数理解析研究所講究録
1129 巻 2000 年 42-53

42

Error analysis of finite element solutions ofconvection problems and its application to
the density-dependent Stokes problems
Satoshi KAIZU*
Department of Mathematics, Faculty of Education
Ibaraki University
April 29, 1999

1

Introduction. Let $T>0$ andor a polyhedron in
be a bounded polygon in$Q=\Omega \mathrm{x}(0,T)$ and. We would like to propose a convergent
finite element scheme approximating the following convection problem (P): findsatisfying
of some material:
unknown densityLet$\mathrm{R}^{3}$$\mathrm{R}^{2}$$\Omega$$S=\partial\Omega \mathrm{x}(0, T)$$\Omegaarrow \mathrm{R}$$\rho(x, t)$in$\ovalbox{\tt\small REJECT}^{\partial}+(u\cdot \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d})p=0$

on$\rho(x, 0)=\rho^{0}(x)$

$Q$

,

(1)
(2)

$\Omega$with respect to known velocity $u(.x, t)$ of some incompresible flows:$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$u(x, t)=0$

,

(3)

on S.

(4)

in

$Q$

In the construction of a convergent finite element scheme for the problem (P),of courese, we assume some regulaity and boundedness on the velocities besides
the incompressibility (3). But, what regularity on velocities we have to assume?
There may be many possibilties for the reply. Still, the most important one isdepend on its aim, certainly, to which the advection problem (P) is applied.
$u$$*\mathrm{g}\mathrm{a}\mathrm{k}/\mathrm{a}\mathrm{d}90426,\mathrm{t}\mathrm{e}\mathrm{x}$43

1 INTRODUCTION, the incomIn fact, the problem (P) is a subtarget of another propblem
pressible Navier-Stokes problems with nonhomogeneous density, . , which is a
-Stokes equation:system governed by the density-dependent$(\hat{\mathrm{Q}})$$\mathrm{i}$$\mathrm{e}.$$\mathrm{N}\mathrm{a}‘ \mathrm{v}\mathrm{i}\mathrm{e}$$p \{\frac{\partial u}{\partial t}(u\cdot \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d})u\}-\mu\triangle u+\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}p=\rho f$

in

$Q$

,

together with two other equations, (1) and (3), where $u,p$ and are unknown
velocities, pressures and densities of the non-homogeneous liquids, respectively.
admits at least a weak solution $u,p$ satisfying
This problem$\rho$$(\hat{\mathrm{Q}})$

$p,$

,

$u\in L^{\infty}(0,$ $T;\{L^{2}(\Omega)\}^{d})\cap L^{2}(0,$ $T;\{H_{0}^{1}(\Omega)\}^{d})$for the initial velocity
and$u_{0}\in\{L^{2}(\Omega)\}^{d}$, the outer force $f\in L^{1}(0,$(5)$T;\{L^{2}(\Omega)\}^{d})$(6)$0<M_{1}\leq\rho^{0}\leq M_{2}<\infty$

where$i=1,2$ , are constants (cf., for example, A. S. N. Antontsev, A. V.
So, in our paper
Kazhikhov and V. N. Monakhov [1] and P. L. Lions [10]we shall choose conditions (5) and (6) on the velocity and the density ,respectively.
Beside, there are another conditions in our paper to be needed, because wehave to construct a finite element scheme for (P). Thus, conditions on discrete
velocities are also needed. Unfortunately, there may be no convergent scheme,. So, from a convergent implicit
which approximates solutions of the problem
scheme, which approximates solutions of the classical Navier-Stokes equationsin the incompressible case with homogeneous density, we shall choose a suitablecondition on discrete velocities. For references it is useful to see the book [11] by
R. Temam.
$0=$
be an approximation of
In fact, let $\tau>0$ be a time mesh and let$\deltaU^{n}=U^{n}-U^{n-1}$$N=[T/\tau]$ . Let$t_{0}<t_{1}<t_{2}<\cdots<t_{n}=n\tau,$. Then we
consider the condition$M_{i},$$)$$u$$\rho^{0}$$(\hat{\mathrm{Q}})$$u(t_{n}),$$U^{n}$$\cdots,$$||U^{n}||^{2}+\Sigma_{n=1}^{N}(||\delta U^{n}||^{2}+\tau||U^{n}||_{h}^{2})\leq c_{0}<\infty$,(7)

