Error analysis of finite element solutions of
convection 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 $\Omega$ be a bounded polygon in $\mathrm{R}^{2}$ or a polyhedron in $\mathrm{R}^{3}$. Let $T>0$ and
$Q=\Omega \mathrm{x}(0,T)$ and $S=\partial\Omega \mathrm{x}(0, T)$. We would like to propose a convergent
finite element scheme approximating the following convection problem (P): find
unknown density $\rho(x, t)$ ofsome material: $\Omegaarrow \mathrm{R}$ satisfying
$\ovalbox{\tt\small REJECT}^{\partial}+(u\cdot \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d})p=0$ in $Q$, (1)
$\rho(x, 0)=\rho^{0}(x)$ on $\Omega$ (2)
with respect to known velocity $u(.x, t)$ of some incompresible flows:
$\mathrm{d}\mathrm{i}\mathrm{v}u=0$ in $Q$, (3)
$u(x, t)=0$ on S. (4)
In the construction of a convergent finiteelement schemefor the problem (P),
of courese, we assume some regulaity and boundedness on the velocities $u$ 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 is depend on its aim, certainly, to which the advection problem (P) is applied.
1 INTRODUCTION
In fact, the problem (P) is a subtarget of another propblem $(\hat{\mathrm{Q}})$, the
incom-pressible Navier-Stokes problems with nonhomogeneous density, $\mathrm{i}$.
$\mathrm{e}.$, which is a
system governed by the density-dependent $\mathrm{N}\mathrm{a}‘ \mathrm{v}\mathrm{i}\mathrm{e}$-Stokes equation:
$p \{\frac{\partial u}{\partial t}(u\cdot \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d})u\}-\mu\triangle u+\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}p=\rho f$ in $Q$,
together with two other equations, (1) and (3), where $u,p$ and $\rho$ are unknown
velocities, pressures and densities of the non-homogeneous liquids, respectively.
This problem $(\hat{\mathrm{Q}})$ admits at least a weak solution
$p,$$u,p$ satisfying
$u\in L^{\infty}(0,$$T;\{L^{2}(\Omega)\}^{d})\cap L^{2}(0,$$T;\{H_{0}^{1}(\Omega)\}^{d})$ , (5)
for the initial velocity $u_{0}\in\{L^{2}(\Omega)\}^{d}$, the outer force $f\in L^{1}(0,$$T;\{L^{2}(\Omega)\}^{d})$
and
$0<M_{1}\leq\rho^{0}\leq M_{2}<\infty$ (6)
where $M_{i},$ $i=1,2$, are constants (cf., for example, A. S. N. Antontsev, A. V.
Kazhikhov and V. N. Monakhov [1] and P. L. Lions [10]$)$ So, in our paper
we shall choose conditions (5) and (6) on the velocity $u$ and the density $\rho^{0}$
,
respectively.
Beside, there are another conditions in our paper to be needed, because we
have to construct a finite element scheme for (P). Thus, conditions on discrete
velocities are also needed. Unfortunately, there may be no convergent scheme,
which approximates solutions of the problem $(\hat{\mathrm{Q}})$. So, froma convergent implicit
scheme, which approximates solutions of the classical Navier-Stokes equations
in the incompressible case with homogeneous density, we shall choose a suitable
condition on discrete velocities. For references it is useful to see the book [11] by
R. Temam.
In fact, let $\tau>0$ beatime mesh and let $U^{n}$ be an approximation of$u(t_{n}),$$0=$
$t_{0}<t_{1}<t_{2}<\cdots<t_{n}=n\tau,$$\cdots,$$N=[T/\tau]$
.
Let $\delta U^{n}=U^{n}-U^{n-1}$. Then weconsider the condition
$||U^{n}||^{2}+\Sigma_{n=1}^{N}(||\delta U^{n}||^{2}+\tau||U^{n}||_{h}^{2})\leq c_{0}<\infty$, (7)
where$c_{0}$ is independent ofthetime meshes $\tau$ and the spacemeshes $h>0$ us$e\mathrm{d}$ in
triangulations $\{\mathcal{T}_{h}\}_{h}$ of the domain $\Omega$ but depend on the initial value of $u_{0}$ and
the force term $f$. Here the notation $||\cdot||$ is $\mathrm{t}_{\mathrm{w}}\mathrm{h}\mathrm{e}L^{2}$ norm over the domain $\Omega$ and
In this paper we propose a finite element scheme $(\mathrm{P})_{h}$ : (12), and prove the
1 INTRODUCTION
another paper [6] under these conditons from (5) to (7). The order of the error
of discrete solutions of (12) is studied under sufficient regularity conditions on $u$
and $p$ in this paper.
