数理解析研究所講究録
1129 巻 2000 年 83-86

83

Unconventional DDM&parallel method for fluid computation
Shi Dongyang$(\mathrm{T}\mathrm{l}\mathrm{T}\prime \mathrm{z}\mathrm{z}\mathrm{U})$Wu Yuchun

Ichiro Hagiwara$(\mathrm{T}\mathrm{l}\mathrm{T}/\mathrm{H}\mathrm{l}\mathrm{T})$A new finite element Domain decomposition rnethod, which is bascd on a$\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\iota$$(\mathrm{T}\mathrm{l}\mathrm{T})$-by-point scheme. domain

decomposition method (DDM) and the matrix-storage frce formulation. is developed and implemented to themodel equation for Navicr-Stokes equations, convection-diffusion equation. Numerical experiments demonstrated
that the proposed method is efficiently to solvc the modcl$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{i}_{0}\iota 1$‘

Key words: Domain Decomposition Method, Locally lmplicit Finite

Matrix$\mathrm{f}_{t\mathrm{C}\mathrm{C}},$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{C}(\mathrm{i}\mathrm{o}\mathrm{n}\cdot \mathrm{D}\mathrm{i}i\mathrm{f}_{\mathrm{U}}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$1. IntroductIon$\mathrm{E}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\ulcorner \mathrm{l}\mathrm{t}$Scheme. Point by Point, Stiffncss

Equation

2. Governing Equation and Discretization

The finite element method (FEM) has been widcly usedin engineering science computing for solving
equations arisen from many field such as

wc discuss the following modeI equation, convcction-$\mathrm{d}\dot{\downarrow}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{l}$$\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}_{0}\mathrm{n}$andfluid etc..Thc cxcellcnt properlies of FEM compared withequation. for Navicr.Stokcs:$\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{u}\Gamma \mathrm{e}\mathrm{h}*\mathrm{e}8\mathrm{t}$the finitc difference method arc its applicability to an(2.1)$(0\leq x\leq L)$$\frac{\partial u}{\partial t}+\frac{\partial(\alpha u)}{\partial x}=v\frac{\partial^{2}u}{\partial x^{2}}$with the time initial condition is.

unstructured mesh, simplicity for managing boundary
conditions and etc.. But. the general FEM should create the(2.2)$u(x.0)=H(x)$stiffness matrix, and there arc somc strict limits to create thestiffness matrix. The defects of FEM are larger storage
rcquircmcnts and highu CPU cost than the finite difference

method and would especially become a barrier whenhandling large scale problems.
Domain decomposition method (DDM) is interesting forseveral reasons. One is the possibility to use different
models on the subdomains in order to gct a more

and boundary conditions$u(\mathrm{O}, 1)=a,$$u(L.t)=b$(2.3)$u^{\mathrm{t}}(0,1)=0,u^{1}(L, t)=0$

$a,$

$b$

arc two constants.

Assume$[0, \mathrm{L}]$is divided uniformly by$0=x\leq x0\}\underline{<<}x_{M_{1^{*u}}},-1=L,$$\psi\}=1-\xi$,$\psi_{2}=\xi$$M_{1}+M_{2}$pointslet$\Delta \mathrm{r}=X_{(}-x_{-},1$(2.5)$(0\leq\xi\leq 1)$$\mathrm{p}\mathrm{l}\iota \mathrm{y}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}$accurate$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}$]$\mathrm{i}\mathrm{Z}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}n}$. It also rcduccs complex geometry to

For cvcry interval

simpler subregions. Last but not least, as this method iseasily parallclizablc, we can take advantage of paralld$[x_{J\cdot JX_{J}}.]$, the coordinate transformation is(2.6)$x=x_{j-\mathrm{I}}\psi\+x_{j}\psi_{2}$Let$\varphi_{j}$be the shape function;$r\mathrm{g}\mathrm{s}_{\mathrm{P}^{\mathrm{o}\mathrm{n}}\mathrm{g}}\mathrm{d}$to point

computcrs.

