考虑如下的抛物积分微分方程

(1)$\begin{equation}\left\{ {\begin{array}{ll}u_t -\Delta u-\int_0^t {\Delta u\left( {X, s} \right)} {\rm d}s=f\left( {X, t}\right), & \left( {X, t} \right)\in \Omega \times \left( {0, T} \right], \\[2mm] u\left( {X, t} \right)=0, & \left( {X, t} \right)\in \partial \Omega \times\left( {0, T} \right], \\ u\left( {X, 0} \right)=u_0 \left( X \right), & X\in \Omega , \end{array}} \right.\end{equation}$

其中$\Omega \in {\mathbb R}^2$是一个矩形区域, $\partial \Omega $为$\Omega $的边界, $T>0$为一常数, $X=\left({x, y} \right)$, $f\left({X, t} \right)$及$u_0 (X)$是已知光滑函数.

抛物积分微分方程来源于许多物理和工程实际问题,例如,具有记忆性质材料的热传导,粘弹性体压缩,流动流体核衰变等.积分项的出现使其与传统的抛物方程有着本质的区别,数值求解也更为困难.关于此类方程的有限元方法研究已经有了一些结果^[1-16].文献[1-2]讨论了双线性元和类Wilson元的超收敛性.文献[3]得到了最佳的误差估计和解的正则性条件.文献[4]通过构造适当的插值后处理算子,得到了各向异性网格下的整体超收敛结果.文献[5]利用插值与投影相结合,在降低对解的光滑度要求下,得到了与以往文献相同的超逼近结果.文献[6-7]探讨了带弱奇异核的抛物积分微分方程的$EQ_1^{rot} $非协调元和Hermite型各向异性矩形元逼近.文献[8-9]研究了非线性抛物积分微分方程的Wilson元和类Wilson元逼近问题.文献[10-11]借助于Ritz-Volterra投影分别讨论了非协调Carey元的收敛性和一维情况下的点态超收敛现象.文献[12]利用$H^1$-Galerkin非协调有限元方法,在不需要LBB条件的情况下,得到了与传统混合元相同的收敛阶.文献[13-16]分别构造了新的混合有限元逼近格式.然而,我们发现上述文献提高收敛阶的方法大多是利用插值后处理技术,外推或者构造新的单元.文献[17-18]在研究二阶抛物和椭圆问题时,借鉴了文献[19-20]的思想方法,利用内部惩罚方法构造了一个新的离散变分形式,并在双线性型中添加了稳定项,使得Wilson元相容误差项变成了0,收敛阶提高到$O\left({h^2} \right)$.

本文的主要目的是将文献[17-18]的思想应用于问题(1.1)的收敛性研究.首先,证明了半离散格式逼近解的存在唯一性.其次,利用Wilson元在新范数意义下的插值估计和双线性型的性质导出了半离散格式下原始变量$u$在新范数意义下$O\left({h^2} \right)$阶的收敛结果.再次,通过构造一个具有一阶精度的全离散格式,得到了相应的$O\left({h^2+\tau } \right)$阶的高精度结果,这是文献[17]和[18]未涉及的.最后,给出了一个数值算例,计算结果验证了理论分析的正确性.

本文中, $W^{s, p}\left(D \right)\left({D\subset \Omega } \right)$表示通常的Sobolev空间, $\left\| \cdot \right\|_{s, p, D} $和$\left| \cdot \right|_{s, p, D} $分别表示其上的范数和半范,其中$s$为非负整数, $1\le p\le \infty $.当$p=2$时,记$W^{s, 2}\left(D \right)=H^s\left(D \right)$, $\left\| \cdot \right\|_{s, D} $和$\left| \cdot \right|_{s, D} $分别表示$H^s\left(\Omega \right)$上的范数和半范,当$D=\Omega $时,省略下标$D$.约定

$\left\| g\right\|_{L^2\left( {0, T;X} \right)} =\bigg( {\int_0^T {\left\| {g\left({\cdot , t} \right)} \right\|_X^2 {\rm d}t} }\bigg)^{\frac{1}{2}}, \ \ \left\| \cdot\right\|_{L^\infty \left( {0, T;X} \right)} =\mathop {\sup }\limits_{0\let\le T} \left\| \cdot \right\|_X , $

其中$X$为Banach空间.本文中, $C$是与$h$和$\tau $无关的正常数, $C$以及$\varepsilon $在不同的地方取值可以不同.

设$\Omega $的边界$\partial \Omega $分别平行于$x$轴与$y$轴, $\Gamma _h $是$\Omega $的矩形单元剖分族,满足正则性假设.对任何$K\in \Gamma _h $,设其中心为$\left({x_K, y_K } \right)$,平行于$x$轴与$y$轴的边分别是$l_1 $, $l_3 $及$l_2 $, $l_4 $,边长分别为2$h_1 $, 2$h_2 $,记$h_K =\mathop {\max }\limits_{K\in \Gamma _h } \left\{ {h_1, h_2 } \right\}$, $h=\mathop {\max }\limits_{K\in \Gamma _h } \left\{ {h_K } \right\}$.

设$\varepsilon _h $表示所有单元边界所成的集合, $E$表示单元边界, $h_E $表示$E$的长度.规定函数$f$在相邻单元$K$和${K}'$的相交边上的跳跃值和平均值分别为$\left[{\left[f \right]} \right]=\left. f \right|_K -\left. f \right|_{{K}'} $和$\left\{ f \right\}=\frac{1}{2}\left({\left. f \right|_K +\left. f \right|_{{K}'} } \right)$.

Wilson元的形函数空间为$P_2 (K)$,有限元空间为

$V_h =\Big\{ {v_h}\ \mbox{定义在$\Omega $上, ${v_h } |_K \in P_2 (K)$, $v_h $在内节点处连续,在边界$\partial \Omega $上节点处为$0\Big\} $}, $

其中$\left. {v_h } \right|_K $由四顶点函数值以及$\frac{h_1^2 }{h_1 h_2 }\int_K {\frac{\partial ^2v_h }{\partial x^2}} {\rm d}X$和$\frac{h_2^2 }{h_1 h_2 }\int_K {\frac{\partial ^2v_h }{\partial y^2}} {\rm d}X$确定.