It may be difficult to extend the scheme (12) to a finite element scheme
ap-proximating the density-dependent Navier-Stokes problem, although it is
possi-ble to extend (12) to a finite element scheme $(\overline{\mathrm{Q}})_{g,h},$ $g=pf$, which approximates
the density-dependent Stokes $\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}^{r}\mathrm{m}(\mathrm{Q})_{j}$ governed by the density-dependent
Stokes equation
$p \frac{\partial u}{\partial t}-\mu\triangle u+\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}p=\rho f$ in $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 so-lutions. Note that, in the density-dependent Stokes problem, $u,p$ and $p$ are
unknown velocities, pressures and densities ofnon-homogeneous liquids,
respec-tively. We shall consider a scheme $(\overline{\mathrm{Q}})_{g,h}$ : (12) and (21), approximating the
density-dependent Stokes problem, and show that theapproximation $U^{n}$ satisfies
$\tau\sum_{n=1}^{N}||\frac{\delta U^{n}}{\tau}||^{2}\leq c_{0}<\infty$, (8)
where $c_{0}$ is independent of the time meshes$\tau$ and the space lneshes $h$, but depend
on the initial value of $u_{0}$ and the force term $f$
.
If we assume the condition (8)adding to the conditions (5), (6) and (7), then we get an increas$e$ in the order of
the error ofdiscrete densities of our scheme (12).
Notation
For a domain $G$ we write $(f,$$g)_{G}= \int_{G}$fgdx. In particular, we write $(f, g)=$ $\int_{\Omega}$fgdx. We denote by $||f||_{p,G}$ the $L^{p}$ norm of $f$ over $G$
.
In paticular, for$G=\Omega$, we write $||f||_{p}=||f||_{p,\Omega}$
.
For a $d-1$ dimensional simplex $F$, wewrite $\langle f,g\rangle_{F}=\int_{F}fgd\sigma$. Let $H=L^{2}(\Omega)$ and $\mathrm{H}=\{L^{2}(\Omega)\}^{d},$ $X=\{H_{0}^{1}(\Omega)\}^{d}$, $V=\{v\in X|\mathrm{d}\mathrm{i}\mathrm{v}v=0\}$
.
Further, for the usual Sobol$e\mathrm{v}$ space, $W^{1,p}(G)$ with $1\leq p\leq\infty$ we need
semi-norms, $|v|_{l,p,G}= \{\sum_{|a|=l}||D^{a}v||_{p,G}^{p}\}^{1/\mathrm{P}}$, and a norm, $||v||_{m,p,G}= \{\sum_{l=0}^{m}|v|_{l,p,G}^{p}\}^{1/p}$,
where $D^{a}$ are differential operators of $\mathrm{o}\mathrm{r}\mathfrak{a}^{1}\mathrm{e}\mathrm{r}a,$ $a$ denote multi-indexes. For the
case $G=\Omega$ we drop the letter $G$ from the surfixes of $|v|_{l,p,G}$ and $||v||_{m,\mathrm{p},G}$ and let
$Z^{m,p}=\{v\in L^{p}(\Omega)|v|_{K}\in W^{m,p}(K)\},$ $m$ denotes a non-negative integer and $p$ satisfies $1<p<\infty$. Generally, we call a function $v\in Z^{m,p}$ for some $m$ and $p$ a
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
$D^{s}v$ denotes s-th derivative of$v$ in the sense ofdistribution.
2
Discrete densities and velocities
Let $\mathcal{T}_{h}=\{K\}$ be a decomposition of $\Omega$ by tr\’iangulations for $d=2$ or by
tetra-hedrons for $d=3$
.
Here we denote by $h$ the maximum ofthe diameters $h_{K}$ of$K\in \mathcal{T}_{h}$
.
The sequence of decompositons of $\Omega,$ $\{\mathcal{T}_{h}\}$ is regular: there exists aconstant $c_{1}$ such that
$\lim\sup\sup\underline{h_{I\{}\cdot}\leq c_{1}<\infty$
,
(9) $harrow 0$ $K\in\tau_{h}h_{0,K}$
where $h_{0,K}$ is the largest diameter of spheres included in $K$.