The locally implicit finite element$\mathrm{s}\mathrm{c}\downarrow$)$\mathrm{e}\mathrm{m}\mathrm{e}$is a uniquekind of finite element method. The distinct fcaturc of this$\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{G}\mathrm{m}\mathrm{e}_{*}\mathrm{C}\mathrm{o}\mathrm{m}_{\mathrm{P}^{\mathrm{a}}\mathrm{g}}\mathrm{r}\mathrm{i}\}\tau$with the general FEM, is not need tomatrix and the numerical computational
procedure carried with)’-point schme,
provides us with more space to improve numericalbuild the$u=u(/xf)/- 1’\psi\}(+uX1)j’\psi_{2}$$\forall x\in[x_{j-1’ J}x]$$y=u(/x_{J},’)\Downarrow l1^{+u}(_{X,t}*1’)\psi_{2}$$\forall x\mathrm{e}[x_{j}.x_{r\downarrow]}*$$\mathrm{t}\mathrm{h}\mathrm{e}\Gamma|$$x_{j}$.
(2.7)

the finite clement approximation problem of (2.1) is$\mathrm{s}\mathrm{t}\mathrm{i}l\tau \mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}$$\mathrm{p}\mathrm{o}\dot{1}\mathrm{n}1- \mathrm{b}_{\}}$

computation procedure and morc flexibility in$\frac{\partial}{\partial t}\int_{0^{u_{j}\varphi_{J}\ }}^{L}- \int^{L}0(\alpha u_{J}-\mathcal{V}\frac{\partial u_{J}}{\partial_{X}})\frac{\partial\varphi_{J}}{\partial x}\ =0$(2.8)$\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\downarrow$$\mathrm{a}\mathrm{p}\mathrm{p}^{\}\mathrm{i}\mathrm{c}}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$.

on the other hand, Thcorctical analysis and numerical

After integration. we have$\frac{\partial}{\partial t}(\frac{1}{6}u/’ 1\frac{2}{3}u+\frac{1}{6}j-+u_{j\mathrm{I}})+(\frac{\alpha}{2\Delta \mathrm{x}})(?l_{j}-k1u_{j-\iota})(2.9)$

simulations show that the proposed locally implicit finite
clement scheme has good stability.

$-( \frac{v}{\Delta \mathrm{x}^{t}})(u_{/\mathrm{l}}*-2u_{J}+u_{j-}|)=0$

Discretizing the time by.

$\mathrm{t}\mathrm{l}\iota \mathrm{e}$$[\mathrm{t}_{0},1_{n}+1],$$\Delta t=t_{\hslash}-l_{\hslash-1}$

Euler implicit scheme for every

84$LHS(u)=\mathrm{R}\mathrm{c}S_{J^{-}}^{n}$$\frac{|}{6}(1l_{\mathit{1}}’|;^{\iota J}1^{-u}\text{ノ}.\’)’+\frac{2}{3}(ul^{\star}’-u_{J}^{\hslash})$(2.10)$+ \frac{C}{2}(\Delta_{l\mathit{1}^{\mathrm{t}m\mathrm{I}}}’)-\Delta lj)/*|j-1-(m)Q(\Delta.u^{(m}-j+\downarrow*1)2\Delta u_{j}+\Delta \mathrm{t}m)u^{\mathrm{t}m)}j-1)]$$+ \frac{1}{6}(u_{/}^{n\}-1-u\mathrm{I}1n+/- 1(\frac{\alpha}{2\Delta \mathrm{x}}\mathrm{I}^{(}u_{J 1}^{n*1}-u^{n\dagger 1}\int)j-$

Resume$du_{J-1}\approx d_{ll_{J}}$and let$|\mathrm{q}$instead of thc$C$$-( \frac{v}{\Delta_{\lambda^{2}}}.\mathrm{I}(u’-2u_{J}^{\hslash}+u^{\hslash}- j\^{*\iota*1}1/1)*|0=$$\langle$$( \frac{5}{6}-\frac{1\mathrm{q}}{2}+Q)dl\mathit{1}=LHS(ju)$

2.16)

where $u_{J}’=7r(x_{/}, f_{l})$So we get the locally implicit finite element scheme$j=0,1,2,\cdots,M=(M_{\mathrm{l}}+M_{?^{-}}1)$$u^{0},=H(x)/$$(2.1]),(2.]4)$$n=0,1_{:} \cdots,[\frac{T}{\Delta t}]$$u_{0}^{n}=\mathit{0}$$C=\alpha\Delta \mathit{1}/\Delta \mathrm{x}$,and (2.16) for the convection-diffusion

equation.

