数理解析研究所講究録
第 1782 巻 2012 年 51-61

51

Analysis of a pressure-stabilized characteristics finite element
scheme for a linearized Navier-Stokes equations
Hirofumi Notsu *Waseda Institute for Advanced Study, Waseda University

1 Introduction
Let be a bounded domain in $\mathbb{R}^{d}(d=2,3),$ be a positive constant,be a velocity
and :be a scalar function. We consider a trajectory of fluid particle, which is importantfor flow problems. Let $X=X(.;x,t^{n})$ :be a solution of the ordinary differential equation,$\Omega$$\phi$$u:\Omega\cross(0, T)arrow \mathbb{R}^{d}$$T$$\Omega\cross(0, T)arrow \mathbb{R}$$(0, T)arrow \mathbb{R}^{d}$(1)$\frac{dX}{dt}=u(X,t)$with a condition $X(t^{n})=x$, where$n\in \mathbb{N}\cup\{0\},$ $\Delta t$

is a time increment and$\frac{D\psi}{Dt}(x,t)=\frac{d}{dt}\phi(X(t),t)$$t^{n}\equiv n\Delta t$,. Then, it holds that
(2)

where
(3)

$\frac{D}{Dt}\equiv\frac{\partial\psi}{\partial t}+u\cdot\nabla$

is a material derivation. Therefore, we can consider a first order approximation of the material derivative at$t=t^{n}(n\geq 1)$

as follows;$\frac{D\phi}{Dt}(x,t^{n})\approx\frac{\phi(X(t^{n}),t^{n})-\psi(X(t^{n-1}),t^{n-1})}{\Delta I}$$\approx\frac{\phi(x,t^{n})-\phi(X_{1}(u^{n},\Delta t)(x),t^{n-1})}{\Delta I}$$= \frac{\phi^{n}-\psi^{n-1}oX_{1}(u^{n},\Delta t)}{\Delta t}(x)$where we have used notations,$\psi^{n}\equiv\phi(\cdot,t^{n})$, for$w:\Omegaarrow \mathbb{R}^{d}$$\psi:\Omegaarrow \mathbb{R}$(4),$X_{1}(w,\Delta t)(x)\equiv x-w(x)\Delta t$and, for

,

,

(5)

,
$\psi oX_{1}(w,$ $\Delta l)(x)\equiv\psi(X_{1}(w,\Delta t)(x))$*Email: h.notsu@aoni.waseda.jp, Web; http: //scheme.hn/

.

(6)

52
The approximation (4) of $D\phi/Dt$ is a basic idea to devise numerical schemes based on the method ofcharacteristics. The idea can be combined with finite element and difference methods (e.g., see [5, 10, 12,
13, 15] . In this paper, we introduce a finite element scheme based on the method of characteristics fora lineanized Navier-Stokes equations, and give stability and convergence results of the scheme. Although
the system of the equations is linear and simpler than one of the Navier-Stokes equations, the results to beshown are useful for an analysis of a scheme for the Navier-Stokes equations [9, 11].
$)$2 Statement of the problem and a characteristics finite element scheme
We consider a linearized Navier-Stokes problem; find$(u,p):\Omega\cross(0, T)arrow \mathbb{R}^{d}\cross \mathbb{R}$such that(7)$\{\begin{array}{ll}\frac{Du}{Dt_{w}}-\nabla(2vD(u))+\nabla p=f, in \Omega\cross(0,T),\nabla\cdot u=0, in \Omega\cross(0,T),u=0, on \Gamma\cross(0,T),u=u^{0}, in \Omega, at t=0,\end{array}$is a given extemal force,
where is the velocity, is the pressure,given initial velocity, $v(>0)$ is a viscosity, $D(u)$ is a strain-rate tensor defined by$u$$f:\Omega\cross(0, T)arrow \mathbb{R}^{d}$$p$$(i,j=1, \cdots,d)$ ,$D_{ij}(u) \equiv\frac{1}{2}(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial_{X_{i}}})$$D/Dt_{w}$$u^{0}:\Omegaarrow \mathbb{R}^{d}$(8)

is a material derivation defined by(9)$\frac{D}{Dt_{w}}\equiv\frac{\partial}{\partial t}+w\cdot\nabla$

is a given velocity. If
to represem
We use the same notation
functions, and define a bilinear forms on
and

is a

$w;\Omega\cross(0, T)arrow \mathbb{R}^{d}$

$($

.,

$\cdot$$)$$w$

is replaced by (7) is the Navier-Stokes problem.
inner product for scalar-, vector- and matrix-valuedbyand on$u,$$L^{2}(\Omega)$$H^{1}(\Omega)^{d}\cross H^{1}(\Omega)^{d}$$a$$a(u,v)\equiv 2v(D(u), D(v))$ ,

and

$b$

$H^{1}(\Omega)^{d}\cross L^{2}(\Omega)$$b(v,q)\equiv-(\nabla\cdot v, q)$

,

(10)

respectively. The weak form of the problem (7) is written as follows; find $\{(u,p)(t)\}_{t\in(0,T)}\subset V\cross Q$ suchthat$\{\begin{array}{l}(\frac{Du}{Dt_{w}},v)+a(u,v)+b(v,p)+b(u,q)=(f,v), \forall(v,q)\in V\cross Q,u(0)=u^{0},\end{array}$where$V\equiv H_{0}^{1}(\Omega)^{d}$and$Q\equiv L_{0}^{2}(\Omega)$.