Let $k$ be anon-negativeintegerand $P_{k}$ be the totalityofpolynomialsofdegree
$k$. Our approximation method for discrete densities relies on the discontinuous, finite element method induced by L. Lesaint and P.-A. Raviart [7], analyzed by
C. Johnson and J. Pitk\"aranta [5]. So we choose an approximating space $G_{h}$ of
discrete densities as follows.
$G_{h}=\{\alpha:\Omegaarrow \mathrm{R}|\alpha|_{K}\in P_{k}K\in \mathcal{T}_{h_{b}}\}$
.
We should construct an approximation space $X_{h}$ which approximates
veloci-ties well. For such a space we shall choose a special kind of spaces introduced by
M. Crouzeix and P.-A. Raviart [4].
Let $l$ be a non-negative integer and let us consider a family of subspaces
$\{P_{K}|K\in \mathcal{T}_{h}\}$ such that $P_{l}\subset P_{I\{^{r}}\subset C^{1}(K)$. We introduce a space defined by $\Phi_{h}=\{\phi:\Omegaarrow \mathrm{R}|\phi|_{K}\in P_{l-1}\}$
.
We call $\Phi_{h}$ a discontinuous finit$e\mathrm{e}1e$ment spac$e$ of degree $l-1$
.
Let $W_{h}$ be thetotality of functions $v$ such that $v|_{K}\in P_{K}$ for all $K\in \mathcal{T}_{h}$ and
$\int_{F}(v_{1}-v_{2})\phi d\sigma=0$ $\forall\phi\in\Phi_{h}$ (10)
for $v_{i}=v|I\acute{\mathrm{t}}_{i},$$I\mathrm{f}_{i}\in \mathcal{T}_{h},$ $i=1,2$, such that $F=\partial I\{\mathrm{i}_{1}\cap\partial I\mathrm{f}_{2}$
.
Further, let $W_{h,0}$ bethe totality offunctions $v$ of$W_{h}$ such that
$\int_{F}v\phi d\sigma=0$ $\forall\phi\in\Phi_{h}$ (11)
for $K\in \mathcal{T}_{h}$ and $F$ such that $F=\partial\Omega\cap\partial K$. We define an approximation space
$X_{h}=\{W_{h,0}\}^{d}$, which approximate velocities. For $v\in X_{h}$ we define $\mathrm{d}\mathrm{i}\mathrm{v}_{h}$ : $X_{h}arrow$
$\Phi_{h}$ defined by
$( \mathrm{d}\mathrm{i}\mathrm{v}_{h}v, \phi)=\sum(\mathrm{d}\mathrm{i}\mathrm{v}v,$$\phi)_{K}$ $\forall\phi\in\Phi_{h}$
.
3 A SCHEME AND ITS STABILITY
By this operator $\mathrm{d}\mathrm{i}\mathrm{v}_{h}$ we define $V_{h}=\{v\in X_{h}|\mathrm{d}\mathrm{i}\mathrm{v}_{h}v=0\}$, which approximates
the solenoidal space $V$
.
Here we call an element $v$ of $X_{h}$ a Crouzeix-Raviartvelocity ofdegree $l$
.
In our paper, the Crouzeix-Raviart velocities ofdegree $l=2k+1$ are applied
to discrete velocities in the scheme described below. Further let us notice that
the linear span of$\{\alpha_{1}\alpha_{2}|\alpha_{i}\in G_{h}, i=1,2\}$ coincides with a discontinuous finite
element space $\Phi_{h}$ of degree $l-1=dk$ .
The upwind element and $\mathrm{d}\mathrm{o}\mathrm{w}$
.nwind
elementFor a discrete velocity $U\in X_{h}$ and a face element $F,$ $F\subset\partial K,$ $K\in \mathcal{T}_{h}$, we can
choose a constant vector $U_{F\nu}$ and a unit normal $\iota\nearrow F$ to $F$ determined uniquely by
$\int_{F}U\cdot\nu_{F}d\sigma=\int_{F}U_{F\nu F}$
.
$\iota/d\sigma\geq 0$for the sake of conditions (10) and (11). Here, if$\int_{F}U\cdot\nu d\sigma=0$, then $U_{F\nu}=0\mathrm{a}\mathrm{n}\mathrm{d}arrow$
$\nu_{F}$ can be set any of unit normals to $F$
.