where is independent of the time meshes and the space meshes $h>0$ us inandof the domain but depend on the initial value oftriangulationsisnorm over the domain andthe force term . Here the notation$e\mathrm{d}$$\tau$$c_{0}$$\{\mathcal{T}_{h}\}_{h}$$f$$\Omega$$u_{0}$$||\cdot||$$\Omega$$\mathrm{t}_{\mathrm{w}}\mathrm{h}\mathrm{e}L^{2}$: (12), and prove theIn this paper we propose a finite element scheme
convergence of their discrete solutions to a weak solution of the problem (P) in$(\mathrm{P})_{h}$441INTRODUCTION

another paper [6] under these conditons from (5) to (7). The order of the errorof discrete solutions of (12) is studied under sufficient regularity conditions on
and in this paper.
It may be difficult to extend the scheme (12) to a finite element scheme approximating the density-dependent Navier-Stokes problem, although it is possi$g=pf$ , which approximates
ble to extend (12) to a finite element scheme
governed by the density-dependent
the density-dependent Stokes
Stokes equation
$u$

$p$

$(\overline{\mathrm{Q}})_{g,h},$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}^{r}\mathrm{m}(\mathrm{Q})_{j}$in$p \frac{\partial u}{\partial t}-\mu\triangle u+\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}p=\rho f$$Q$,

together with two other equations (1) and (3) under the two conditions (2) and(4). Further, it is possible to give the order of the error of their discrete solutions. Note that, in the density-dependent Stokes problem, $u,p$ and are
unknown velocities, pressures and densities of non-homogeneous liquids, respectively. We shall consider a scheme: (12) and (21), approximating thedensity-dependent Stokes problem, and show that the approximation
satisfies$p$$(\overline{\mathrm{Q}})_{g,h}$$U^{n}$,(8)$\tau\sum_{n=1}^{N}||\frac{\delta U^{n}}{\tau}||^{2}\leq c_{0}<\infty$

where is independent of the time meshes and the space lneshes , but dependon the initial value ofand the force term . If we assume the condition (8)adding to the conditions (5), (6) and (7), then we get an increas in the order ofthe error of discrete densities of our scheme (12).
$h$$\tau$$c_{0}$$f$$u_{0}$$e$

Notation
For a domain$G$

we write $(f,$ $g)_{G}= \int_{G}$ fgdx. In particular, we write $(f, g)=$We denote bythenorm of over . In paticular, for$G=\Omega$ , we write $||f||_{p}=||f||_{p,\Omega}$ . For a $d-1$ dimensional simplex, we. Let $H=L^{2}(\Omega)$ andwrite,$\int_{\Omega}$

fgdx.

$L^{p}$$||f||_{p,G}$

$f$

$G$

$F$

$\mathrm{H}=\{L^{2}(\Omega)\}^{d},$ $X=\{H_{0}^{1}(\Omega)\}^{d}$$\langle f,g\rangle_{F}=\int_{F}fgd\sigma$$V=\{v\in X|\mathrm{d}\mathrm{i}\mathrm{v}v=0\}$.

Further, for the usual Sobol

norms,$e\mathrm{v}$space, $W^{1,p}(G)$ with $1\leq p\leq\infty$ we need semi-$|v|_{l,p,G}= \{\sum_{|a|=l}||D^{a}v||_{p,G}^{p}\}^{1/\mathrm{P}}$, and a norm,$||v||_{m,p,G}= \{\sum_{l=0}^{m}|v|_{l,p,G}^{p}\}^{1/p}$,

are differential operators of
wheredenote multi-indexes. For theandand letcase $G=\Omega$ we drop the letter from the surfixes of$Z^{m,p}=\{v\in L^{p}(\Omega)|v|_{K}\in W^{m,p}(K)\},$ $m$ denotes a non-negative integer andsatisfies $1<p<\infty$ . Generally, we call a function $v\in Z^{m,p}$ for some $m$ and apiecewise smooth function. The dual space of a Banach space is denoted by .
$D^{a}$$\mathrm{o}\mathrm{r}\mathfrak{a}^{1}\mathrm{e}\mathrm{r}a,$

$G$

$a$

$|v|_{l,p,G}$$||v||_{m,\mathrm{p},G}$

$p$

$p$

$Z$

$Z^{*}$

45

2 DISCRETE DENSITIES AND VELOCITIES

Further, $W^{m,p}(0, T;L^{q}( \Omega))=\{v\in L^{p}(0, T;L^{q}(\Omega))|\int_{0}^{T}||D^{s}v||_{L^{q}(\Omega)}^{\mathrm{p}}ds<\infty\}$ where$D^{s}v$

2denotes s-th derivative of$v$

in the sense of distribution.