$u_{\mathrm{A}\mathit{4}M\iota’-}^{n}=b\star\iota$$Q=v\Delta 1/\Delta x^{2}$Equation (2.10) is similar to the finite difference scheme.
be written as
then (
let3$\mathrm{B}\mathrm{r}\mathrm{r}\mathrm{o}r\mathrm{E}s\mathrm{t}\mathrm{i}\mathrm{m}\mathfrak{g}\mathrm{t}\mathrm{i}\mathrm{o}\mathfrak{n}$$\mathrm{M}$$\overline{u}_{/^{\mathrm{i}\mathrm{s}}}^{n}$$\Delta u_{j}=u^{n\mathrm{s}}-/u_{j}^{\hslash}|$$\rho_{J}’=\overline{u}_{J^{-}}u_{j}’’$solutionexact$\mathrm{t}\mathrm{l})\mathrm{e}$$2.10\rangle \mathrm{c}\mathrm{a}\mathrm{n}$,of$(2.1)-(2.3)$ ,,$e_{/}^{\hslash}=O(\Delta t+\Delta\chi^{\mathrm{a}})$$1_{\Delta u_{l}}6s\mathrm{t}+^{2_{\Delta u.+}1_{\ }}36l_{-},\mathrm{t}$$\rho^{j}=(\rho|’\rho_{2},\ldots, \rho_{N^{-1}})1jjT$$+^{C}2(^{\Delta u}4’- 1)^{-O}-\Delta_{\mathcal{U}}.(\Delta u,\wedge’-2\Delta u,$

.

lhen

$+\Delta u_{-[])}$(2..1])$={\rm Res}$ ’, satisfies the follow$\rho^{l}$,$e’=(e^{j}, e,\ldots, e12’|N^{-}\mathrm{t})^{T}$$\dot{t}1$equation:$J4\rho^{n+1}=B\rho^{n}+\Delta^{(}e^{n},$
$\{$

with

$)jQ\mathrm{r}(+u_{)l1}^{\hslash}-2ul+u_{/-1}^{n})$$\mathrm{R}es_{j}^{n}=-\frac{C}{2}(u_{j\star 1^{-}}^{\hslash}u-$}

We construct the$\mathrm{f}\mathrm{o}$}$\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$iteration scheme. For thc$m+$ ]

where A. are$\check{B}$iteration, let$\Delta u_{j}^{\mathrm{t}}m+1)=\Delta u_{J}^{\mathrm{t}}m)+du_{J}*$$\Delta u_{j}^{(0)}=0,$$(N_{\mathrm{I}}-])\mathrm{x}(N-|1)$matrices$du_{0}=0$(2.12)$(\mathrm{m}=0,1.2, \ldots)$$l4=[_{0}^{\frac{1}{6}-} \frac{2}{3}+\frac{c}{2,0}.-2QQ\frac{]}{6}+\frac{c}{2}\frac{1}{6}-\frac{c}{2}Q\frac{2}{3}+.2Q0^{-}-Q\frac{1}{6}+\frac{t}{2}.-Q\frac{2}{3}+0_{2Q}0$

and

(3.1)

$(n=0,1,..,[ \frac{T}{\Delta t}]-1)$$\rho^{0}=0$$du_{J}$

computed by the following formulas (2.14) and(2.16).

$0^{\cdot}$

From (2.11). the lcfl to right scheme is(2.13)$\frac{2}{3}d_{ll_{j}}+\frac{1}{6}du_{J}*\downarrow+\frac{C}{2}du_{J\star}1$$B=$$-Q(-2duj+d\iota \mathit{1}_{j+\mathrm{I}})=RHS(u)$here$uIs_{(u)=}{\rm Re} s_{j}^{\hslash}-$$+ \frac{C}{2}(\Delta u_{j\dashv}^{\mathrm{t}m}-)\Delta u_{j-}\}\mathrm{I}\mathrm{t}’|\star 1)-Q(\Delta u_{J1}^{\mathrm{t}m)}-2\Delta \mathrm{t}\mathit{4}_{J/}^{\mathrm{t}m}+\Delta\rangle u)*-1]\mathrm{t}h*1)$

and so$\rho^{n+\}}=\Delta t[A^{-}1en+A^{-\downarrow BAe^{n-1}}-1$

Resume $du_{j}=d\iota$’and let ] instead of the$|\mathrm{C}$$( \frac{5}{6}+\frac{1\mathrm{q}}{2}+Q)du=RHS\dot{\text{ノ}}(u)$$+(A^{-)}B)2\mathrm{A}^{-|}en-2+\ldots+(\mathrm{A}^{-\iota_{B}10})^{n_{A^{-}e}}]$$C$(2.14)$\mathrm{t}$lt is easy to see that$||B||_{-}=1$.