By $\nu_{F}$ both of the upwind element $IC_{U}$and the downwind associated with $F$ aredetermined. Here, we see $F=I\mathrm{f}_{U}\cap K_{D}$
.
By thes$e$ elements, $I\mathrm{f}_{U}$ and $K_{D}$, we can define the gap of $\alpha$ on $F$ by
$[\alpha]_{D}^{U}=\alpha|_{K_{U}}-\alpha|_{K_{D}}$.
3
A
scheme and
its
stability
First wepropose afinite element scheme which approximates advection equations well. Let $s_{h},t_{h}$ : $V_{h}\cross G_{h}\cdot \mathrm{x}G_{h}arrow \mathrm{R}$ be trilinear functionals defined by
$\{$
$s_{h}(v, \alpha_{1}, \alpha_{2})=\sum_{K\in \mathcal{T}_{h}}\sum_{j=1}^{d}(v_{j\neq_{x_{j}}},$$\alpha_{2})_{K}\partial\alpha$,
$t_{h}(v, \alpha_{1}, \alpha_{2})=\sum_{F\subset\Omega}\langle v_{MF},$ $[\alpha_{1}]_{U}^{D}\alpha_{2D}\rangle_{F}$. Now, our scheme is described as the following way.
$(\mathrm{P})_{h}$: For $U^{n-1}\in V_{h}$ and $r^{n-1}\in G_{h}$ find $r^{n}\in G_{h_{b}}$ such that
$\frac{(\delta r^{n},\beta)}{\tau}+s_{h}(U^{n-1}, r^{n}, \alpha)$
$+t_{h}(U^{n-1}.r^{n}, \alpha)’=<g^{n},$$\alpha>$ $\forall\alpha\in G_{h}$, (12)
where $g^{n}\mathrm{b}$elongs to the dual space of $G_{h}$ for each $\mathrm{n}$. For the case: $g^{n}\equiv 0$ we
have a discrete sch$e$me approximating to the problem (P) as $h-+\mathrm{O}$ and
’ $\tauarrow 0$.
The stability to the solutions $r^{n},$$n=1,2,3,$$\cdots,$$N$, is described $\mathrm{b}$elow. We omit
4 TRUNCATION TERMS
Lemma 1 (The discrete maximum principle) For the case : $g^{n}\equiv 0$, and
for
each $m=1,2,3,$ $\cdots,$$N$, together with $k=0,$ $\uparrow ve$ have
$\min\{r^{m-1}|_{K}|K\in \mathcal{T}_{h}\}\leq\min\{r^{m}|_{K}|K\in \mathcal{T}_{h}\}$
$\leq\max_{I\mathrm{f}}\{r^{m}|_{K}|K\in \mathcal{T}_{h}\}\leq\max_{I\mathrm{f}}\{r^{m}|_{K}|K\in \mathcal{T}_{h}\}$
.
$\mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}_{}\mathrm{a}2$ (
$L^{2}$-stabiliiy) For each $m=1,2,3,$
$\cdots,$$N$, together with $0\leq k$ we have
$||r^{m}||^{2}+ \sum_{n=1}^{m}||\delta r^{n}||^{2}+\tau\sum_{n=1}^{m}\langle U_{F}^{m-1},$ $|[r^{m}]_{U}^{D}|^{2}\rangle_{F}=||r^{0}||^{2}$
$+2 \tau\sum_{n=1}^{m}<g^{n},$$r^{n}>$ .
4
Truncation
terms
Let $\pi_{h}$ be the orthogonal projection from $L^{2}(\Omega)$ onto the subspace $G_{h}$. We write
$\overline{p}^{m}=\pi_{h}(\rho^{n})$
.
Le us consider the errors $e^{n}=r^{n}-\overline{p}^{n}$ of$r^{n}$.To represent trucation terms it is convienient to introduce trilinear forms $s_{h}^{*}$
and $t_{h}^{*}$ adjoint formally to $s_{h}$ and $t_{h}$
,
respectively. In fact, they are defined by$\{$
$s_{h}^{*}(v, \alpha_{1},\alpha_{2})$ $=- \sum_{I\{’\in \mathcal{T}_{h}}\sum_{j=1}^{d}(v_{j}\alpha_{1},$ $\frac{\partial}{\partial}\alpha B)_{K}x_{j}$,
$t_{h}^{*}(v, \alpha_{1}, \alpha_{2})$
$= \sum_{F\subset\Omega}\langle v_{MF},$
$\alpha_{1U}[\alpha_{2}]_{D}^{U}\rangle_{F}$.