定义$V_h $上的模和双线性型^[17]分别为

(2.1)$\begin{equation}\left\| {v_h } \right\|_h^2 = \sum\limits_{K\in \Gamma _h } {\left| {v_h }\right|_{1, K}^2 } +\sum\limits_{E\in \varepsilon _h } {\left\{ {\frac{1}{h_E}\left\| {\left[ {\left[ {v_h } \right]} \right]} \right\|_{0, E}^2 }\right\}} , \end{equation}$

(2.2)$\begin{eqnarray}a_h \left( {u_h , v_h } \right) &= & \left( {\nabla u_h , \nabla v_h } \right)_h+\sum\limits_{E\in \varepsilon _h } {\left\{ {\frac{\alpha }{h_E}\left\langle {\left[ {\left[ {u_h } \right]} \right], \left[ {\left[ {v_h }\right]} \right]} \right\rangle _E } \right\}} \nonumber \\&& -\left\langle {\left\{ {\frac{\partial u_h }{\partial n}} \right\}, \left[{\left[ {v_h } \right]} \right]} \right\rangle _h -\left\langle {\left[{\left[ {u_h } \right]} \right], \left\{ {\frac{\partial v_h }{\partial n}}\right\}} \right\rangle _h , \end{eqnarray}$

其中$\left({\cdot, \cdot } \right)_h =\sum\limits_{K\in \Gamma _h } {\left({\cdot, \cdot } \right)_K } $, $\left\langle {\cdot, \cdot } \right\rangle _h =\sum\limits_{E\in \varepsilon _h } {\left\langle {\cdot, \cdot } \right\rangle _E } $, $\alpha $是待定常数. $I_h: H^2\left(\Omega \right)\to V_h $为相应的插值算子.

文献[17-18]已证明如下三个结论:

引理2.1  设$u\in H^3\left(\Omega \right)$,则有

$\left\| {u-I_h u} \right\|_h \le Ch^2\left\| u \right\|_3 .$

引理2.2  设$\left\{ {\Gamma _h } \right\}$是区域$\Omega $的正则矩形剖分,即存在$\sigma >0$,使得$\forall K\in \Gamma _h $,有$\frac{h_K }{\rho _K }\le \sigma $,则存在$C_0 >0$,使得

$h_K \left\| {\frac{\partial v_h }{\partial n}} \right\|_{0, \partial K}^2 \leC_0 \left| {v_h } \right|_{1, K}^2 , \forall v_h \in V_h , $

其中$C_0 =\sigma \left({1+\frac{2}{\varepsilon_{0}^2}+6\varepsilon_{0}^2} \right)$, $\varepsilon_{0} >0$.

引理2.3   $a_h \left({\cdot, \cdot } \right)$: $V_h \times V_h \to \mathbb{R}$是连续的, $V$ -椭圆的双线性型,即$\exists C_1, \alpha _0 >0$,使得

$a_h \left( {u_h , v_h } \right)\le C_1 \left\| {u_h } \right\|_h \left\| {v_h} \right\|_h , \forall u_h , v_h \in V_h , $

$a_h \left( {v_h , v_h } \right)\ge \alpha _0 \left\| {v_h } \right\|_h^2 , \forall v_h \in V_h .$

与(1.1)式等价的变分问题为:求$u\in H_0^1 \left(\Omega \right)$,使得

(3.1)$\begin{equation}\left\{ {\begin{array}{ll} \left( {u_t , v} \right)+\left( {\nabla u, \nabla v} \right)+\int_0^t {\left({\nabla u\left( {X, s} \right), \nabla v} \right)} {\rm d}s=\left( {f, v}\right), & \forall v\in H_0^1 \left( \Omega \right), \\[2mm] u\left( {X, 0} \right)=u_0 \left( X \right), & X\in \Omega . \end{array}} \right.\end{equation}$

(3.1)式的传统半离散格式为:求$u_h \in V_h $,使得

(3.2)$\begin{equation}\left\{ {\begin{array}{ll} \left( {u_{ht} , v_h } \right)+\left( {\nabla u_h , \nabla v_h }\right)+\int_0^t {\left( {\nabla u_h \left( {X, s} \right), \nabla v_h }\right)} {\rm d}s=\left( {f, v_h } \right), &\forall v_h \in V_h , \\[2mm] u_h \left( {X, 0} \right)=I_h u_0 \left( X \right), & X\in \Omega . \end{array}} \right.\end{equation}$

为提高逼近精度,我们引进新的半离散格式为:求$u_h \in V_h $,使得

(3.3)$\begin{equation}\left\{ {\begin{array}{ll} \left( {u_{ht} , v_h } \right)+a_h \left( {u_h , v_h } \right)+\int_0^t {a_h\left( {u_h \left( {X, s} \right), v_h } \right)} {\rm d}s=\left( {f, v_h }\right), & \forall v_h \in V_h , \\[2mm] u_h \left( {X, 0} \right)=I_h u_0 \left( X \right), & X\in \Omega . \end{array}} \right.\end{equation}$

定理3.1  问题(3.3)存在唯一解.