We state a characteristics finite element scheme for (7). Let$\Omega_{h}\equiv$

We define function spaces$X_{h},$ $M_{h},$$V_{h}$

int

and$( \bigcup_{K\in.7_{h}}K)$$Q_{h}$

and$\ovalbox{\tt\small REJECT}_{h}=\{K\}$$\Gamma_{h}\equiv\partial\Omega_{h}$(11)be a triangulation,

.

(12)

by

$X_{h}\equiv\{v_{h}\in C^{0}(\Omega_{h}^{-})^{d};v_{h}|_{K}\in P_{1}(K)^{d}, \forall K\in\ovalbox{\tt\small REJECT}_{h}\}$

,

(13)

53
$M_{h}\equiv\{q_{h}\in C^{0}(\Omega_{h}^{-});q_{h}|_{K}\in P_{1}(K),$ $\forall K\in F_{h}\}$and$V_{h}\equiv X_{h}\cap H_{0}^{1}(\Omega)$$Q_{h}\equiv M_{h}\cap L_{0}^{2}(\Omega_{h})$

,

(14)

,

(15)

respectively, where $P_{1}(K)$ is a function space of piecewise linear functions on an elemem $K\in F_{h},$a nodal poim on , andis a subspace ofdefined by$L_{0}^{2}(\Omega)\equiv\{q\in L^{2}(\Omega);(q, 1)=0\}$

be a total number of time steps,
be an inner product in. For anda bilinear form on
byLet$N_{T}\equiv[T/\Delta t]$$L^{2}(K)^{d}$$\mathscr{C}_{h}$

means

$P$

$L^{2}(\Omega)$$L_{0}^{2}(\Omega)$$\Gamma_{h}$$\delta$.

(16)

be a positive constant,be a diameter of, we define a linear form
$h_{K}$$w\in H^{l}(\Omega)^{d}$$u,\tilde{u}$$K$$\mathscr{M}_{h}(u,\tilde{u};At, w)$

and

on

$(\cdot,\cdot)_{K}$$V_{h}$

and$H^{1}(\Omega)\cross H^{1}(\Omega)$$\langle \mathscr{M}_{h}(u,\tilde{u};\Delta I,w),$$v_{h} \}\equiv(\frac{u-\tilde{u}oX_{1}(w,\Delta t)}{\Delta t},$respectively. Let an approximate function
element scheme for (7) is to find$u_{h}^{0}\in V_{h}$$v_{h})$

of$u^{0}$

and$\mathscr{C}_{h}(p,q)\equiv-\delta\sum_{K\in\sqrt{h}}h_{K}^{2}(\nabla p, \nabla q)_{K}$(17)

be given. A pressure-stabilized characteristics finite,
such that, for $n=1,$$\{(u_{h}^{n}, p_{h}^{n})\}_{n=1}^{N_{\mathcal{T}}}\subset V_{h}\cross Q_{h}$$\langle \mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};\Delta t,w^{n-1}),$,$\cdots,N_{T}$$v_{h}\}+a(u_{h}^{n},v_{h})+b(v_{h},p_{h}^{n})+b(u_{h}^{n},q_{h})+\mathscr{C}_{h}(p_{h}^{n},q_{h})$$=(f^{n}, v_{h})$,$\forall(v_{h},q_{h})\in V_{h}\cross Q_{h}$

,

(18)

The scheme (18) can deal with high Reynolds number (small viscosity) problems by the method ofcharacteristics. The material derivative term $(Du/Dt_{w}, v)$ in (11) is approximated by.
When we findin the scheme (18), a composite function
is a known function
and a coefficient matrix of the system of the linear equations is symmetric. The advantage enables us to use$\{\mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};\Delta t, w^{n-1}), v_{h}\}$$u_{h}^{n-1}oX_{1}(w_{h}^{n-1},\Delta t)$$(u_{h}^{n}, p_{h}^{n})$symmetric linear iterative solvers, i.e., CG, CR, MINRES [2]. Since the coefficient matnix is independentof step number , it is enough to make the matrix at only the first time step. The scheme employs a cheap
element Pl $/P1$ , it is useful for large scale computation, especially in $3D$ . Although Pl $lP1$ element does notsatisfy the inf-sup condition [8], the scheme works by a pressure-stabilization tem .
We impose assumption for a given velocity and review a proposition in [15].
$n$$\mathscr{C}_{h}$$w$Hypothesis 2.1. Afunction$w$

satisfies$\{\begin{array}{l}w\in C^{0}([0, T];W^{1,\infty}(\Omega)),w=0 on \Gamma,divw=0 in \Omega.\end{array}$(19)

Proposition 2.2 ([15]). Under Hypothesis 2.1 and an inequality;$\Delta I<\frac{1}{\Vert w\Vert_{C^{0}(W^{1.\infty})}}$

,

(20)

it holds that, for any $t\in[0, T]$ ,$X_{1}(w(\cdot,t),\Delta t)(\Omega)=\Omega$

.