Discrete densities and velocitiesLet
be a decomposition of by tr\’iangulations for $d=2$ or by tetra$d=3$. Here we denote by the maximum of the diameters ofhedrons for. The sequence of decompositons ofis regular: there exists a
constant such that$\Omega$$\mathcal{T}_{h}=\{K\}$$h$$h_{K}$$K\in \mathcal{T}_{h}$$\Omega,$$\{\mathcal{T}_{h}\}$$c_{1}$$\lim\sup\sup\underline{h_{I\{}\cdot}\leq c_{1}<\infty$$harrow 0$,(9)$K\in\tau_{h}h_{0,K}$

where
is the largest diameter of spheres included in $K$ .
Let be a non-negative integer and be the totality of polynomials of degree. Our approximation method for discrete densities relies on the discontinuous,
finite element method induced by L. Lesaint and P.-A. Raviart [7], analyzed byof
C. Johnson and J. Pitk\"aranta [5]. So we choose an approximating space
discrete densities as follows.
$h_{0,K}$$k$$P_{k}$$k$$G_{h}$$G_{h}=\{\alpha:\Omegaarrow \mathrm{R}|\alpha|_{K}\in P_{k}K\in \mathcal{T}_{h_{b}}\}$.

We should construct an approximation space
which approximates velocities well. For such a space we shall choose a special kind of spaces introduced by
M. Crouzeix and P.-A. Raviart [4].
Let be a non-negative integer and let us consider a family of subspaces
such that. We introduce a space defined by$X_{h}$$l$$P_{l}\subset P_{I\{^{r}}\subset C^{1}(K)$$\{P_{K}|K\in \mathcal{T}_{h}\}$$\Phi_{h}=\{\phi:\Omegaarrow \mathrm{R}|\phi|_{K}\in P_{l-1}\}$.

a discontinuous finit
ment spac of degree $l-1$ . Let
We call
totality of functions such that $v|_{K}\in P_{K}$ for all
and$e\mathrm{e}1e$$\Phi_{h}$$e$be the$K\in \mathcal{T}_{h}$$v$$\int_{F}(v_{1}-v_{2})\phi d\sigma=0$$F=\partial I\{\mathrm{i}_{1}\cap\partial I\mathrm{f}_{2}$$I\mathrm{f}_{i}\in \mathcal{T}_{h},$

$v$

(10)

$\forall\phi\in\Phi_{h}$$i=1,2$ , such thatfor
the totality of functions ofsuch that$v_{i}=v|I\acute{\mathrm{t}}_{i},$$W_{h}$. Further, let$W_{h,0}$be$W_{h}$$\int_{F}v\phi d\sigma=0$(11)$\forall\phi\in\Phi_{h}$. We define an approximation spaceand such that
$X_{h}=\{W_{h,0}\}^{d}$ , which approximate velocities. For $v\in X_{h}$ we define:
defined by

for$K\in \mathcal{T}_{h}$$F$$F=\partial\Omega\cap\partial K$$\mathrm{d}\mathrm{i}\mathrm{v}_{h}$$\Phi_{h}$$( \mathrm{d}\mathrm{i}\mathrm{v}_{h}v, \phi)=\sum(\mathrm{d}\mathrm{i}\mathrm{v}v,$ $\phi)_{K}$$K\in \mathcal{T}_{h}$$\forall\phi\in\Phi_{h}$.

$X_{h}arrow$

46

3 A SCHEME AND ITS STABILITY, which approximates
we defineBy this operatora Crouzeix-Raviart
the solenoidal space . Here we call an element ofvelocity of degree .
In our paper, the Crouzeix-Raviart velocities of degree $l=2k+1$ are applied
to discrete velocities in the scheme described below. Further let us notice thatcoincides with a discontinuous finite
the linear span ofof degree $l-1=dk$ .
element space
The upwind element and.nwind element, we canFor a discrete velocity $U\in X_{h}$ and a face element
to determined uniquely by
and a unit normalchoose a constant vector$V_{h}=\{v\in X_{h}|\mathrm{d}\mathrm{i}\mathrm{v}_{h}v=0\}$$\mathrm{d}\mathrm{i}\mathrm{v}_{h}$$V$$X_{h}$$v$$l$$\{\alpha_{1}\alpha_{2}|\alpha_{i}\in G_{h}, i=1,2\}$$\Phi_{h}$$\mathrm{d}\mathrm{o}\mathrm{w}$$F,$$U_{F\nu}$$F\subset\partial K,$$K\in \mathcal{T}_{h}$$F$$\iota\nearrow F$.