证  设$\left\{ {\varphi _i \left(X \right)} \right\}_{i=1}^r $是$V_h $的基函数,则

$u_h =\sum\limits_{i=1}^r {l_i \left( t \right)\varphi _i \left( X \right)} , I_h u_0 =u_h \left( 0 \right)=\sum\limits_{i=1}^r {l_i \left( 0 \right)\varphi _i \left( X \right)} , \left( {X, t} \right)\in \Omega \times \left( {0, T} \right].$

在(3.3)式中选取$v_h =\varphi _j, j=1, 2, \cdots, r$,可得

(3.4)$\begin{equation}\left\{ {\begin{array}{l}A\frac{d\vec {L}\left( t \right)}{{\rm d}t}+B\vec {L}\left( t \right)+\int_0^t{B\vec {L}\left( s \right)} {\rm d}s=\vec {F}, \\\vec L\left( 0 \right)=L_0 , \\ \end{array}} \right.\end{equation}$

其中$L_0 $由$u_h \left({X, 0} \right)=I_h u_0 \left(X \right)$决定, $\vec {L}\left(t \right)=\left({l_1 \left(t \right), l_2 \left(t \right), \cdots, l_r \left(t \right)} \right)^\prime $, $A=\left({\left({\varphi _i, \varphi _j } \right)} \right)_{r\times r} $, $B=\left({a_h \left({\varphi _i, \varphi _j } \right)} \right)_{r\times r} $, $\vec {F}=\left({\left({f, \varphi _j } \right)} \right)_{r\times 1} $.

从而(3.4)式是关于向量$\vec {L}\left(t \right)$的一个常微分方程组.注意到$A$, $B$是正定矩阵,根据常微分方程的理论,当$t>0$时, $\vec {L}\left(t \right)$是唯一存在的.因而问题(3.3)的解存在唯一.证毕.

下面先讨论上述问题的收敛性.

定理3.2  设$u$, $u_h $分别是(1.1)和(3.3)式的解.若$u, u_t \in L^2\left({0, T; H^3\left(\Omega \right)} \right)$,则有

$\left\| {u-u_h } \right\|_h \le Ch^2\left( {\left\| u \right\|_3 +\left\|{u_t } \right\|_{L^2\left( {0, T;H^3\left( \Omega \right)} \right)} +\left\|u \right\|_{L^2\left( {0, T;H^3\left( \Omega \right)} \right)} } \right).$

证  令$u-u_h =\left({u-I_h u} \right)-\left({u_h -I_h u}\right)\dot {=}\omega -\theta $.

由于$u\in H^3\left( \Omega \right)$,则有

$a_h \left( {u, v_h } \right)=\left( {\nabla u, \nabla v_h } \right)_h-\left\langle {\left\{ {\frac{\partial u}{\partial n}} \right\}, \left[{\left[ {v_h } \right]} \right]} \right\rangle _h , \forall v_h \in V_h .$

根据(1.1)和(3.3)式有下面误差方程

(3.5)$\begin{equation}\left( {\theta _t , v_h } \right)+a_h \left( {\theta , v_h } \right)=\left( {\omega _t , v_h } \right)+a_h \left( {\omega , v_h } \right)+\int_0^t{a_h \left( {u\left( {X, s} \right)-u_h \left( {X, s} \right), v_h } \right)}{\rm d}s .\end{equation}$

在(3.5)式中令$v_h =\theta _t $得

(3.6)$\begin{equation}\left( {\theta _t , \theta _t } \right)+a_h \left( {\theta , \theta _t }\right)=\left( {\omega _t , \theta _t } \right)+a_h \left( {\omega , \theta _t }\right)+\int_0^t {a_h \left( {u\left( {X, s} \right)-u_h \left( {X, s}\right), \theta _t } \right)} {\rm d}s\dot {=}\sum\limits_{i=1}^3 {G_i } .\end{equation}$

首先注意到

(3.7)$\begin{eqnarray}a_h \left( {\theta , \theta _t } \right) & = &\left( {\nabla \theta , \nabla \theta_t } \right)_h +\sum\limits_{E\in \varepsilon _h } {\left\{ {\frac{\alpha }{h_E}\left\langle {\left[ {\left[ \theta \right]} \right], \left[ {\left[ {\theta_t } \right]} \right]} \right\rangle _E } \right\}} -\left\langle {\left\{{\frac{\partial \theta }{\partial n}} \right\}, \left[ {\left[ {\theta _t }\right]} \right]} \right\rangle _h -\left\langle {\left[ {\left[ \theta\right]} \right], \left\{ {\frac{\partial \theta _t }{\partial n}} \right\}}\right\rangle _h \nonumber \\& =&\frac{\rm d}{{\rm d}t}\left\{ {\frac{1}{2}\sum\limits_{K\in \Gamma _h } {\left\|{\nabla \theta } \right\|_{0, K}^2 +\sum\limits_{E\in \varepsilon _h }{\left[ {\frac{\alpha }{2h_E }\left\| {\left[ {\left[ \theta \right]}\right]} \right\|_{0, E}^2 -\left\langle {\left\{ {\frac{\partial \theta}{\partial n}} \right\}, \left[ {\left[ \theta \right]} \right]}\right\rangle _E } \right]} } } \right\}.\end{eqnarray}$

下面对$G_1 \sim G_3 $逐项进行估计

$G_1 \le Ch^2\left\| {u_t } \right\|_2 \left\| {\theta _t } \right\|_0 \leCh^4\left\| {u_t } \right\|_2^2 +\varepsilon \left\| {\theta _t }\right\|_0^2 , $

$G_2 =\frac{\rm d}{{\rm d}t}a_h \left( {\omega , \theta } \right)-a_h \left( {\omega _t, \theta } \right)\le \frac{\rm d}{{\rm d}t}a_h \left( {\omega , \theta } \right)+Ch^4\left\| {u_t }\right\|_3^2 +\varepsilon \left\| \theta \right\|_h^2 , $

$\begin{eqnarray*}G_3 & =&\frac{\rm d}{{\rm d}t}\int_0^t {a_h \left( {u\left( {X, s} \right)-u_h \left( {X, s}\right), \theta } \right)} {\rm d}s-a_h \left( {u-u_h, \theta } \right) \\& \le& \frac{\rm d}{{\rm d}t}\int_0^t {a_h \left( {u\left( {X, s} \right)-u_h \left( {X, s}\right), \theta } \right)} {\rm d}s+Ch^4\left\| u \right\|_3^2 +C\left\|\theta \right\|_h^2 .\end{eqnarray*}$