(21)

In the following sections, we assume that Hypothesis 2.1 and the inequality (20) hold, and that, for thesake of simplicity, is convex polygonal domain.
$\Omega$$(\Omega=\Omega_{h})$54
3 Stability and convergence
In this section, we consider stability and convergence of the scheme (lS). We use to represent the generic
positive constant independent of discretization parameters and solutions, which can take different values atand have particulardifferent places, and $c(A)$ is a positive constant, which depends on . Constants
meanuings in this paper,
$c$

$A$

$c0=co(\Vert w\Vert_{\sigma(L^{\infty})}))$,$c_{1}=c_{1}(\Vert w\Vert_{\sigma(W^{1.\infty})}))$

and$c_{0},$$c_{1}$$c_{2}$$c_{2}=c_{2}(\Vert w\Vert_{C^{\langle)}(W^{1,\infty})\cap C^{1}(L^{\infty})})$

,

(22)

respectively.

3.1 Stability
This subsection is devoted to the stability of the scheme (18). We use norms and seminorms,$\Vert\cdot\Vert_{H^{k}(\Omega)}(k=0,1,2),$$\Vert\cdot\Vert_{V}\equiv\Vert\cdot\Vert_{1},$,$\Vert(v,q)\Vert_{v\cross Q}\equiv\{\Vert v\Vert_{V}^{2}+\Vert q\Vert_{Q}^{2}\}^{1/2}$$\Vert u\Vert_{l_{(\prime}^{\infty}(L^{2})}\equiv\max_{n\hslash=m}.\Vert u^{n}\Vert_{0}$,,Setting a bilinear form

$d_{h}$

on$|q|_{h} \equiv\{\sum_{K\in\sqrt{h}}h_{K}^{2}(\nabla q,\nabla q)_{K}\}^{I/2}$$\Vert_{k}\equiv$,,$\Vert u\Vert_{l^{\infty}(X)}\equiv\Vert u\Vert_{l_{(0)}^{\infty}(X)}(X=L^{2},H^{1})$$\Vert u\Vert_{l_{(,n)}^{2}(H^{1})}\equiv\{\Delta t\sum_{n=m}^{N_{7}}\Vert u^{n}\Vert_{1}^{2}\}^{1/2}$

and

I

,

$\Vert u\Vert_{l_{(\prime}^{\infty}(H^{1})}\equiv\max_{n\hslash=m}\Vert u^{n}\Vert_{1}$$\Vert u\Vert_{l_{(,n)}^{2}(L^{2})}\equiv\{\Delta t\sum_{n=m}^{N,}\Vert u^{n}\Vert_{0}^{2}\}^{1/2}$$|p|_{l_{(,n)}^{2}(M_{h})} \equiv\{\Delta t\sum_{n=m}^{N_{7}}|p^{n}|_{h}^{2}\}^{1/2}$$\Vert\cdot\Vert_{Q}\equiv\Vert\cdot\Vert_{0}$$|p|_{l^{2}(M_{h})}\equiv|p|_{l_{(I)}^{2}(M_{h})}$$(V_{h}\cross Q_{h})\cross(V_{h}\cross Q_{h})$,,$\Vert u\Vert_{l^{2}(X)}\equiv\Vert u\Vert_{l_{(1)}^{2}(X)}(X=L^{2},H^{1})$

.

(23)

by

$d_{h}((u,p);(v,q))\equiv a(u,v)+b(v,p)+b(u,q)+\mathscr{C}_{h}(p,q)$

,

(24)

we have another representation of the scheme (18);,$\langle \mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};\Delta t, w^{n-1}),v_{h}\rangle+_{h}((u_{h}^{n},p_{h}^{n});(v_{h},q_{h}))=(f^{n},v_{h})$$\forall(v_{h},q_{h})\in V_{h}\cross Q_{h}$

.

(25)

We prepare two lemmas for the stability. First one is an estimate of a composite function appearing

Lemma 3.1. For any$v\in L^{2}(\Omega)^{d}$

.

(26)

We omit the proof, because it is similar to the proof of Lemma 1 in [15].
Pl $lP1$ element is employed in the scheme (18), i.e., both finite dimensional function spaces
consist of Pl element, and Pl $/P1$ element does not satisfy the inf-sup condition [8];$\inf_{q_{h}\in Q_{h}}\sup_{v_{h}\in V_{h}}\frac{b(v_{h},q_{h})}{\Vert v_{h}\Vert_{V}||q_{h}\Vert_{Q}}\geq\beta^{*}>0$

$b$

and

$\mathscr{C}_{h}$;and $t\in[0,T]$ , it holds that$\Vert voX_{1}(w(\cdot,t),\Delta t)\Vert_{0}\leq(1+c_{1}\Delta t)\Vert v\Vert_{0}$However, $a,$$\mathscr{M}_{h}$.

$V_{h}$

and$Q_{h}$(27)

satisfies the following inf-sup condition.

Lemma 3.2 ([7]). There is a positive constant such that$\gamma$$inf\sup$$(u_{h},p_{h}) \in V_{h}\cross Q_{h(v_{h},q_{h})\in V_{h}\cross Q_{h}}\frac{d_{h}((u_{h},p_{h});(v_{h},q_{h}))}{\Vert(u_{h},p_{h})\Vert_{V\cross Q}\Vert(v_{h},q_{h})\Vert_{V\cross Q}}\geq\gamma$

.

(28)

,

55
From the above lemmas, we obtain a stability result.
Theorem 3.3 (stability). Let$\Delta t_{0}(<1)$(i) It holds thatbe a fixed number. Assume$\Delta t\in(0,\Delta t_{0}]$.