$||A||_{rightarrow}\geq 1,$$||A^{-)}||*\leq 1$.

,$||A^{-\mathrm{I}}B||_{rightarrow}\leq||4^{- 1}||-||B||-\leq 1$

Similarly, }) right to left scheme is$\mathrm{e}$$\frac{2}{3}du_{!^{+}}\frac{1}{6}du_{j^{-}\}}-\frac{C}{2}du_{J-\}}$$\langle$

2.. 15)$||A- 1en||_{\infty}\leq||A^{-1}\#rightarrow||e|n|_{\alpha}=O(\Delta t+\Delta \mathrm{x}^{2})$$\mathrm{b}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{u}\mathrm{S}\mathrm{e}0\leq N\leq\frac{T}{\Delta t}$.

,$-Q(-2du_{/}+du_{j-}1)=LHS(u)$here$\frac{1}{6}-\frac{c}{2}-Q\frac{2}{3}+2Q000)$$\frac{1}{6}+\frac{c}{2}000^{-Q}$$||\rho^{n}||_{\infty}\leq\Delta tNO(\Delta f+\Delta \mathrm{r}^{2})\leq O(\Delta t+\Delta x^{2})$$0^{\cdot}$

85

So when step ratio$s= \frac{\Delta t}{\Delta \mathrm{x}^{2}}=coh\mathrm{r}$.

$\mathrm{S}\mathrm{O}|\rho_{j}^{j}|=o(\Delta\chi^{2})$.

$||\rho^{n}||_{\sim}\leq O(\Delta x^{2})$.

$( \frac{5}{6}+\frac{|C|}{2}+Q\mathrm{I}du_{1}^{k}(j^{])},=RHS(_{\mathrm{t}(_{1}}k-\})$(3.2)$(\mathrm{d}3)$$\mathrm{r}\frac{5}{6}-\frac{|C\{}{2}+Q)du_{1}^{t}(j,1)=LHS(\mathrm{t}_{1}k-1)$
$\{$

(3.6)

that approximation scbeme (2.14) is$\mathrm{i}\gamma_{1}\mathrm{d}\mathrm{i}\mathrm{C}\mathrm{a}\mathrm{t}\mathrm{G}$convcrgcnt for arbitrary value$S$. So the error of the locally

implicit finite element scheme is first-order for time, and is$u \int(M_{1}-\mathrm{J}.1)=u_{2^{-}}^{k\}}(M_{l}-],])$$u_{1}^{1}(M_{1},1)=1l_{2^{-}}^{tt}(M_{1},1)$$u_{1}^{\mathrm{t}1}-(0,1)=\mathit{0}$forsecond-order for space. As to the stability analysis , werefer to reference [4].

$X\in \mathrm{q}$$( \frac{5}{6}+\frac{|C|}{2}+Q)du_{2}^{k}(j,1)=RHS(u_{2}^{t1}-)$(44)$( \frac{5}{6}-\frac{|C|}{2}+Q)du_{2}^{k}(j,1)=LHs_{(u_{2}}\mathrm{g}- \mathrm{t})$4. Domaln Decomposition Method
Suppose the initial domain

domains. denoted by$\Omega,$$\Omega$

is divided into two sub-$u_{I}^{1}(M,*1)=u \int(M_{1},1)$$=[X,$$X0M_{t}$ ], and$u_{2}^{k}(M_{1}+M1,1)2^{-}=b$$\Omega_{\overline{\Gamma}}[_{XX}\lambda\prime_{1}’ M1\star M2-1].\Omega \mathit{3}=[xu,-]’ Mx]1$

overlapping area.

$ll_{2}^{\lambda}(M_{1}-],])=u^{k}(|-],1M_{1})$$u_{i}=t\mathit{4}_{\Omega},$$(i=1,2)\mathrm{F}\mathrm{o}\mathrm{r}$node number is $M,$ , and for the

the

is

$\Omega_{\mathit{1}}$Case 1.