将$G_1 \sim G_3 $的估计及(3.7)式代入(3.6)式,可将(3.6)式化为

(3.8)$\begin{eqnarray}&& \left\| {\theta _t } \right\|_0^2 +\frac{\rm d}{{\rm d}t}\left\{{\frac{1}{2}\sum\limits_{K\in \Gamma _h } {\left\| {\nabla \theta }\right\|_{0, K}^2 } +\sum\limits_{E\in \varepsilon _h } {\left[ {\frac{\alpha}{2h_E }\left\| {\left[ {\left[ \theta \right]} \right]} \right\|_{0, E}^2-\left\langle {\left\{ {\frac{\partial \theta }{\partial n}} \right\}, \left[{\left[ \theta \right]} \right]} \right\rangle _E } \right]} } \right\} \nonumber \\&\le & Ch^4\left( {\left\| {u_t } \right\|_2^2 +\left\| u_t \right\|_3^2 +\left\|{u } \right\|_3^2 } \right)+ C\left\| \theta \right\|_h^2+\frac{\rm d}{{\rm d}t}a_h \left( {\omega , \theta } \right) \nonumber \\&& +\frac{\rm d}{{\rm d}t}\int_0^t {a_h\left( {u\left( {X, s} \right)-u_h \left( {X, s} \right), \theta } \right)}{\rm d}s.\end{eqnarray}$

对(3.8)式两边对变量$t$从0到$t$求定积分,注意到$\theta \left(0 \right)=0$得

(3.9)$\begin{eqnarray}&& \int_0^t {\left\| {\theta _t } \right\|_0^2 }{\rm d}s+\frac{1}{2}\sum\limits_{K\in \Gamma _h } {\left\| {\nabla \theta }\right\|_{0, K}^2 } +\sum\limits_{E\in \varepsilon _h } {\left[ {\frac{\alpha}{2h_E }\left\| {\left[ {\left[ \theta \right]} \right]} \right\|_{0, E}^2-\left\langle {\left\{ {\frac{\partial \theta }{\partial n}} \right\}, \left[{\left[ \theta \right]} \right]} \right\rangle _E } \right]} \nonumber \\&\le & Ch^4\int_0^t {\left( {\left\| {u_t } \right\|_2^2 +\left\| {u_t }\right\|_3^2 +\left\| u \right\|_3^2 } \right)} {\rm d}s+ C\int_0^t{\left\| \theta \right\|_h^2 } {\rm d}s \nonumber \\&& +a_h \left( {\omega , \theta } \right)+\int_0^t {a_h \left( {u\left( {X, s}\right)-u_h \left( {X, s} \right), \theta } \right)} {\rm d}s.\end{eqnarray}$

由于

(3.10)$\begin{equation}a_h \left( {\omega , \theta } \right)\le Ch^4\left\| u \right\|_3^2+\varepsilon \left\| \theta \right\|_h^2 , \end{equation}$

(3.11)$\begin{eqnarray}&& \int_0^t {a_h \left( {u\left( {X, s} \right)-u_h \left( {X, s} \right), \theta} \right)} {\rm d}s \nonumber \\&\le & C\int_0^t {\left( {\left\| {\omega \left({X, s} \right)} \right\|_h +\left\| {\theta \left( {X, s} \right)} \right\|_h} \right) \left\| \theta \right\|_h {\rm d}s} \nonumber \\&\le & Ch^4\int_0^t {\left\| {u\left( {X, s} \right)} \right\|_3^2 {\rm d}s}+ C\int_0^t {\left\| {\theta \left( {X, s} \right)} \right\|_h^2{\rm d}s} +\varepsilon \left\| \theta \right\|_h^2 .\end{eqnarray}$

将(3.10)和(3.11)式代入(3.9)式,则(3.9)式可化为

(3.12)$\begin{eqnarray}&& \int_0^t {\left\| {\theta _t } \right\|_0^2 {\rm d}s}+\frac{1}{2}\sum\limits_{K\in \Gamma _h } {\left\| {\nabla \theta }\right\|_{0, K}^2 } +\sum\limits_{E\in \varepsilon _h } {\left[ {\frac{\alpha}{2h_E }\left\| {\left[ {\left[ \theta \right]} \right]} \right\|_{0, E}^2-\left\langle {\left\{ {\frac{\partial \theta }{\partial n}} \right\}, \left[{\left[ \theta \right]} \right]} \right\rangle _E } \right]} \nonumber \\&\le & Ch^4\left[ {\left\| u \right\|_3^2 +\int_0^t {\left( {\left\| {u_t }\right\|_2^2 +\left\| {u_t } \right\|_3^2 +\left\| u \right\|_3^2 } \right)}{\rm d}s} \right]+ C\int_0^t {\left\| \theta \right\|_h^2 {\rm d}s} +\varepsilon \left\|\theta \right\|_h^2 .\end{eqnarray}$

由文献[17]知(3.12)式左端可化为

$\begin{eqnarray*}&& \int_0^t {\left\| {\theta _t } \right\|_0^2 {\rm d}s}+\frac{1}{2}\sum\limits_{K\in \Gamma _h } {\left\| {\nabla \theta }\right\|_{0, K}^2 +\sum\limits_{E\in \varepsilon _h } {\left[ {\frac{\alpha}{2h_E }\left\| {\left[ {\left[ \theta \right]} \right]} \right\|_{0, E}^2-\left\langle {\left\{ {\frac{\partial \theta }{\partial n}} \right\}, \left[{\left[ \theta \right]} \right]} \right\rangle _E } \right]} } \\&\ge & \int_0^t {\left\| {\theta _t } \right\|_0^2 {\rm d}s} +\min \left\{{\frac{1}{2}-\frac{3C{ }_0}{8\varepsilon _1 }, \frac{\alpha}{2}-\frac{\varepsilon _1 }{2}} \right\}\left\| \theta \right\|_h^2 .\end{eqnarray*}$