0$$\int_{F}U\cdot\nu_{F}d\sigma=\int_{F}U_{F\nu$\iota/d\sigma\geqF}$, thenfor the sake of conditions (10) and (11). Here, if
both of the upwind element
can be set any of unit normals to . By
and the downwind associated with are determined. Here, we see, we can define the gap of on byand
By thes elements,$U_{F\nu}=0\mathrm{a}\mathrm{n}\mathrm{d}arrow$$\int_{F}U\cdot\nu d\sigma=0$$F$$\nu_{F}$$\nu_{F}$$F=I\mathrm{f}_{U}\cap K_{D}$$F$$I\mathrm{f}_{U}$$e$$K_{D}$.

$F$$\alpha$$[\alpha]_{D}^{U}=\alpha|_{K_{U}}-\alpha|_{K_{D}}$3$IC_{U}$.

A scheme and its stability

First we propose a finite element scheme which approximates advection equations
be trilinear functionals defined by:well. Let$s_{h},t_{h}$$V_{h}\cross G_{h}\cdot \mathrm{x}G_{h}arrow \mathrm{R}$$\alpha_{2})_{K}\partial\alpha$$s_{h}(v, \alpha_{1}, \alpha_{2})=\sum_{K\in \mathcal{T}_{h}}\sum_{j=1}^{d}(v_{j\neq_{x_{j}}},$

,

$\{$

$[\alpha_{1}]_{U}^{D}\alpha_{2D}\rangle_{F}$$t_{h}(v, \alpha_{1}, \alpha_{2})=\sum_{F\subset\Omega}\langle v_{MF},$.

Now, our scheme is described as the following way.
such that: For $U^{n-1}\in V_{h}$ and $r^{n-1}\in G_{h}$ find$r^{n}\in G_{h_{b}}$$(\mathrm{P})_{h}$$\frac{(\delta r^{n},\beta)}{\tau}+s_{h}(U^{n-1}, r^{n}, \alpha)$$+t_{h}(U^{n-1}.r^{n}, \alpha)’=<g^{n},$ $\alpha>$$\forall\alpha\in G_{h}$

(12)

,

for each . For the case:elongs to the dual space of
where
and
have a discrete sch me approximating to the problem (P) as$G_{h}$$g^{n}\mathrm{b}$$h-+\mathrm{O}$$e$

The stability to the solutions
the proofs of lemmas.

$g^{n}\equiv 0$$\mathrm{n}$$r^{n},$$n=1,2,3,$$\cdots,$

$N$

we

’

$\tauarrow 0$.

, is described elow. We omit$\mathrm{b}$47

4

TRUNCATION TERMS

Lemma 1 (The discrete maximum principle) For the case :$N$ , together with $k=0,$
each $m=1,2,3,$
have$g^{n}\equiv 0$, and for$\uparrow ve$$\cdots,$$\min\{r^{m-1}|_{K}|K\in \mathcal{T}_{h}\}\leq\min\{r^{m}|_{K}|K\in \mathcal{T}_{h}\}$$\leq\max_{I\mathrm{f}}\{r^{m}|_{K}|K\in \mathcal{T}_{h}\}\leq\max_{I\mathrm{f}}\{r^{m}|_{K}|K\in \mathcal{T}_{h}\}$( -stabiliiy) For each $m=1,2,3,$$L^{2}$$\mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}_{}\mathrm{a}2$$\cdots,$$N$.

, together with $0\leq k$ wehave$||r^{m}||^{2}+ \sum_{n=1}^{m}||\delta r^{n}||^{2}+\tau\sum_{n=1}^{m}\langle U_{F}^{m-1},$$|[r^{m}]_{U}^{D}|^{2}\rangle_{F}=||r^{0}||^{2}$$+2 \tau\sum_{n=1}^{m}<g^{n},$ $r^{n}>$4.

Truncation terms

be the orthogonal projection fromonto the subspace. We write. Le us consider the errorsof .
To represent trucation terms it is convienient to introduce trilinear forms
and adjoint formally to and , respectively. In fact, they are defined byLet$L^{2}(\Omega)$$\pi_{h}$$\overline{p}^{m}=\pi_{h}(\rho^{n})$$G_{h}$$e^{n}=r^{n}-\overline{p}^{n}$$r^{n}$$s_{h}^{*}$$t_{h}^{*}$$s_{h}$$s_{h}^{*}(v, \alpha_{1},\alpha_{2})$$t_{h}$$=- \sum_{I\{’\in \mathcal{T}_{h}}\sum_{j=1}^{d}(v_{j}\alpha_{1},$$\frac{\partial}{\partial}\alpha B)_{K}x_{j}$

,

$\{$

$t_{h}^{*}(v, \alpha_{1}, \alpha_{2})$$\alpha_{1U}[\alpha_{2}]_{D}^{U}\rangle_{F}$$= \sum_{F\subset\Omega}\langle v_{MF},$.