Clearly, these trilinear forms may be defined for piecewise smooth functions $\beta_{1}$
and $\beta_{2}$ instead of $\alpha_{1}$ and $\alpha_{2}$, respectively. Other trilinear forms $L_{h}$ and $l_{h}$ are
also useful to describe our truncation terms for discret$e$ densities $r^{n}$.
$\{$ $L_{h}(v_{1}, v_{2}, v_{3})$ $= \sum_{K\in \mathcal{T}_{h}}(v_{1},$$v_{2}v_{3})$, $l_{h}(v_{1}, v_{2}, v_{3})$ $= \sum_{F\grave{\subset}\Omega}..\langle v_{1},$ $v_{2}v_{3}\rangle_{F}$,
where $v_{1},$$v_{2}$ and $v_{3}$ are piecewise smooth
soalar
functions over domain $\Omega$. Byusing these trilinear forms we can descrive truncation terms as follows.
Lemma 3 Let $\rho$ be the exact solution
of
the problelm $(P)$ and $p^{n}(x)=\rho(x, t_{n})$.
Then
$\frac{(\delta e^{n},\alpha)}{\tau}+s_{h}^{*}(U^{n-1}, e^{n}, \alpha)+t_{h}^{*}(U^{n-1}, e^{n}, \alpha)=<g^{n},$
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)$
$+s_{h}^{*}(U^{n-1},\rho^{n}-\overline{\rho}^{n}, \alpha)+t_{h}^{*}(U^{n-1},p^{n}-\overline{\rho}^{n},\alpha)$
$-t_{h}(U^{n-1}, \rho^{n}, \alpha)-L_{h}(\mathrm{d}\mathrm{i}\mathrm{v}_{h}U^{n-1}, \rho^{n}-\overline{\rho}^{n}, \alpha)$
$+l_{h}([U^{n-1}]_{F\nu}^{U\nu},\rho^{n}-\overline{\rho}^{n},$ $\mu)+l_{h}([U^{n-1}]_{D\nu}^{F\nu},p^{n}-\overline{\rho}^{n},$ $\alpha)$
.
5
Error analysis
We can provethe order oferrors for the above discrete densities $r^{n}$ provided that
the exact solution $\rho$ is smooth enough.
For velocities $v\in\{H^{2}(\Omega)\}^{d}$ we assume that
(H.1) there exists an operator
$\Pi_{h}\in \mathcal{L}(\{H^{2}(\Omega)\}^{d}$ ;$\{W_{h}\}^{d})\cap \mathcal{L}(\{H^{2}(\Omega)\}^{d}\cap(H_{0}^{1}(\Omega)\}^{d}$ ;$\{W_{h,0}\}^{d})$
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},$$1\leq m\leq 2k+1$, where $c_{2}$ is independent of$h$ and $\tau$, but
depend on $c_{1}$.
The hypothesis (H.1) is due to M. Crouzeix and P.-A. Raviart [4] and the
existence of finite elements velocities for $k=1,3$ satisfying this hypothesis see the examples in [4]. Let $\tilde{u}^{n}=\Pi_{h}(u^{n})$
.
For the error $E^{n}$ of the discrete velocity $U^{n}$ defined by$E^{n}=U^{n}-\tilde{u}^{n},$$n=1,2,3,$$\cdots,$$N$
.
We assume that the discrete velocities $U^{n}$ satisfy the estimates
(H.2).
$||E^{m}||_{h}^{2}\leq||E^{0}||_{h}^{2}+C_{1}h^{2(2k+1)}+C_{2}\tau^{2}$ $m=1,2,3,$$\cdots$
,
$N$, (15)wh$e\mathrm{r}\mathrm{e}C_{i},$$i=1,2$, are independent of $\tau$ and $h$.
When we apply an implicit standard finite element scheme to the Stokes
equation we get the above estimates (15). Considering each truncation terms in
Lemma 3 the following lemma holds true.