从而当$\alpha >\varepsilon _1 >\frac{3}{4}C_0 $时有

$\left\| \theta \right\|_h^2 +\int_0^t {\left\| {\theta _t } \right\|_0^2 {\rm d}s}\le Ch^4\left[ {\left\| u \right\|_3^2 +\int_0^t {\left( {\left\| {u_t }\right\|_2^2 +\left\| {u_t } \right\|_3^2 +\left\| u \right\|_3^2 } \right)}{\rm d}s} \right]+ C\int_0^t {\left\| \theta \right\|_h^2 } {\rm d}s.$

应用Gronwall引理有

$\left\| \theta \right\|_h \le Ch^2\left[ {\left\| u \right\|_3^2 +\int_0^t{\left( {\left\| {u_t } \right\|_2^2 +\left\| {u_t } \right\|_3^2 +\left\| u\right\|_3^2 } \right)} {\rm d}s} \right]^{\frac{1}{2}}.$

因此

$\begin{eqnarray*}\left\| {u-u_h } \right\|_h &\le & \left\| {u-I_h u} \right\|_h +\left\| \theta\right\|_h \\&\le & Ch^2\left\{ {\left\| u \right\|_3 +\left[ {\left\| u \right\|_3^2+\int_0^t {\left( {\left\| {u_t } \right\|_2^2 +\left\| {u_t } \right\|_3^2+\left\| u \right\|_3^2 } \right)} {\rm d}s} \right]^{\frac{1}{2}}} \right\}.\end{eqnarray*}$

证毕.

在本节中,将给出问题(3.1)的全离散逼近格式及相应的误差估计.将时间区间$\left[{0, T} \right]$进行$N$等分,即$0=t_0 < t_1 < \cdots < t_N =T$,则时间步长$\tau =\frac{T}{N}$, $t_n =n\tau $, $n=0, 1, \cdots, N$. $U^n$表示$t=t_n =n\tau $在$V_h $中的逼近,定义向后差商$\partial _t U^n=\frac{1}{\tau }\left({U^n-U^{n-1}} \right)$,对于积分项的处理运用数值积分公式:

$\int_0^{t_n } {\nabla u\left( s \right){\rm d}s} \approx \tau\sum\limits_{j=0}^{n-1} {\nabla u^j} .$

建立(3.1)式的全离散逼近格式如下:求$U^n\in V_h $,使得

(4.1)$\begin{equation}\left\{ {\begin{array}{l} \left( {\partial _t U^n, v_h } \right)+a_h \left( {U^n, v_h } \right)+\tau\sum\limits_{j=0}^{n-1} {a_h \left( {U^j, v_h } \right)} =\left( {f^n, v_h }\right), \forall v_h \in V_{h, } \\[2mm] U^0=I_h u_0 \left( X \right). \end{array}} \right.\end{equation}$

定理4.1  设$u^n$, $U^n$分别是(1.1)和(4.1)式的解,若$u, u_t \in L^\infty \left({0, T; H^3\left(\Omega \right)} \right)$, $u_{tt} \in L^\infty \left({0, T; L^2\left(\Omega \right)} \right)$,则有

$\left\| {u^n-U^n} \right\|_h =O\left( {h^2+\tau } \right).$

证  记$u^n-U^n=\left({u^n-I_h u^n} \right)+\left({I_h u^n-U^n} \right)\doteq\rho ^n+\xi ^n$. $\forall v_h \in V_h, $根据(1.1)和(4.1)式有下面误差方程

(4.2)$\begin{eqnarray}\left( {\partial _t \xi ^n, v_h } \right)+a_h \left( {\xi ^n, v_h }\right)&= & -\left( {\partial _t \rho ^n, v_h } \right)-a_h \left( {\rho ^n, v_h }\right) \nonumber \\&& -\tau \sum\limits_{j=0}^{n-1} {a_h \left( {u^j-U^j, v_h } \right)} +\left({R_1^n , v_h } \right)+a_h \left( {R_2^n , v_h } \right), \end{eqnarray}$

其中, $R_1^n =\partial _t u^n-u_t^n $, $R_2^n =\tau \sum\limits_{j=0}^{n-1} {u^j} -\int_0^{t_n } {u\left({X, s} \right){\rm d}s} $.

在(4.2)式中令$v_h =\partial _t \xi ^n$得

(4.3)$\begin{eqnarray}\left( {\partial _t \xi ^n, \partial _t \xi ^n} \right)+a_h \left( {\xi^n, \partial _t \xi ^n} \right)&= & -\left( {\partial _t \rho^n, \partial _t \xi ^n} \right)-a_h \left( {\rho ^n, \partial _t \xi ^n} \right)-\tau \sum\limits_{j=0}^{n-1} {a_h \left( {u^j-U^j, \partial _t \xi ^n}\right)} \nonumber \\&& +\left( {R_1^n , \partial _t \xi ^n} \right)+a_h \left( {R_2^n, \partial _t \xi ^n} \right)\doteq\sum\limits_{i=1}^5 {B_i } .\end{eqnarray}$

首先注意到

$\left\| {\partial _t \rho ^n} \right\|_0^2\le \frac{Ch^4}{\tau }\int_{t_{n-1} }^{t_n } {\left\| {u_t } \right\|_2^2{\rm d}s} , $

