} } {\left| {D^{2}(u-u_{h} )(O_{T} )} \right|^{2}h_{x}^{2}h_{y}^{2}} \right)^{\frac{1}{2}}\leq ch^{4}\left| u \right|_{5, \Omega }. $

证

$\begin{array}{l} {\left( {\sum\limits_{T \in {J_h}} {{{\left| {{D^2}(u - {u_h})({O_T})} \right|}^2}h_x^2h_y^2} } \right)^{\frac{1}{2}}} \le {\left( {\sum\limits_{T \in {J_h}} {{{\left| {{D^2}(u - {\Pi _T}u)({O_T})} \right|}^2}h_x^2h_y^2} } \right)^{\frac{1}{2}}}\\ + {\left( {\sum\limits_{T \in {J_h}} {{{\left| {{D^2}({\Pi _T}u - {u_h})({O_T})} \right|}^2}h_x^2h_y^2} } \right)^{\frac{1}{2}}}\mathop = \limits^\Delta M + N. \end{array}$

(3.1)

首先估计$M$.设$\hat{{O}}_{\hat{{T}}} $为在参考单元$\hat{{T}}$上与单元$T$上$O_{T} $对应的点, 则

$ \left| {D^{2}(u-\Pi_{T} u)(O_{T} )} \right|^{2}\leq \left| {\frac{\partial ^{2}(u-\Pi_{T} u)}{\partial x^{2}}(O_{T} )} \right|^{2}+\left| {\frac{\partial^{2}(u-\Pi_{T} u)}{\partial x\partial y}(O_{T} )} \right|^{2}+\left| {\frac{\partial^{2}(u-\Pi_{T} u)}{\partial y^{2}}(O_{T} )} \right|^{2}. $

令$\hat{{Q}}(\hat{{u}})=\left| \displaystyle{\frac{\partial^{2}(\hat{{u}}-\hat{{\Pi }}_{\hat{{T}}} \hat{{u}})}{\partial \varepsilon^{2}}(\hat{{O}}_{\hat{{T}}} )} \right|$, 则$\forall \displaystyle\frac{\partial^{2}\hat{{u}}}{\partial \varepsilon^{2}}\in P_{2} (\hat{{T}})$, 当$3\varepsilon ^{2}-1=0$时, 由引理2.1的证明可知$\hat{{Q}}(\hat{{u}})=0$, 再由Bramble-Hilbert引理可得

${\left| {\frac{{{\partial ^2}(u - {\Pi _T}u)}}{{\partial {x^2}}}({O_T})} \right|^2} = h_x^{ - 4}{\left| {\widehat Q(\widehat u)} \right|^2} \le c{({h_x}{h_y})^{ - 2}}h_T^2\sum\limits_{\left| \alpha \right| = 3} {h_T^{2\alpha }\left\| {{D^\alpha }\frac{{{\partial ^2}u}}{{\partial {x^2}}}} \right\|_{0, T}^2} .$

(3.2)

同理可证

${\left| {\frac{{{\partial ^2}(u - {\Pi _T}u)}}{{\partial {y^2}}}({O_T})} \right|^2} \le c{({h_x}{h_y})^{ - 2}}h_T^2\sum\limits_{\left| \alpha \right| = 3} {h_T^{2\alpha }\left\| {{D^\alpha }\frac{{{\partial ^2}u}}{{\partial {y^2}}}} \right\|_{0, T}^2} .$

(3.3)

令$\hat{{L}}(\hat{{u}})=\left| \displaystyle{\frac{\partial^{2}(\hat{{u}}-\hat{{\Pi }}_{\hat{{T}}} \hat{{u}})}{\partial \varepsilon \partial \eta }(\hat{{O}}_{\hat{{T}}} )} \right|$, 则$\forall \displaystyle\frac{\partial ^{2}\hat{{u}}}{\partial \varepsilon \partial \eta }\in P_{2} (\hat{{T}})$, 由引理2.1的证明过程知$\hat{{L}}(\hat{{u}})=0$, 又由Bramble-Hilbert引理可得

${\left| {\frac{{{\partial ^2}(u - {\Pi _T}u)}}{{\partial x\partial y}}({O_T})} \right|^2} = h_x^{ - 2}h_y^{ - 2}{\left| {\widehat L(\widehat u)} \right|^2} \le c{({h_x}{h_y})^{ - 2}}h_T^2\sum\limits_{\left| \alpha \right| = 3} {h_T^{2\alpha }\left\| {{D^\alpha }\frac{{{\partial ^2}u}}{{\partial x\partial y}}} \right\|_{0, T}^2} .$

(3.4)

将(3.2)-(3.4) 式代入$M$得$M\leq ch^{4}\left| u \right|_{5, \Omega } $.

下面估计$N$, 由有限维空间的模等价原理知

$\begin{array}{l} {\left| {{D^2}({\Pi _T}u - {u_h})({O_T})} \right|^2}\\ \le {\left| {\frac{{{\partial ^2}({\Pi _T}u - {u_h})}}{{\partial {x^2}}}({O_T})} \right|^2} + {\left| {\frac{{{\partial ^2}({\Pi _T}u - {u_h})}}{{\partial x\partial y}}({O_T})} \right|^2} + {\left| {\frac{{{\partial ^2}({\Pi _T}u - {u_h})}}{{\partial {y^2}}}({O_T})} \right|^2}\\ = h_x^{ - 4}{\left| {\frac{{{\partial ^2}({{\widehat \Pi }_{\widehat T}}\widehat u - {{\widehat u}_h})}}{{\partial {\varepsilon ^2}}}({{\widehat O}_{\widehat T}})} \right|^2} + h_x^{ - 2}h_y^{ - 2}{\left| {\frac{{{\partial ^2}({{\widehat \Pi }_{\widehat T}}\widehat u - {{\widehat u}_h})}}{{\partial \varepsilon \partial \eta }}({{\widehat O}_{\widehat T}})} \right|^2} + h_y^{ - 4}{\left| {\frac{{{\partial ^2}({{\widehat \Pi }_{\widehat T}}\widehat u - {{\widehat u}_h})}}{{\partial {\eta ^2}}}({{\widehat O}_{\widehat T}})} \right|^2}\\ \le h_x^{ - 4}\left\| {\frac{{{\partial ^2}({{\widehat \Pi }_{\widehat T}}\widehat u - {{\widehat u}_h})}}{{\partial {\varepsilon ^2}}}} \right\|_{0, \widehat T}^2 + h_x^{ - 2}h_y^{ - 2}\left\| {\frac{{{\partial ^2}({{\widehat \Pi }_{\widehat T}}\widehat u - {{\widehat u}_h})}}{{\partial \varepsilon \partial \eta }}} \right\|_{0, \widehat T}^2 + h_y^{ - 4}\left\| {\frac{{{\partial ^2}({{\widehat \Pi }_{\widehat T}}\widehat u - {{\widehat u}_h})}}{{\partial {\eta ^2}}}} \right\|_{0, \widehat T}^2\\ \le c{({h_x}{h_y})^{ - 2}}h_T^2\left| {{\Pi _T}u - {u_h}} \right|_{2, \widehat T}^2. \end{array}$

由定理2.1知$N\leq ch^{4}\left| u \right|_{5, \Omega } $, 将$M, N$代入(3.1) 式, 定理得证.