$\Vert u_{h}\Vert_{l^{\infty}(L^{2})}+\sqrt{v}\Vert u_{h}\Vert_{l^{2}(H^{1})}+\sqrt{\delta}|p_{h}|_{l^{2}(M_{h})}\leq c_{1}(\Vert u_{h}^{0}\Vert_{0}+\Vert f\Vert_{l^{2}(L^{2})})$

.

(29)

(ii) The following two inequalities hold;$\sqrt{v}\sqrt{\delta}\sqrt{\delta\Delta\prime}|\overline{D}_{N}p_{h}|_{l_{(2)}^{2}(M_{h})}$

,

(30)

.

(31)

$\leq cl(\Vert u_{h}^{1}\Vert_{1}+|p_{h}^{1}|_{h}+\Vert f\Vert_{l_{(2)}^{2}(L^{2})})$$\Vert p_{h}\Vert_{l_{(2)}^{2}(L^{2})}\leq c_{1}$Proof. First we prove (i). Substituting$(Iu_{h}^{1}\Vert_{1}+|p_{h}^{1}|_{h}+\Vert f\Vert_{l_{(2)}^{2}(L^{2})})$$(u_{h}^{n}, -p_{h}^{n})\in V_{h}\cross Q_{h}$into$(v_{h},q_{h})$in the scheme (18), we have$\langle \mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};\Delta I,w^{n-1}),u_{h}^{n}\rangle+a(u_{h}^{n},u_{h}^{n})+\mathscr{C}_{h}(p_{h}^{n}, -p_{h}^{n})=(f^{n},u_{h}^{n})$

.

(32)

We evaluate the four terms;$\{\mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};\Delta t, w^{n-1}),u_{h}^{n}\rangle=\frac{1}{\Delta t}\{\frac{1}{2}(\Vert u_{h}^{n}\Vert_{0}^{2}-\Vert u_{h}^{n-1}oX_{1}(w^{n-1},\Delta t)\Vert_{0}^{2})+\frac{1}{2}\Vert u_{h}^{n}-u_{h}^{n-1}oX_{1}(w^{n-1},\Delta t)\Vert_{0}^{2}\}$$(\cdot.\cdot(a-b)a=(a^{2}-b^{2})/2+(a-b)^{2}/2)$$\geq\frac{1}{\Delta t}(\frac{1}{2}\Vert u_{h}^{n}\Vert_{0}^{2}-\frac{1}{2}\Vert u_{h}^{n-1}\Vert_{0}^{2})-c_{1}\Vert u_{h}^{n-1}\Vert_{0}^{2}$( Lemma 3. 1)$\cdot.\cdot$$= \overline{D}_{N}(\frac{1}{2}\Vert u_{h}^{n}\Vert_{0}^{2})-c_{1}\Vert u_{h}^{n-1}\Vert_{0}^{2}$,($a(u_{h}^{n},u_{h}^{n})=2v\Vert D(u_{h}^{n})\Vert_{0}^{2}\geq cv\Vert u_{h}^{n}\Vert_{1}^{2}$$\mathscr{C}_{h}(p_{h}^{n}, -p_{h}^{n})=\delta|p_{h}^{n}|_{h}^{2}$(33a)$\cdot.\cdot$

Kom’s inequality [6]),

(33b)

,

(33c)

and
$(\cdot.\cdot ab\leq(a^{2}+b^{2})/2))$$(f^{n},u_{h}^{n}) \leq\frac{1}{2}(\Vert f^{n}\Vert_{0}^{2}+\Vert u_{h}^{n}\Vert_{0}^{2})$where$\overline{D}_{N}a^{n}\equiv(a^{n}-a^{n-1})/\Delta t$

,

(33d)

. The inequalities (33) implies$\overline{D}_{\Delta l}(\frac{1}{2}\Vert u_{h}^{n}\Vert_{0}^{2})+cv\Vert u_{h}^{n}\Vert_{1}^{2}+\delta|p_{h}^{n}|_{h}^{2}\leq\frac{1}{2}(\Vert f^{n}\Vert_{0}^{2}+\Vert u_{h}^{n}\Vert_{0}^{2})$and (29) holds from the discrete Gronwall lemma [20].
Next, we prove (30) of (ii). Substituting
into$0\in V_{h}$$b(u_{h}^{n},q_{h})+\mathscr{C}_{h}(p_{h}^{n},q_{h})=0$$v_{h}$$(n=1, \cdots,N_{T})$

,

(34)

in the scheme (18), we have$(\forall q_{h}\in Q_{h}, n=1, \cdots,N_{T})$

,

(35)

and, then,$b(\overline{D}_{N}u_{h}^{n},q_{h})+\mathscr{C}_{h}(\overline{D}_{[be]}p_{h}^{n},q_{h})=0$

Substituting$(\overline{D}_{\Delta},u_{h}^{n},0)\in V_{h}\cross Q_{h}$into$(v_{h},q_{h})$$(\forall q_{h}\in Q_{h}, n=2, \cdots,N_{T})$

.

(36)

in the scheme (18), we have$\langle \mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};\Delta t,w^{n-1}),\overline{D}_{N}u_{h}^{n}\}+a(u_{h}^{n},\overline{D}_{N}u_{h}^{n})+b(\overline{D}_{N}u_{h}^{n},p_{h}^{n})=(f^{n},\overline{D}_{N}u_{h}^{n})$$(n=1, \cdots,N_{T})$

,

(37)

56
and, by (36),$\langle \mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};\Delta t,\sqrt{}^{\iota-1}),\overline{D}_{N}u_{h}^{n}\rangle+a(u_{h}^{n},\overline{D}_{N}u_{h}^{n})-\mathscr{C}_{h}(\overline{D}_{N}p_{h}^{n},p_{h}^{n})=(f^{n},\overline{D}_{N}u_{h}^{n})$$(n=2, \cdots,N_{T})$

.