(4.4)$\begin{eqnarray}\left\| {\partial _t \rho ^n} \right\|_h^2 & =&\sum\limits_{K\in \Gamma _h } {\left| {\frac{1}{\tau }\left( {\rho ^n-\rho^{n-1}} \right)} \right|_{1, K}^2 } +\sum\limits_{E\in \varepsilon _h }{\left\{ {\frac{1}{h_E }\left\| {\left[ {\left[ {\frac{1}{\tau }\left( {\rho^n-\rho ^{n-1}} \right)} \right]} \right]} \right\|_{0, E}^2 } \right\}} \nonumber \\& =&\sum\limits_{K\in \Gamma _h } {\left| {\frac{1}{\tau }\left( {\nabla \rho^n-\nabla \rho ^{n-1}} \right)} \right|_{0, K}^2 } +\sum\limits_{E\in\varepsilon _h } {\left\{ {\frac{1}{h_E }\left\| {\frac{1}{\tau }\left({\left[ {\left[ {\rho ^n} \right]} \right]-\left[ {\left[ {\rho ^{n-1}}\right]} \right]} \right)} \right\|_{0, E}^2 } \right\}} \nonumber \\& =&\frac{1}{\tau ^2}\left\{ {\sum\limits_{K\in \Gamma _h } {\int_K {\bigg({\int_{t_{n-1} }^{t_n } {\nabla \rho _t } {\rm d}s} \bigg)^2} {\rm d}X}+\sum\limits_{E\in \varepsilon _h } {\frac{1}{h_E }\int_E {\bigg({\int_{t_{n-1} }^{t_n } {\left[ {\left[ \rho \right]} \right]_t } {\rm d}s}\bigg)^2} {\rm d}l} } \right\} \nonumber \\& \le &\frac{1}{\tau }\left\{ {\sum\limits_{K\in \Gamma _h } {\int_{t_{n-1}}^{t_n } {\left\| {\nabla \rho _t } \right\|_0^2 } {\rm d}s} +\sum\limits_{E\in\varepsilon _h } {\frac{1}{h_E }\int_{t_{n-1} }^{t_n } {\int_E {\left[{\left[ {\rho _t } \right]} \right]^2} {\rm d}l} {\rm d}s} } \right\} \nonumber \\& =&\frac{1}{\tau }\int_{t_{n-1} }^{t_n } {\bigg( {\sum\limits_{K\in \Gamma _h} {\left| {\rho _t } \right|_{1, K}^2 } +\sum\limits_{E\in \varepsilon _h }{\frac{1}{h_E }\left\| {\left[ {\left[ {\rho _t } \right]} \right]}\right\|_{0, E}^2 } }\bigg)} {\rm d}s \nonumber \\& =&\frac{1}{\tau }\int_{t_{n-1} }^{t_n } {\left\| {\rho _t } \right\|_h^2 }{\rm d}s\le \frac{Ch^4}{\tau }\int_{t_{n-1} }^{t_n } {\left\| {u_t } \right\|_3^2{\rm d}s} , \end{eqnarray}$

$\left\| {R_1^n } \right\|_0^2 \le C\tau \int_{t_{n-1} }^{t_n } {\left\|{u_{tt} (X, s)} \right\|_0^2 {\rm d}s} \le C\tau ^2\left\| {u_{tt} }\right\|_{L^\infty \left( {0, T;L^2\left( \Omega \right)} \right)}^2 , $

$\left\| {R_2^n } \right\|_h^2 \le C\tau ^2\int_0^{t_n } {\left\| {u_t (X, s)}\right\|_h^2 {\rm d}s} \le C\tau ^2\left\| {u_t } \right\|_{L^\infty \left({0, T;H^3\left( \Omega \right)} \right)}^2 , $

$\begin{eqnarray*}\left\| {\partial _t R_2^n } \right\|_h^2 & =&\bigg\| {\frac{1}{\tau }\bigg({\int_{t_{n-1} }^{t_n } {u_t \left( {X, \zeta } \right)\left( {t_{n-1} -s}\right){\rm d}s} } \bigg)} \bigg\|_h^2 \\& \le& \bigg\| {\frac{1}{\tau }\bigg( {\int_{t_{n-1} }^{t_n } {u_t^2 \left({X, \zeta } \right){\rm d}s} }\bigg)^{\frac{1}{2}}\bigg( {\int_{t_{n-1} }^{t_n }{\left( {t_{n-1} -s} \right)^2{\rm d}s} }\bigg)^{\frac{1}{2}}}\bigg\|_h^2 \\& \le &C\tau ^2\left\| {u_t } \right\|_{L^\infty \left( {0, T;H^3\left( \Omega\right)} \right)}^2 , \end{eqnarray*}$

其中$t_{n-1} < \zeta < s$.

下面对$B_1 \sim B_5 $逐项进行估计

$B_1\le \frac{Ch^4}{\tau }\int_{t_{n-1} }^{t_n } {\left\| {u_t } \right\|_2^2{\rm d}s} +\varepsilon \left\| {\partial _t \xi ^n} \right\|_0^2 , $

$\begin{eqnarray*}B_2 & =&-\frac{1}{\tau }a_h \left( {\rho ^n, \xi ^n} \right)+\frac{1}{\tau }a_h\left( {\rho ^{n-1}, \xi ^{n-1}} \right)+a_h \left( {\partial _t \rho ^n, \xi^{n-1}} \right) \\& \le &\frac{1}{\tau }\left[ {a_h \left( {\rho ^{n-1}, \xi ^{n-1}} \right)-a_h\left( {\rho ^n, \xi ^n} \right)} \right]+\frac{Ch^4}{\tau }\int_{t_{n-1}}^{t_n } {\left\| {u_t } \right\|_3^2 {\rm d}s} +\varepsilon \left\| {\xi ^{n-1}}\right\|_h^2 , \end{eqnarray*}$

$\begin{eqnarray*}B_3 &= & -\tau \sum\limits_{j=0}^{n-1} {a_h \left( {\rho ^j, \partial _t \xi ^n}\right)} -\tau \sum\limits_{j=0}^{n-1} {a_h \left( {\xi ^j, \partial _t \xi^n} \right)} \\&= & -\tau \partial _t \bigg( {a_h \bigg( {\sum\limits_{j=0}^{n-1} {\rho ^j}, \xi ^n} \bigg)} \bigg)+a_h \left( {\rho ^{n-1}, \xi ^{n-1}} \right) -\tau\partial _t \bigg( {a_h \bigg( {\sum\limits_{j=0}^{n-1} {\xi ^j} , \xi ^n}\bigg)} \bigg)+a_h \bigg( {\xi ^{n-1}, \xi ^{n-1}} \bigg) \\&\le & Ch^4\left\| u \right\|_{L^\infty \left( {0, T;H^3\left( \Omega\right)} \right)}^2 + C \left\| {\xi ^{n-1}} \right\|_h^2 -\tau \partial _t \bigg( {a_h \bigg( {\sum\limits_{j=0}^{n-1}{\rho ^j} , \xi ^n} \bigg)} \bigg)-\tau \partial _t \bigg( {a_h \bigg({\sum\limits_{j=0}^{n-1} {\xi ^j} , \xi ^n} \bigg)}\bigg), \end{eqnarray*}$