Clearly, these trilinear forms may be defined for piecewise smooth functions
andinstead ofand , respectively. Other trilinear forms
and arealso useful to describe our truncation terms for discret densities .
$\beta_{1}$$\beta_{2}$$L_{h}$$\alpha_{2}$$\alpha_{1}$$e$$L_{h}(v_{1}, v_{2}, v_{3})$$v_{2}v_{3})$$= \sum_{K\in \mathcal{T}_{h}}(v_{1},$$l_{h}$$r^{n}$,$\{$$l_{h}(v_{1}, v_{2}, v_{3})$$v_{2}v_{3}\rangle_{F}$$= \sum_{F\grave{\subset}\Omega}..\langle v_{1},$

,

where
and
are piecewise smooth soalar functions over domainusing these trilinear forms we can descrive truncation terms as follows.
$v_{1},$ $v_{2}$$v_{3}$

Lemma 3 Let
Then$\rho$$\Omega$. By

be the exact solution of the problelm $(P)$ and $p^{n}(x)=\rho(x, t_{n})$ .

$\frac{(\delta e^{n},\alpha)}{\tau}+s_{h}^{*}(U^{n-1}, e^{n}, \alpha)+t_{h}^{*}(U^{n-1}, e^{n}, \alpha)=<g^{n},$$\alpha>$

,

(13)

48

5 ERROR$AN\mathrm{A}$LYSIS$<g^{n},$ $\alpha>=(\dot{p}-\frac{\delta\overline{p}^{n}}{\tau},$ $\alpha)+s_{h}(u^{n}-U^{n-1},p^{n}, \alpha)$$+s_{h}^{*}(U^{n-1},\rho^{n}-\overline{\rho}^{n}, \alpha)+t_{h}^{*}(U^{n-1},p^{n}-\overline{\rho}^{n},\alpha)$$-t_{h}(U^{n-1}, \rho^{n}, \alpha)-L_{h}(\mathrm{d}\mathrm{i}\mathrm{v}_{h}U^{n-1}, \rho^{n}-\overline{\rho}^{n}, \alpha)$$+l_{h}([U^{n-1}]_{F\nu}^{U\nu},\rho^{n}-\overline{\rho}^{n},$5$\mu)+l_{h}([U^{n-1}]_{D\nu}^{F\nu},p^{n}-\overline{\rho}^{n},$$\alpha)$.

Error analysis

We can prove the order of errors for the above discrete densities
the exact solution is smooth enough.
we assume that
For velocities(H.1) there exists an operator$r^{n}$

provided that$\rho$$v\in\{H^{2}(\Omega)\}^{d}$$\Pi_{h}\in \mathcal{L}(\{H^{2}(\Omega)\}^{d}$;$\{W_{h}\}^{d})\cap \mathcal{L}(\{H^{2}(\Omega)\}^{d}\cap(H_{0}^{1}(\Omega)\}^{d}$;$\{W_{h,0}\}^{d})$such that$\mathrm{d}\mathrm{i}\mathrm{v}_{h}\Pi_{h}v=\mathrm{d}\mathrm{i}\mathrm{v}_{h}v$,$||\Pi_{h}v-v||_{h}\leq c_{2}h^{m}|v|_{m+1}$

,

(14)

for all $v\in\{H^{m+1}(\Omega)\}^{d},$ $1\leq m\leq 2k+1$ , where is independent of and , but
depend on .
The hypothesis (H.1) is due to M. Crouzeix and P.-A. Raviart [4] and theexistence of finite elements velocities for $k=1,3$ satisfying this hypothesis seeof the discrete velocity. For the error
the examples in [4]. Let
defined by$h$$c_{2}$$\tau$$c_{1}$$E^{n}$$\tilde{u}^{n}=\Pi_{h}(u^{n})$$U^{n}$$E^{n}=U^{n}-\tilde{u}^{n},$We assume that the discrete velocities(H.2).