(38)

We evaluate the four terms in (38);$\{\mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};At, w^{n-1}),\overline{D}_{N}u_{h}^{n}\}=(\overline{D}_{N}u_{h}^{n}+\frac{1}{\Delta t}(u_{h}^{n-1}-u_{h}^{n-1}\circ X_{1}(w^{\rho-1},\Delta t)),\overline{D}_{N}u_{h}^{n})$$= \Vert\overline{D}_{N}u_{h}^{n}\Vert_{0}^{2}+\frac{1}{\Delta\prime}(u_{h}^{n-1}-u_{h}^{n-1}\circ X_{1}(w^{n-1},\Delta t),\overline{D}_{N}u_{h}^{n})$$\geq\Vert\overline{D}_{N}u_{h}^{n}\Vert_{0}^{2}-\frac{1}{\Delta t}\Vert u_{h}^{n-1}-u_{h}^{n-1}oX_{1}(w^{n-1},\Delta t)\Vert_{0}\Vert\overline{D}_{N}u_{h}^{n}||0$$\geq\Vert\overline{D}_{N}u_{h}^{n}\Vert_{0}^{2}-c_{0}|u_{h}^{n-1}|_{1}\Vert\overline{D}_{N}u_{h}^{n}\Vert_{0}$$\geq\Vert\overline{D}_{N}u_{h}^{n}\Vert_{0}^{2}-(c_{0}|u_{h}^{n-1}|_{1}^{2}+\frac{1}{4}\Vert\overline{D}_{N}u_{h}^{n}\Vert_{0}^{2})$($\geq\frac{3}{4}\Vert\overline{D}_{N}u_{h}^{n}\Vert_{0}^{2}-c_{0}\Vert D(u_{h}^{n-1})\Vert_{0}^{2}$$\cdot.\cdot$

Kom の不等式),(39a)$a(u_{h}^{n}, \overline{D}_{N}u_{h}^{n})=\overline{D}_{N}(\frac{1}{2}a(u_{h}^{n},u_{h}^{n}))+\frac{\Delta t}{2}a(\overline{D}_{N}u_{h}^{n},\overline{D}_{N}u_{h}^{n})$

$=$

万

$N$

$(v\Vert D(u_{h}^{n})\Vert_{0}^{2})+v\Delta t\Vert D(\overline{D}_{N}u_{h}^{n})\Vert_{0}^{2}$

,

(39b)

$- \mathscr{C}_{h}(\overline{D}_{N}p_{h}^{n},p_{h}^{n})=\overline{D}_{\Delta t}(-\frac{1}{2}\mathscr{C}_{h}(p_{h}^{n},p_{h}^{n}))-\frac{\Delta t}{2}\mathscr{C}_{h}(\overline{D}_{N}p_{h}^{n},\overline{D}_{\Delta l}p_{h}^{n})$$= \overline{D}_{\Delta I}(\frac{\delta}{2}|p_{h}^{n}|_{h}^{2})+\frac{\delta\Delta t}{2}|\overline{D}_{N}p_{h}^{n}|_{h}^{2}$

,

(39c)

and
$(f^{n}, \overline{D}_{N}u_{h}^{n})=\Vert f^{n}\Vert_{0}^{2}+\frac{1}{4}\Vert\overline{D}_{\Delta!}u_{h}^{n}\Vert_{0}^{2}$

.

(39d)

Combining (39) with (38), we have$\overline{D}_{N}(v\Vert D(u_{h}^{n})\Vert_{0}^{2}+\frac{\delta}{2}|p_{h}^{n}|_{h}^{2})+\frac{1}{2}\Vert\overline{D}_{N}u_{h}^{n}\Vert_{0}^{2}+v\Delta t\Vert D(\overline{D}_{\Delta!}u_{h}^{n})\Vert_{0}^{2}+\frac{\delta\Delta t}{2}|\overline{D}_{N}p_{h}^{n}|_{h}^{2}$$\leq\Vert f^{n}\Vert_{0}^{2}+c0\Vert D(u_{h}^{n-1})\Vert_{0}^{2}$

which implies (30) by the discrete Gronwall lemma.
Finally we prove (31) of (ii). From Lemma 3.2,$\Vert p_{h}^{n}\Vert_{0}$$(n=2, \cdots,N_{T})$

,

(40)

is evaluated as follows;$\Vert p_{h}^{n}\Vert_{0}\leq\Vert(u_{h}^{n},p_{h}^{n})\Vert_{V\cross Q}$$\leq\frac{1}{\gamma}\sup_{(v_{h},q_{h})\in V_{h}\cross Q_{h}}\frac{d_{h}((u_{h}^{n},p_{h}^{n});(v_{h},q_{h}))}{||(v_{h},q_{h})\Vert_{V\cross Q}}$$\leq\frac{1}{\gamma}\sup_{(v_{h},q_{h})\in V_{h}\cross Q_{h}}\frac{(f^{n},v_{h})-\langle \mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};\Delta l,w^{n-1}),v_{h}\}}{\Vert(v_{h},q_{h})\Vert_{V\cross Q}}$$\leq\frac{1}{\gamma}(\Vert f^{n}\Vert_{V_{h}’}+\Vert \mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};\Delta t,d^{-1})\Vert_{V_{h}’})$$\leq\frac{1}{\gamma}(\Vert f^{n}\Vert_{0}+\Vert\overline{D}_{N}u_{h}^{n}\Vert_{0}+co|u_{h}^{n-1}|_{1})$