$B_4 \le C\left\| {R_1^n } \right\|_0^2 +\varepsilon \left\| {\partial _t \xi^n} \right\|_0^2\le C\tau ^2\left\| {u_{tt} } \right\|_{L^\infty \left( {0, T;L^2\left(\Omega \right)} \right)}^2 +\varepsilon \left\| {\partial _t \xi ^n}\right\|_0^2 , $

$B_5 \le \frac{1}{\tau }\left[ {a_h \left( {R_2^n , \xi ^n} \right)-a_h \left({R_2^{n-1} , \xi ^{n-1}} \right)} \right]+C\tau ^2\left\| {u_t }\right\|_{L^\infty \left( {0, T;H^3\left( \Omega \right)} \right)}^2+\varepsilon \left\| {\xi ^{n-1}} \right\|_h^2 .$

注意到(4.3)式左端

(4.5)$\begin{equation}\left( {\partial _t \xi ^n, \partial _t \xi ^n} \right)+a_h \left( {\xi^n, \partial _t \xi ^n} \right)\ge \alpha _0 \left\| {\partial _t \xi ^n} \right\|_0^2 +\frac{1}{2\tau}\left\| {\xi ^n} \right\|_h^2 -\frac{1}{2\tau }\left\| {\xi ^{n-1}}\right\|_h^2 .\end{equation}$

将$B_1 \sim B_5 $的估计及(4.5)式代入(4.3)式,把$n$换成$i$,两边同乘以$2\tau $,再关于$i$从2至$n$求和得

(4.6)$\begin{eqnarray}\left\| {\xi ^n} \right\|_h^2 &\le & Ch^4\left[ {\int_{t_1 }^{t_n } {\left({\left\| {u_t } \right\|_2^2 +\left\| {u_t } \right\|_3^2 } \right){\rm d}s}+\left\| u \right\|_{L^\infty \left( {0, T;H^3\left( \Omega \right)}\right)}^2 } \right] \nonumber \\& &+C\tau ^2\left( {\left\| {u_{tt} } \right\|_{L^\infty \left( {0, T;L^2\left(\Omega \right)} \right)}^2 +\left\| {u_t } \right\|_{L^\infty \left({0, T;H^3\left( \Omega \right)} \right)}^2 } \right)+ C \tau \sum\limits_{i=1}^{n-1} {\left\| {\xi ^i} \right\|_h^2 } \nonumber \\&& -2a_h \left( {\rho ^n, \xi ^n} \right)+2a_h \left( {\rho ^1, \xi ^1} \right)+2a_h \left( {R_2^n , \xi ^n} \right)-2a_h \left( {R_2^1 , \xi ^1} \right) \nonumber \\&& -2\tau a_h \bigg( {\sum\limits_{i=0}^{n-1} {\rho ^i} , \xi ^n} \bigg)+2\taua_h \left( {\rho ^0, \xi ^1} \right)-2\tau a_h \bigg( {\sum\limits_{i=0}^{n-1} {\xi ^i} , \xi ^n} \bigg)+\left\|{\xi ^1} \right\|_h^2 .\end{eqnarray}$

下面先估计上式最后一项,在(4.3)式中令$n=1$得

(4.7)$\begin{eqnarray}\left( {\partial _t \xi ^1, \partial _t \xi ^1} \right)+a_h \left( {\xi^1, \partial _t \xi ^1} \right)&= & -\left( {\partial _t \rho ^1, \partial _t \xi^1} \right)-a_h \left( {\rho ^1, \partial _t \xi ^1} \right)-\tau a_h \left( {\rho ^0, \partial _t \xi ^1} \right) \nonumber \\&& + \left( {R_1^1, \partial _t \xi ^1} \right)+a_h \left( {R_2^1 , \partial _t \xi ^1}\right).\end{eqnarray}$

注意到

(4.8)$\begin{equation}-\left( {\partial _t \rho ^1, \partial _t \xi ^1} \right)\le \frac{Ch^4}{\tau}\int_0^{t_1 } {\left\| {u_t } \right\|_2^2 {\rm d}s} +\varepsilon \left\|{\partial _t \xi ^1} \right\|_0^2 , \end{equation}$

(4.9)$\begin{equation}-a_h \left( {\rho ^1, \partial _t \xi ^1} \right)=-\frac{1}{\tau }a_h \left( {\rho ^1, \xi ^1} \right), \end{equation}$

(4.10)$\begin{equation}-\tau a_h \left( {\rho ^0, \partial _t \xi ^1} \right)=-a_h \left( {\rho^0, \xi ^1} \right), \end{equation}$

(4.11)$\begin{equation}\left( {R_1^1 , \partial _t \xi ^1} \right)\le C\tau ^2\left\| {u_{tt} } \right\|_{L^\infty \left( {0, T;L^2\left(\Omega \right)} \right)}^2 +\varepsilon \left\| {\partial _t \xi ^1}\right\|_0^2 , \end{equation}$

(4.12)$\begin{equation}a_h \left( {R_2^1 , \partial _t \xi ^1} \right)=\frac{1}{\tau }a_h \left({R_2^1 , \xi ^1} \right).\end{equation}$

将(4.8)-(4.12)式代入(4.7)式,两边再同乘以$2\tau $,得

(4.13)$\begin{eqnarray}\left\| {\xi ^1} \right\|_h^2 &\le & Ch^4\int_0^{t_1 } {\left\| {u_t }\right\|_2^2 {\rm d}s} +C\tau ^3\left\| {u_{tt} } \right\|_{L^\infty \left({0, T;L^2\left( \Omega \right)} \right)}^2 \nonumber \\&& -2a_h \left( {\rho ^1, \xi ^1} \right)-2\tau a_h \left( {\rho ^0, \xi ^1}\right)+2a_h \left( {R_2^1 , \xi ^1} \right).\end{eqnarray}$