$n=1,2,3,$$U^{n}$$\cdots,$$N$.

satisfy the estimates$||E^{m}||_{h}^{2}\leq||E^{0}||_{h}^{2}+C_{1}h^{2(2k+1)}+C_{2}\tau^{2}$$m=1,2,3,$$\cdots$

, ,

(15)

$N$

and .
When we apply an implicit standard finite element scheme to the Stokes
equation we get the above estimates (15). Considering each truncation terms inLemma 3 the following lemma holds true.

wh$e\mathrm{r}\mathrm{e}C_{i},$$i=1,2$ , are independent of$\tau$$h$be the right hand side of (13). Under the assumptions on in
Lemma 4 Let
Theorem 1 there exists a constant , which is indepenof and , such that$g^{n}$$p$$C$$2 \tau\sum_{n=1}^{m}\langle g^{n},$$e^{n} \rangle\leq\tau\sum_{F\subset\Omega}\langle U_{F\nu}^{n-1},$$‘\tilde{t}ent$$h$$\tau$$|[e^{n}]_{D}^{U}|^{2}\rangle_{F}$$+C( \tau^{2a}+h^{2(2k+1)}+\tau\sum_{n=1}^{m}||E^{n}||_{h}^{2})$

,

(16)

49

6 THE STOKES EQUATIONS
where a denotes a constant described in the next theorem.
After substituing
into the first identity in Lemma 3 and applying
the proof of Lemma 2 , adding from $n=1$ to we get$\alpha=2\tau e^{n}$$m$$||e^{m}||^{2}+ \sum_{n=1}^{m}||\delta e^{n}||^{2}+\tau\sum_{F\subset\Omega}\langle U_{F\nu}^{n-1},$ $|[e^{n}]_{D}^{U}|^{2}\rangle_{F}$

(17)

.

$\leq||e^{0}||^{2}+2\tau\sum_{n=1}^{m}<g^{n},$ $e^{n}>$Applying Lemma 4 and the discrete Gronwall inequality we can prove the next
theorem below.
Theorem 1 Assume (5), (7) (15) and that a solutionof the problem$p$satisfies$\rho\in C([0, T];C^{k+1}(\overline{\Omega}))\cap C^{1}([0, T];W^{k+1,4}(\Omega))\cap C^{2}([0,T];H)$In the case: $k=0_{f}$ we further assume (6).
independent of and such that

(18)

Then there exists a constant$C_{f}$$h$$\tau$$m=1,2,3,$$||e^{m}||^{2}\leq C(||e^{0}||^{2}+||E^{0}||_{h}^{2}+h^{2(2k+1\rangle}+\tau^{2a})$

where

$C$

is independents of

where $a=1/2$ for the case: (8) does notholds.

6

.

$(P)$

$hold_{J}$and$\tau$$h$,$\cdots,$

$N$

,
(19)

and $a=1$ for the other case: (8)The Stokes equations

Here we consider a scheme for the modified density-dependent Stokes problem
governed by$(\overline{\mathrm{Q}})_{g}$in$\rho\frac{\partial u}{\partial t}+\mu\triangle u+\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}p=g$$\{$$\neq_{i}^{\partial}+(u\cdot \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d})\rho=0$$\mathrm{d}\mathrm{i}\mathrm{v}u=0$

in

$Q$

. in

$Q$

$Q$

,

,
(20)

,

with the boundary condition: $u=0$ on and the initial conditions: $p(x, 0)=$and $u(x, 0)=u^{0}(x)$ on , where $g\in L^{2}(0, T;X^{*})$ . The modified densitydependent Stokes problem
with the casereduces to the densitydependent Stokes problem
previously described in the introduction.
$S$$p^{0}(x)$$\Omega$$(\overline{\mathrm{Q}})_{g}$$(\mathrm{Q})_{f}$$g\equiv\rho f$