,

(41)

57
where we have used the following inequality in the last inequality of (41);$\Vert \mathscr{M}_{h}(u_{h^{u_{h};\Delta I}}^{nn-\iota-1}\parallel)\Vert_{V_{h}’}\leq\frac{1}{\Delta t}\Vert u_{h}^{n}-u_{h}^{n-1}\circ X_{1}(w^{n-1},\Delta t)\Vert_{0}$$= \Vert\overline{D}_{N}u_{h}^{n}+\frac{1}{\Delta t}(u_{h}^{n-1}-u_{h}^{n-1}oX_{1}(w^{n-1},\Delta t))\Vert_{0}$$\leq\Vert\overline{D}_{\Delta t}u_{h}^{n}\Vert_{0}+\frac{1}{\Delta t}\Vert u_{h}^{n-1}-u_{h}^{n-1}oX_{1}(w^{n-1},\Delta t))\Vert_{0}$$\leq\Vert\overline{D}_{N}u_{h}^{n}\Vert_{0}+c_{0}|u_{h}^{n-1}|_{1}$

.

(42)

Combining (41) with (30), we have$\Vert p_{h}\Vert_{l_{(2)}^{2}(L^{2})}\leq c_{0}(\Vert f\Vert_{l_{(2)}^{2}(L^{2})}+\Vert\overline{D}_{\Delta!}u_{h}^{n}\Vert_{l_{(2)}^{2}(L^{2})}+\Vert u_{h}\Vert_{l_{(1)}^{2}(H^{1})})$

.

(43)

$\leq c_{1}(\Vert u_{h}^{1}\Vert_{1}+|p_{h}^{1}|_{h}+\Vert f\Vert_{l_{(2)}^{2}(L^{2})})$口

Remark 3.4. In the right hand side of (31),evaluation is required.

1 IIl can be estimated by (30), and, for$u_{h}^{1}$$|p_{h}^{1}|_{h}$, a detailed

3.2 ConvergenceThis subsection is devoted to convergence of the scheme (18). At first we define a Stokes projection.

Definition 3.5 (Stokes projection). For $(u,p)\in(V\cap H^{2}(\Omega)^{d})\cross(Q\cap H^{1}(\Omega)),$$(w_{h},r_{h})\in V_{h}\cross Q_{h}$is a Stokes

,

(44)

projection of $(u,p)pro\mathcal{V}ued$$a(w_{h}, v_{h})+b(v_{h}, r_{h})+b(w_{h},q_{h})+^{\zeta}\mathscr{E}_{h}(r_{h},q_{h})=\langle \mathscr{G}_{h},$ $(v_{h},q_{h})\}$

where$\ovalbox{\tt\small REJECT}_{h}$is a linearfom on$V_{h}\cross Q_{h}$,$\forall(v_{h},q_{h})\in V_{h}\cross Q_{h}$defined by$\{\ovalbox{\tt\small REJECT}_{h}, (v_{h},q_{h})\}\equiv(g,v_{h})-\delta\sum_{K\in J_{h}}h_{K}^{2}(g, \nabla q_{h})_{K}$with$g\equiv-2v\nabla D(u)+\nabla p$

.

(45)

For the Stokes projection, the following error estimate holds.
Proposition 3.6. Suppose $(u,p)\in(V\cap H^{2}(\Omega)^{d})\cross(Q\cap H^{2}(\Omega))$, and$b(u,q)=0$

Let$(w_{h},r_{h})$$(q\in Q)$$u$satisfies.

(46)

be a Stokes projection of $(u,p)$ by (44). Then, there exists a positive constantof , such that$c_{s}$, independent$h$$\Vert u-w_{h}\Vert_{1}+\Vert p-r_{h}\Vert 0\leq c_{s}h(\Vert u\Vert_{2}+h\Vert p\Vert_{2})$

.

(47)

We omit the proof, because papers [3] and [7] give the result.
Theorem 3.7. Let, and $(u,p)$ andbe a fixed positive number,be solutionsof (11) and (18), respectively, where is a first component of the Stokes projection of $(u^{0},0)$ . Suppose$u\in C^{0}([0, T];V\cap H^{2})\cap H^{2}(0, T;L^{2})\cap H^{1}(0, T;H^{1})$and $p\in C^{0}([0, T];Q\cap H^{1})\cap H^{1}(0,T;H^{2})$. Then,holds that$\Delta t_{0}(<1)$$\Delta 1\in(0,\Delta t_{0}]$$(u_{h},p_{h})$$u_{h}^{0}$$it$$\Vert u-uh\Vert_{l^{\infty}(L^{2})}+\sqrt{v}\Vert u-u_{h}\Vert_{l^{2}(H^{1})}+\sqrt{\delta}|p-p_{h}|_{l^{2}(M_{h})}$58$\leq C2\{\Delta t(\Vert u\Vert_{H^{2}(0,T:L^{2})}+\Vert u\Vert_{H^{1}(0,T;H^{1})}+\Vert u\Vert_{L^{2}(0,\tau:H^{2})})$$+h(\Vert u^{0}\Vert_{2}+\sqrt{\delta}(v\Vert u\Vert_{l^{2}(H^{2})}+\Vert p\Vert_{l^{2}(H^{1})})+\Vert(u,p)\Vert_{H^{1}(0,T:H^{2}\cross H^{2})})\}$

Proof. Let$(\hat{u}h,\hat{p}_{h})(t)$(48)

be the Stokes projection of $(u,p)(t)\in H^{2}(\Omega)^{d}\cross H^{2}(\Omega)$ ,$e_{h}^{n}\equiv u_{h}^{n}-\hat{u}_{h}^{n}$

For any.