, the, the node number is$\Omega_{\mathit{2}}$2). When$\iota \mathrm{h}\mathrm{o}$$M_{2}$$n=\mathit{2}_{r}\mathit{3},\ldots$

$k=l$

.

$x\in\Omega_{1}$.

$( \frac{5}{6}+\lfloor \mathrm{q}_{+Q}2)du_{1}^{k}(j, n)=RHS(u, )$Then we construct the iterations$(u_{1}^{l}, u_{2}^{k})$for arbitrary

time interval$[l_{\emptyset-}r , t_{n}]$$\{$$( \frac{5}{\mathit{6}}-\frac{1\mathrm{q}}{2}+Q)du_{1}^{k}(j, n)=LHS(y_{\mathrm{I}})$

with the locally implicit finite element scheme asfollow:1).

$u_{1}‘(M_{1}-1,n)=\overline{u}|(M, -1,n-1)+c(u_{1}\sim(M_{1}-2, n-])-y_{2}\wedge(M_{1}-1,n-]))$$u_{1}^{k}(M_{1},n)=u_{1}\sim(M,, n-1)+c(y_{1}\sim(M_{\}}-1, n-])-\overline{u}_{l}(M_{\iota}.n-1))$

When $n=$ ], Case $1.k=1$ ,

(4.5)

Where

$x\in\Omega_{\mathrm{I}}$$\sim y_{1},$$\sim u_{l}$are the approximation values of abovc

step for several iteration steps.

for$x\in\Omega_{2}$$\{$$u_{1}(x)=H(x)$$u_{1}^{k}(M_{\mathrm{I}^{-}}1,n)=u_{1}(M_{1}-1,0)+c(u(\mathrm{I}M_{1}-2,\mathrm{o})-u(|\mathrm{A}i_{1}-1,\mathrm{o}))$$tl \int(Mn\downarrow’)=u_{1}(M_{\mathrm{t}}.\mathrm{o})+C(u_{1}(M_{1}-1,0)-u_{1}(M01’))$$\{$$u_{\mathrm{I}}(\mathrm{o}_{n},)=_{O}$

(4.1)

for

$x\in\Omega_{2}$$u_{2}^{k}(M_{1^{-}}1, n)=y_{1}^{k}(M_{1}-1,n)$$u_{l}^{l}(M_{1}, n)=u^{\kappa},(\mathrm{A}J_{1}, \prime l)$$u_{l}^{k}(M_{1}+M_{2^{-\mathrm{i}}},$

Case 2.
for$k=\mathit{2},\mathit{3}\ldots$$n_{\text{ノ}^{})_{=b}}$.

$\Omega_{J}$$\{$$u_{2}(x)arrow-H(x)$$u_{2}^{k}(M_{\mathrm{t}}-1, n)=?i_{1}^{k}(M_{1}-1, n)$$u_{2}^{k}(M_{\mathfrak{l}}, n)=u_{1}^{l}(M_{1}, n)$$lJ_{2}^{\mathrm{A}}(M_{\mathrm{t}}+M_{2}-1, n)=b$

Case 2.
for$k\triangleright$].

$x\in\Omega_{1}$$\{$$u_{1}^{k}(M_{1}-1, n)=u_{l^{-}}^{k1}(M_{1}-1, n)$$u_{\mathrm{I}}^{k}(M_{\}}, ’?)=uIk-|(M_{1}, n)$$u_{1}^{k}(0, n)=O$

86

for

$x\in\Omega_{2}$discontinuity crossed to the par12$(\ulcorner-150)$. the cu$r\mathrm{v}\mathrm{e}$issmooth. But accumulating the historical effect, and under$( \frac{5}{6}+\frac{1\mathrm{q}}{2}+Q)du_{2}(kj,n)=RHs_{(u^{k}}-\iota)l$the dissipation and the convection, there is a wave on the(4.8)$( \frac{5}{6}-\lfloor \mathrm{q}_{+}2Q\mathrm{I}du_{\mathrm{z}}k(j,n)=LHs(u^{k})\mathrm{z}-1$

inlct of the part 1. So the numerical results is satisfied. The$\{^{\{}$

numerical rcsults are coincide with the error analysis.