50

6 THE STOKES EQUATIONS
Let us consider bilinear forms
defined by$a_{h}$:$X_{h}\cross X_{h}arrow \mathrm{R}$and$a_{h}(v,w)= \sum_{K\in \mathcal{T}_{h}}(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}v,\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}w)_{K}$$b_{h}(v, \alpha)=$ $-$$\alpha)_{K}$$\sum_{K\in \mathcal{T}_{h}}(\mathrm{d}\mathrm{i}\mathrm{v}_{h}v,$$b_{h}$:$X_{h}\cross G_{h,0}arrow \mathrm{R}$,,where $G_{h,0}=\{\alpha\in G_{h}|(1, \alpha)=0\}$ .
:we consider a scheme
For the above problem$r^{n}\in
G_{h}$$U^{n-1}\in
V_{h}$
satisfying (12) with the casefind(1) for $r^{n-1}\in G_{h}$ and$k=0$ ,
and $P^{n}\in G_{h,0}$ satisfying
and $U^{n-1}\in V_{h}$ , find(2) then, for$(\overline{\mathrm{Q}})_{g,h}$$(\overline{\mathrm{Q}})_{g}$$U^{n}\in X_{h}$$r^{n-1},$ $r^{n}\in G_{h}$$\frac{(r^{n}\delta U^{n},v)}{\tau}+\mu a_{h}(U^{n}, v)+b_{h}(v, P^{n})=<\hat{g}^{n},$$v>$$\forall v\in X_{h}$

,

(21)

$\{$

$b_{h}(U^{n}, \alpha)=0$

In the above,$\hat{g}^{n}$$\forall\alpha\in G_{h,0}$.

approximates $g(x, t_{n})$ , in some sense, and satisfying$|<\hat{g}^{n},v>|\leq C_{1}^{n}||v||_{h}+C_{0}^{n}||v||$$\forall v\in X_{h}$, $N$ , where each
for $n=1,2,3,$
with some constants
We hav the stabiliy for the above scheme as follows.
$C_{n}$$\cdots$(22),

may depend on$C_{n}$$\hat{g}^{n}$.

$e$

Lemma 5 A ssume (22). $Then_{f}$ for each
. Further, usingof the problem
we have$(Q)_{h}$$P^{n}$$n_{J}$$G_{h}$

we have a unique solution
with $k=0$ and $m=1,2,3,$$r^{n},$$U^{n}$

and$N$ ,$\cdots,$$2M_{1} \tau\sum_{n=1}^{m}||\frac{\delta U^{n}}{\tau}||^{2}+2\mu||U^{m}||_{h}^{2}+\mu\sum_{n=1}^{m}||\delta U^{n}||_{h}^{2}$$\leq 2\mu||U^{0}||_{h}^{2}+C_{1}’\sum_{n=1}^{m}|C_{1}^{n}|^{2}+C_{0}’\tau\sum_{n=1}^{m}|C_{0}^{n}|^{2}$

Here C\’i,$C_{0}’$are independent of

$h$

.

and .
$\tau$

This is obtained by substituting $v=2\delta U^{n}$ into (21) through the discrete
maximum principle on . Applying the discrete Poincar\’e inequality we get$r^{n}$

Lemma 6 There exists a constant, $N$ ,
that, for $m=1,2,3,$$C_{3)}$

which is independent of$h$$\cdots$$||U^{m}||^{2} \leq C_{3}(\mu||U^{0}||_{h}^{2}+C_{1}’\sum_{n=1}^{m}|C_{1}^{n}|^{2}+C_{0}’\tau\sum_{n=1}^{m}|C_{0}^{n}|^{2})$.

and , such$\tau$

51

7 ERRORS ON DISCRETE VELOCITIES

7Errors on discrete velocities

We consider the errors of discrete velocities and densities of the density-dependent. Let be the discrete solution of the scheme
Stokes problem, where$g=pf$ . Already we have define the errorsof discret velocities. The errorof the discrete pressure
is defined by $E_{P}^{n}=P^{n}-p^{n}$ . Then, for $v\in X_{h}$ ,$U^{n}$$(\mathrm{Q})_{f,h}$$(\overline{\mathrm{Q}})_{g_{l}h}$$E^{n}$$U^{n}$$e$$P^{n}$$\frac{(r^{n}\delta E^{n},v)}{\tau}+\mu a_{h}(E^{n}, v)+b_{h}(v, E_{P}^{n})=\sum_{j=1}^{5}\langle\eta_{j}^{n},v\rangle$$<\eta_{1}^{n},$,$v>=((r^{n}-p^{n})f^{n},v)$$<\eta_{2}^{n},$ $v>=( \rho^{n}\dot{u}^{n}-\frac{\delta\tilde{u}^{n}}{\tau},$

$v)$

,

,

$<\eta_{3}^{n},$ $v>=-\mu((\triangle u^{n}, v)+(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}u^{n},\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}v))$$<\eta_{4}^{n},v>=\mu(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}(\tilde{u}^{n}-u^{n}),\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}v)$$<\eta_{5}^{n},v>=(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}p^{n},$ $v)+(\mathrm{d}\mathrm{i}\mathrm{v}v,p^{n})$.