$(v_{h},q_{h})\in V_{h}\cross Q_{h}$,$\epsilon_{h}^{n}\equiv p_{h}^{n}-\hat{p}_{h}^{n}$and$\eta_{h}(t)\equiv(u-\hat{u}_{h})(t)$

.

(49)

, it holds that, from (11) and (18),$\langle \mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};\Delta t,w^{n-1}),v_{h}\rangle+a(e_{h}^{n},v_{h})+b(v_{h},\epsilon_{h}^{n})+b(e_{h}^{n},q_{h})+\mathscr{C}_{h}(\epsilon_{h}^{n},q_{h})$$=(f^{n},v_{h})-(-2v \nabla D(u^{n})+\nabla p^{n},v_{h})+\delta\sum_{K}h_{K}^{2}(-2v\nabla D(u^{n})+\nabla p^{n},\nabla q_{h})_{K}$,$=( \frac{Du^{n}}{Dt_{w}},v_{h})+\delta\sum_{K}h_{K}^{2}(-2v\nabla D(u^{n})+\nabla p^{n},\nabla q_{h})_{K}$(50)

and from an identity,$\mathscr{M}_{h}(u_{h}^{n},u_{h}^{n-1};\Delta t,w^{n-1})=\mathscr{M}_{h}(e_{h}^{n},e_{h}^{n-1};\Delta t,w^{n-1})-\mathscr{M}_{h}(\eta_{h}^{n},\eta_{h}^{n-1};\Delta t,w^{n-1})+\mathscr{M}_{h}(u^{n},u^{n-1};\Delta t, w^{n-1})$(51)

we have$\langle \mathscr{M}_{h}(e_{h}^{n},e_{h}^{n-1};\Delta t,w^{n-1}),$ $v_{h}\rangle+a(e_{h}^{n}, v_{h})+b(v_{h},\epsilon_{h}^{n})+b(l_{h},q_{h})+\mathscr{C}_{h}(\epsilon_{h}^{n},q_{h})$$=\langle \mathscr{M}_{h}(\eta_{h}^{n}, \eta_{h}^{n-1};\Delta t,w^{n-1})-\mathscr{M}_{h}(u^{n},u^{n-1};\Delta t, w^{n-1}),$$v_{h} \rangle+(\frac{Du^{n}}{Dt_{w}},v_{h})+\delta\sum_{K}h_{K}^{2}(-2v\nabla D(u^{n})+\nabla p^{n},\nabla q_{h})_{K}$$=( \frac{\eta_{h}^{n}-\eta_{h}^{n-1}oX_{1}(w^{n-1},\Delta t)}{\Delta t}-\frac{u^{n}-u^{n-1}oX_{1}(w^{n-1},\Delta t)}{\Delta t}+\frac{Du^{n}}{Dt_{w}},$$v_{h})+ \delta\sum_{K}h_{K}^{2}(-2v\nabla D(u^{n})+\nabla p^{n},\nabla q_{h})_{K}$(52)

Substituting$(l_{h}, -\epsilon_{h}^{n})$into$(v_{h},q_{h})$in (52), we have$\{\mathscr{M}_{h}(e_{h}^{n},e_{h}^{n-1};\Delta t,w^{n-1}),e_{h}^{n}\rangle+a(e_{h}^{n},e_{h}^{n})-\mathscr{C}_{h}(\epsilon_{h}^{n},\epsilon_{h}^{n})$$=( \frac{\eta_{h}^{n}-\eta_{h}^{n-1}oX_{1}(w^{n-1},\Delta t)}{\Delta t}-\frac{u^{n}-u^{n-1}oX_{1}(w^{n-1},\Delta\prime)}{\Delta t}+\frac{Du^{n}}{Dt_{w}},e_{h}^{n})-\delta\sum_{K}h_{K}^{2}(-2v\nabla D(u^{n})+\nabla p^{n},\nabla\epsilon_{h}^{n})_{K}$$\equiv I_{1}^{n}+I_{2}^{n}$.

The two terms(53)$I_{1}$

and$I_{2}$

are evaluated as follows. We have$\Vert\frac{\eta_{h}^{n}-\eta_{h}^{n-1}\circ X_{1}(\nu\nearrow^{-1},\Delta t)}{\Delta t}\Vert_{0}=\Vert\int_{0}^{1}(\frac{\partial\eta_{h}}{\partial t}+(w^{n-1}(x)\cdot\nabla)\eta_{h})(x-sw^{n-1}(x)\Delta t,t^{n}-s\Delta t)ds\Vert_{0}$$=[ \int_{\Omega}\{\int_{0}^{1}(\frac{\partial\eta_{h}}{\partial t}+(w^{n-1}(x)\cdot\nabla)\eta_{h})(x-sw^{n-1}(x)[be],t^{n}-s\Delta t)ds\}^{2}dx]^{1/2}$$\leq\{\int_{0}^{1}ds\int_{\Omega}(\frac{\partial\eta_{h}}{\partial\iota}+(w^{n-1}(x)\cdot\nabla)\eta_{h})(x-sw^{n-1}(x)\Delta t,t^{n}-s\Delta t)^{2}dx\}^{1/2}$$\leq C0[\{\int_{0}^{1}ds\int_{\Omega}\frac{\partial\eta_{h}}{\partial t}(x-sw^{n-1}(x)\Delta t,t^{n}-s\Delta t)^{2}dx\}^{1/2}$.