Lemma 4 Let $g^{n}$ be the right hand side
of
(13). Under the assumptions on $p$ inTheorem 1 there exists a constant $C$, which is indepen$‘\tilde{t}ent$
of
$h$ and $\tau$, such that$2 \tau\sum_{n=1}^{m}\langle g^{n},$ $e^{n} \rangle\leq\tau\sum_{F\subset\Omega}\langle U_{F\nu}^{n-1},$ $|[e^{n}]_{D}^{U}|^{2}\rangle_{F}$
6 THE STOKES EQUATIONS
where a denotes a constant described in the next theorem.
After substituing $\alpha=2\tau e^{n}$ into the first identity in Lemma 3 and applying
the proof of Lemma 2 , adding from $n=1$ to $m$ we get
$||e^{m}||^{2}+ \sum_{n=1}^{m}||\delta e^{n}||^{2}+\tau\sum_{F\subset\Omega}\langle U_{F\nu}^{n-1},$ $|[e^{n}]_{D}^{U}|^{2}\rangle_{F}$
$\leq||e^{0}||^{2}+2\tau\sum_{n=1}^{m}<g^{n},$$e^{n}>$ . (17)
Applying Lemma 4 and the discrete Gronwall inequality we can prove the next
theorem below.
Theorem 1 Assume (5), (7) (15) and that a solution $p$
of
the problem $(P)$satisfies
$\rho\in C([0, T];C^{k+1}(\overline{\Omega}))\cap C^{1}([0, T];W^{k+1,4}(\Omega))\cap C^{2}([0,T];H)$
.
(18)In the case: $k=0_{f}$ we
further
assume (6). Then there exists a constant $C_{f}$independent
of
$\tau$ and $h$ such that$||e^{m}||^{2}\leq C(||e^{0}||^{2}+||E^{0}||_{h}^{2}+h^{2(2k+1\rangle}+\tau^{2a})$ $m=1,2,3,$
$\cdots,$$N$,
where $C$ is
independents
of
$\tau$ and $h$, (19)where $a=1/2$
for
the case: (8) does not $hold_{J}$ and $a=1$for
the other case: (8)holds.
6
The Stokes
equations
Here we consider a scheme for the modified density-dependent Stokes problem
$(\overline{\mathrm{Q}})_{g}$ governed by
$\{$
$\rho\frac{\partial u}{\partial t}+\mu\triangle u+\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}p=g$ in $Q$,
$\neq_{i}^{\partial}+(u\cdot \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d})\rho=0$
.
in $Q$,$\mathrm{d}\mathrm{i}\mathrm{v}u=0$ in $Q$,
(20)
with the boundary condition: $u=0$ on $S$ and the initial conditions: $p(x, 0)=$ $p^{0}(x)$ and $u(x, 0)=u^{0}(x)$ on $\Omega$, where $g\in L^{2}(0, T;X^{*})$. The modified
density-dependent Stokes problem $(\overline{\mathrm{Q}})_{g}$ with the case
$g\equiv\rho f$ reduces to the
6 THE STOKES EQUATIONS
Let us consider bilinear forms $a_{h}$ : $X_{h}\cross X_{h}arrow \mathrm{R}$ and $b_{h}$ : $X_{h}\cross G_{h,0}arrow \mathrm{R}$
defined by
$a_{h}(v,w)= \sum_{K\in \mathcal{T}_{h}}(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}v,\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}w)_{K}$ , $b_{h}(v, \alpha)=$ $-$$\sum_{K\in \mathcal{T}_{h}}(\mathrm{d}\mathrm{i}\mathrm{v}_{h}v,$
$\alpha)_{K}$,
where $G_{h,0}=\{\alpha\in G_{h}|(1, \alpha)=0\}$
.
For the above problem $(\overline{\mathrm{Q}})_{g}$ we consider a scheme $(\overline{\mathrm{Q}})_{g,h}$:
(1) for $r^{n-1}\in G_{h}$ and $U^{n-1}\in V_{h}$ find $r^{n}\in G_{h}$ satisfying (12) with the case
$k=0$,
(2) then,for$r^{n-1},$ $r^{n}\in G_{h}$ and $U^{n-1}\in V_{h}$,find $U^{n}\in X_{h}$ and $P^{n}\in G_{h,0}$ satisfying
$\{$
$\frac{(r^{n}\delta U^{n},v)}{\tau}+\mu a_{h}(U^{n}, v)+b_{h}(v, P^{n})=<\hat{g}^{n},$$v>$ $\forall v\in X_{h}$,
$b_{h}(U^{n}, \alpha)=0$ $\forall\alpha\in G_{h,0}$
.