$u_{2}^{k}(M_{1}-1, n)=u^{k}1(M_{\mathrm{I}}-1, n)$

for the space. the error is$u_{2}^{k}(M_{\iota},n)=u_{\mathrm{I}}^{k}(M_{1}, n)$$\sec 0\mathrm{n}\mathrm{d}-\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}_{*}$$\mathrm{i}.\mathrm{c}.$,for the time, thecrror is first-ordcr.

$u_{2}^{\mathrm{A}}(M_{1}+x\mathit{4}-],n)\mathrm{z}=b$Here we mention that the boundary conditions imposedon thc intersection are different from the typical DD method.
order to impose the boundary conditions more exactly
during the procedure, we take into account of the convection

6. Concluding Remarks$\ln$$\mathrm{e}lT\mathrm{e}\mathrm{c}\mathrm{t}$in (45)$\ln$

the present study. we discussed the locally implicit

finite element schcme. which is base on a poinl-by-point

schemc. domain decomposition method (DDM) and the

5. Numerical Results$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{X}\prime \mathrm{S}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}$free formulation, then developcd andTwo numerical experiments were donc for the

implemented it to thc model equation for Navier-Stokes.

equation, convection-diffusion equation. By virtue of the-point scheme, the implmentation is quite

discontinues$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}\mathfrak{l}\mathrm{C}\mathrm{m}$

, and

.

$\alpha=1.0$

is varied about 20.0 toexperiment.
the is varied
the second numericalIn the first numerical experiment,
10$\Delta_{X}=1.0$$\Delta t=0.4,$

$u$

$u$

$C1\mathrm{t}\mathfrak{n}$$\mathrm{P}^{0\mathrm{i}\mathrm{n}\mathrm{t}- \mathrm{b}}\mathrm{y}$simple.
The validity of the Scheme was verified lhrough two
numerical experiments which convcction is dominatcd. Inorder to impose the boundary conditions more exactly

during the procedure, we take into account of thc convection

effect in (3.3) for the DD method. The results of thenumcrical experiments were satisficd. And the error iscoincided with the thcoretical analysis.
Rcferenccs

1. M. C. Ciccoli. Adaptivc Domain Decomposition
Algorithms

andApproximation for
Fig 2

In

$\mathrm{p}$)$\mathrm{g}$.

$1$

Elcment

Equations.

Journal of Scientific Computing. Vol. 11, No. 4. 1996

and Fig. 2, the$\mathrm{v}\mathrm{e}\mathrm{n};_{\mathrm{c}\mathrm{a}}$} line is a symbol of two$\ulcorner-20$.

$\mathrm{t}=80,$$r-150.\mathrm{W}\mathrm{h}\mathrm{c}\mathrm{n}$. the discontinuity is in the part 1, not to cross thevertical line to part 2. The curve is smooth both in part 1 andpart 2. When

VolumelFinite$\mathrm{A}\mathrm{d}_{\mathrm{V}\mathrm{C}}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}-\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}_{\mathrm{U}}\iota \mathrm{a}\mathrm{t}\mathfrak{j}\mathrm{o}\mathrm{n}\mathrm{a}11\{\mathrm{G}\mathrm{S}\mathrm{u}1\iota(\mathrm{u}=\downarrow\cup.\cup\sim\angle\cup \mathrm{U}j$domains. there are three states,$\ulcorner-20$

Finite$\ulcorner-80$, the discontinuity is crossing the$\mathrm{v}\mathrm{e}\hslash \mathrm{i}\mathrm{c}\mathrm{a}\mathrm{I}$line to the part2. There is some undulation in the curve

Because of the error of this scheme, the boundary conditionof part 1 on the vertical line would be imposed to exactly
true value. The imposed value would be undulating aroundthe exact value Then the curve is not very smooth. After the

2. Y. Nakabayashi, H. Okuda, G. Yagawa. Parallel Finite
Element Flutd$\mathrm{A}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{s}\dot{\mathrm{l}}\mathrm{s}$on an$\mathrm{E}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{G}\mathrm{n}\uparrow- \mathrm{b}\mathrm{y}-\mathrm{E}$]$\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}$Basis. Computational Mcchanics 18(1996)

3. K. C. Reddy, J. N. Reddy. Finite Element Solver for
3-D Viscous Flows.

4. Yi Fengming, Wu$\mathrm{N}87\cdot 29763$

Yuchun. Locally Implicit Finite

Elcment Scheme for Viscous Flows. China Joumal ofEngineering Thermophysics, Vol. 17, No. 1, 1996