将(4.13)式代入(4.6)式得

(4.14)$\begin{eqnarray}\left\| {\xi ^n} \right\|_h^2& \le & Ch^4\left[ {\int_0^{t_n } {\left( {\left\|{u_t } \right\|_2^2 +\left\| {u_t } \right\|_3^2 } \right){\rm d}s} +\left\| u\right\|_{L^\infty \left( {0, T;H^3\left( \Omega \right)} \right)}^2 }\right] \nonumber \\&& +C\tau ^2\left( {\left\| {u_{tt} } \right\|_{L^\infty \left( {0, T;L^2\left(\Omega \right)} \right)}^2 +\left\| {u_t } \right\|_{L^\infty \left({0, T;H^3\left( \Omega \right)} \right)}^2 } \right)+ C \tau \sum\limits_{i=1}^{n-1} {\left\| {\xi ^i} \right\|_h^2 } \nonumber \\&& -2a_h \left( {\rho ^n, \xi ^n} \right)+2a_h \left( {R_2^n , \xi ^n} \right)-2\tau a_h \bigg( {\sum\limits_{i=0}^{n-1} {\rho ^i} , \xi ^n} \bigg)-2\tau a_h \bigg( {\sum\limits_{i=0}^{n-1} {\xi ^i} , \xi ^n}\bigg).\end{eqnarray}$

注意到

(4.15)$\begin{equation}-2a_h \left( {\rho ^n, \xi ^n} \right) \le Ch^4\left\| u \right\|_{L^\infty \left({0, T;H^3\left( \Omega \right)} \right)}^2 +\frac{1}{8}\left\| {\xi ^n} \right\|_h^2 , \end{equation}$

(4.16)$\begin{equation}-2\tau a_h \bigg( {\sum\limits_{i=0}^{n-1} {\rho ^i} , \xi ^n}\bigg) \le Ch^4\left\| u\right\|_{L^\infty \left( {0, T;H^3\left( \Omega \right)} \right)}^2+\frac{1}{8}\left\| {\xi ^n} \right\|_h^2 , \end{equation}$

(4.17)$\begin{equation}-2\tau a_h \bigg( {\sum\limits_{i=0}^{n-1} {\xi ^i} , \xi ^n} \bigg)\leC\tau \sum\limits_{i=1}^{n-1} {\left\| {\xi ^i} \right\|_h^2 }+\frac{1}{8}\left\| {\xi ^n} \right\|_h^2 , \end{equation}$

(4.18)$\begin{equation}2a_h \left( {R_2^n , \xi ^n} \right) \le C\tau ^2\left\| {u_t }\right\|_{L^\infty \left( {0, T;H^3\left( \Omega \right)} \right)}^2+\frac{1}{8}\left\| {\xi ^n} \right\|_h^2 .\end{equation}$

将(4.15)–(4.18)式代入(4.14)式得

(4.19)$\begin{eqnarray}\left\| {\xi ^n} \right\|_h^2& \le & Ch^4\left[ {\int_0^{t_n } {\left( {\left\|{u_t } \right\|_2^2 +\left\| {u_t } \right\|_3^2 } \right){\rm d}s} +\left\| u\right\|_{L^\infty \left( {0, T;H^3\left( \Omega \right)} \right)}^2 }\right] \nonumber \\&& +C\tau ^2\left( {\left\| {u_{tt} } \right\|_{L^\infty \left( {0, T;L^2\left(\Omega \right)} \right)}^2 +\left\| {u_t } \right\|_{L^\infty \left({0, T;H^3\left( \Omega \right)} \right)}^2 } \right)+C\tau\sum\limits_{i=1}^{n-1} {\left\| {\xi ^i} \right\|_h^2 } .\end{eqnarray}$

利用离散的Gronwall引理得

(4.20)$\begin{equation}\left\| {\xi ^n} \right\|_h \le C\left( {h^2+\tau } \right).\end{equation}$

因此

$\left\| {u^n-U^n} \right\|_h \le \left\| {u^n-I_h u^n} \right\|_h +\left\|{\xi ^n} \right\|_h \le C\left( {h^2+\tau } \right).$

证毕.

在数值实验中,我们考虑如下抛物积分微分方程

(5.1)$\left\{ \begin{align}\begin{array}{ll}u_t -\Delta u-\int_0^t {\Delta u\left( {X, s} \right)} {\rm d}s=f\left( {X, t}\right), & \left( {X, t} \right)\in \Omega \times \left({0, T} \right], \\[2mm] u\left( {X, t} \right)=0, & \left( {X, t} \right)\in \partial \Omega \times\left( {0, T} \right], \\ u\left( {X, 0} \right)=u_0 \left( X \right), & X\in \Omega , \end{array}\end{align} \right.$

其中$\Omega =\left[{0, 1} \right]\times \left[{0, 1} \right]$, $f\left({X, t} \right)$是由真解$u=e^txy\left({1-x} \right)\left({1-y}\right)$确定的右端项.在表 1, 2中选择$\tau=h^2$,时间节点分别为$t=0.5, 1$.可以看到当$h\to 0$时, $\left\| {u-u_h} \right\|_0 $和$\left\| {u-u_h} \right\|_h $的收敛阶均为$O\left({h^2} \right)$.实验结果和我们的结论是相一致的.

本文通过引入新的双线性型,对抛物积分微分方程分别构造了一种新的半离散格式和全离散格式,并且在比传统的能量模更大的范数意义下,得到了相应的比通常估计方法高一阶精度的收敛结果.在整个分析过程中,对(4.4)式的估计起到了关键作用.本文对其他偏微分方程的数值求解提供了可借鉴的思路和途径.