(21)
In the above, $\hat{g}^{n}$ approximates $g(x, t_{n})$, in some sense, and satisfying
$|<\hat{g}^{n},v>|\leq C_{1}^{n}||v||_{h}+C_{0}^{n}||v||$ $\forall v\in X_{h}$, (22)
with some constants $C_{n}$ for $n=1,2,3,$$\cdots$ , $N$, where each $C_{n}$ may depend on$\hat{g}^{n}$.
We hav$e$ the stabiliy for the above scheme as follows.
Lemma 5 Assume (22). $Then_{f}$
for
each$n_{J}$ we have a unique solution$r^{n},$$U^{n}$ and $P^{n}$
of
the problem $(Q)_{h}$. Further, using $G_{h}$ with $k=0$ and $m=1,2,3,$ $\cdots,$$N$,we have
$2M_{1} \tau\sum_{n=1}^{m}||\frac{\delta U^{n}}{\tau}||^{2}+2\mu||U^{m}||_{h}^{2}+\mu\sum_{n=1}^{m}||\delta U^{n}||_{h}^{2}$
$\leq 2\mu||U^{0}||_{h}^{2}+C_{1}’\sum_{n=1}^{m}|C_{1}^{n}|^{2}+C_{0}’\tau\sum_{n=1}^{m}|C_{0}^{n}|^{2}$ . Here
C\’i,
$C_{0}’$ are independentof
$h$ and $\tau$.This is obtained by substituting $v=2\delta U^{n}$ into (21) through the discrete
maximum principle on $r^{n}$. Applying the discrete Poincar\’e inequality we get
Lemma 6 There exists a constant $C_{3)}$ which is independent
of
$h$ and $\tau$, suchthat,
for
$m=1,2,3,$$\cdots$ ,$N$,7 ERRORS ON DISCRETE VELOCITIES
7
Errors
on
discrete
velocities
We consider the errors of discrete velocities and densities of the density-dependent
Stokes problem$(\mathrm{Q})_{f,h}$
.
Let $U^{n}$ bethe discrete solution of the scheme $(\overline{\mathrm{Q}})_{g_{l}h}$, where$g=pf$
.
Already wehave define the errors $E^{n}$ ofdiscret$e$ velocities $U^{n}$.
The errorofthe discrete pressure $P^{n}$ is defined by $E_{P}^{n}=P^{n}-p^{n}$
.
Then, for $v\in X_{h}$,$\frac{(r^{n}\delta E^{n},v)}{\tau}+\mu a_{h}(E^{n}, v)+b_{h}(v, E_{P}^{n})=\sum_{j=1}^{5}\langle\eta_{j}^{n},v\rangle$,
$<\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}$,
wheretheconstants$C(\eta j),j=1,2,$$\cdots$ , areindependentof$\tau,$$h,$$n=1,2,3,$$\cdots$ ,$N$.
REFERENCES
Lemma 7 Assume (18) on $p$ and
$u\in C([0,T],$$\{C^{2}(\overline{\Omega}^{d})\}^{d})\cap C^{1}([0, T],$ $\{C(\overline{\Omega})\}^{d})\cap W^{2,2}(0,T;\mathrm{H})$
.
(23)Then there exist constants $C_{1},$$C_{2}$ and$C_{3}$, which are independent
of
$h$ and $\tau$, suchthat
$||E^{m}||_{h}^{2} \leq||E^{0}||_{h}^{2}+C_{1}\tau\sum_{n=1}^{m}||e^{n}||^{2}$
$+C_{2}\tau+C_{3^{\frac{h^{2}}{\tau}}}$
.
Combining (16), (17), (23) and the discrete Gronwall inequality, we get the theorem below.
Theorem 2 Assume (18) and (23). Then there exists a constant $C$, which is
independent
of
$m=1,2,3,$ $\cdots,$$N,$$h,$$\tau_{j}$ such that$||e^{m}||^{2}+||E^{m}||_{h}^{2} \leq C(||e^{0}||^{2}+||E^{0}||_{h}^{2}+\tau+\frac{h^{2}}{\tau})$ .
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