Now, we see$|<\eta_{1}^{n},$$v>|\leq||f||_{\infty,Q}(||e^{n}||+hC_{1}(\eta_{1})|p^{n}|_{1,2,\omega,Q})||v||$$|<\eta_{2}^{n},$,$v>|\leq C_{0}(\eta_{2})||\dot{u}||_{\infty,Q}(||\delta r^{n}||+||e^{n}||)||v||$$+hC_{0}’(\eta_{2})||v||(||\dot{u}||_{\infty,Q}|\rho|_{1,2,\infty,Q}+||\dot{u}||_{1,2,\infty,Q}||p||_{\infty,Q})$$|<\eta_{3}^{n},$$v>|\leq\mu hC_{1}(\eta_{3})|u^{n}|_{2,\infty,Q}||v||_{h}$,$|<\eta_{4}^{n},$$v>|\leq\mu hC_{1}(\eta_{4})|u^{n}|_{2,\infty,Q}||v||_{h}$,$|<\eta_{5}^{n},$$v>|\leq hC_{1}(\eta_{5})\uparrow p^{n}|_{1,\infty,Q}||v||_{h}$,where the constants $C(\eta j),j=1,2,$ , are independent of$n=1,2,3,$
Applying Lemma 5 to the above estimates we get the result elow.
$\cdots$$\tau,$$h,$$\mathrm{b}$$\cdots$, $N$ .

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REFERENCES
Lemma 7 Assume (18) on

$p$

and

$u\in C([0,T],$ $\{C^{2}(\overline{\Omega}^{d})\}^{d})\cap C^{1}([0, T],$ $\{C(\overline{\Omega})\}^{d})\cap W^{2,2}(0,T;\mathrm{H})$Then there exist constantsthat$C_{1},$ $C_{2}$and$C_{3}$.

(23), which are independent of and , such$h$$\tau$$||E^{m}||_{h}^{2} \leq||E^{0}||_{h}^{2}+C_{1}\tau\sum_{n=1}^{m}||e^{n}||^{2}$$+C_{2}\tau+C_{3^{\frac{h^{2}}{\tau}}}$.

Combining (16), (17), (23) and the discrete Gronwall inequality, we get thetheorem below.
Theorem 2 Assume (18) and (23). Then there exists a constant$N,$such thatindependent of $m=1,2,3,$$\cdots,$

$h,$

$C$

, which is$\tau_{j}$$||e^{m}||^{2}+||E^{m}||_{h}^{2} \leq C(||e^{0}||^{2}+||E^{0}||_{h}^{2}+\tau+\frac{h^{2}}{\tau})$.

References[1] Antontsev, S. N., A. V. Kazhikov and V. N. Monakohv, Boundary value
problems in mechanics of non-homogeneous fluids,North-Holland, Amsterdam, 1990.

[2] Bardos, C., Probl\‘emes aux limites pour les \’equations aux d\’eriv\’ees partielles
du premier ordre \‘a coeffici e nts reels, Ann. Scient. Ec. Norm. Sup., 4e serie,t. 3, 185-233, 1970.
[3] Ciarlet, P. G. and P.-A. Raviart, General Lagrange and Hermite interpowith applications to finite element methods, Arch. Rational.
lation in
Mech. Anal., 46, pp.177-199, 1972.
$\mathrm{R}^{n}$[4] Crouzeix, M. and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, R. A. I. R. 0.,7, S\’er. Anal. Numer. , pp. 33-76, 1973.

[5] Johnson, C. and J. Pitk\"aranta, An analysis of the discontinuous Galerkinmethod for a scalar hyperbolic equation, Math. Compu. 46, pp. 1-26, 1986.
[6] Kaizu, S., the convergence of finite element solutions of advection problems,in preparation.

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REFERENCES[7] Lesaint, L. and P.-A. Raviart, On a finite element method for solving theneutron transport equation, Aspects of finite elements in partial differential
equations, ed. C. de Boor, Academic Press, New York, 1974.
[8] Lions, Jacque Lui, Probl\‘ems aux limites dans les \’equatioons aux d\’eriv\’eespartielles, Le press de l’universit\’e de Montr\’eal, 1965.
[9] Ohmori, Katsushi, On a modified Lax-Wendroff finite element method forlinear hyperbolic equations, Preprint.
[10] Lions, P.-L. , Mathematical topics in fluid mechanics, Clarendon Press, Oxford, 1996.
[11] Temam, R. , Navier Stokes equations, North-Holland, Amsterdam, New
York, Oxford, 1979.