59$+ \{\int_{0}^{1}ds\int_{\Omega}\nabla\eta_{h}(x-sw^{n-1}(x)\Delta t,t^{n}-s\Delta t)^{2}dx\}^{1/2}]$$\leq c_{1}[\{\int_{0}^{1}\Vert\frac{\partial\eta_{h}}{\partial t}(\cdot,t^{n}-s\Delta t)\Vert_{0}^{2}ds\}^{1/2}+\{\int_{0}^{1}\Vert\nabla\eta_{h}(\cdot,t^{n}-s\Delta t)\Vert_{0}^{2}ds\}^{1/2}]$$\leq\frac{c_{1}}{\sqrt{\Delta t}}[\{\int_{t^{n-1}}^{t^{n}}\Vert\frac{\partial\eta_{h}}{\partial t}(\cdot,t)\Vert_{0}^{2}dt\}^{1/2}+\{\int_{t^{n-1}}^{t^{n}}\Vert\nabla\eta_{h}(\cdot,t)\Vert_{0}^{2}dt\}^{1/2}]$$= \frac{c_{1}}{\sqrt{[be]}}\{\Vert\frac{\partial\eta_{h}}{\partial t}\Vert_{L^{2}(t^{\prime\iota-1},t^{n};L^{2})}+\Vert\nabla\eta_{h}\Vert_{L^{2}(t^{n-1},t^{n};L^{2})}\}$$\leq\frac{c_{1}h}{\sqrt{\Delta t}}\{\Vert(\frac{\partial u}{\partial t},$$\frac{\partial p}{\partial t})\Vert_{L^{2}(t^{n-1},t^{n};H^{2}\cross H^{2})}+\Vert(u,p)\Vert_{L^{2}(t^{n-1},t^{n};H^{2}\cross H^{2})}\}$$= \frac{c_{1}h}{\sqrt{\Delta t}}\Vert(u,p)\Vert_{H^{1}(t^{n-1},t^{n};H^{2}\cross H^{2})}$

,

(54)

and
$\Vert(\frac{Du}{Dt_{w}})^{n}-\frac{u^{n}-u^{n-1}\circ X_{1}(w^{n-1},\Delta t)}{\Delta t}\Vert_{0}\leq c_{2}\sqrt{\Delta t}\{\Vert u\Vert_{H^{2}(t^{n-1},t^{n};\text{乙^{}2})}+\Vert u\Vert_{H^{1}(t^{n-1},t^{n}:H^{1})}+\Vert u\Vert$

乙 2 $(t^{n-1},t^{n};H^{2})\}$

.

(55)

For any $\infty>0$ , it holds that, from the inequalities (54) and (55),$17 \leq\frac{c_{2}}{\infty}\{\Delta t(\Vert u\Vert_{H^{2}(t^{n-1},t^{n};L^{2})}^{2}+\Vert u\Vert_{H^{1}(t^{n-1},t^{n},H^{1})}^{2}+\Vert u\Vert_{\text{乙^{}2}(t^{n-1},t^{n};H^{2})}^{2})+\frac{h^{2}}{\Delta t}\Vert(u,p)\Vert_{H^{1}(^{n-1},t^{n};H^{2}\cross H^{2})}^{2}\}+\infty\Vert e_{h}^{n}\Vert_{0}^{2}$

(56)

For

$I_{2}$, we have$I_{2}^{n} \leq\delta\sum_{K}h_{K}^{2}\Vert\nabla\epsilon_{h}^{n}\Vert_{\text{乙^{}2}(K)}(2v\Vert\nabla D(u^{n})\Vert_{L^{2}(K)}+\Vert\nabla p^{n}\Vert_{L^{2}(K)})$$\leq 2v\delta\sum_{K}h_{K}^{2}\Vert\nabla\epsilon_{h}^{n}\Vert_{\text{乙^{}2}(K)}\Vert u^{n}\Vert_{H^{2}(K)}+\delta\sum_{K}h_{K}^{2}\Vert\nabla\epsilon_{h}^{n}\Vert_{\text{乙^{}2}(K)}\Vert\nabla p^{n}\Vert_{L^{2}(K)}$$\leq v\delta\sum_{K}h_{K}^{2}(\alpha_{1}\Vert\nabla\epsilon_{h}^{n}\Vert_{L^{2}(K)}^{2}+\frac{c}{\alpha_{1}}\Vert u^{n}\Vert_{H^{2}(K)}^{2})+\delta\sum_{K}h_{K}^{2}(\alpha_{2}\Vert\nabla\epsilon_{h}^{n}\Vert_{L^{2}(K)}^{2}+\frac{c}{\alpha_{2}}\Vert\nabla p^{n}\Vert_{\text{乙^{}2}(K)}^{2})$$\leq(\alpha_{1}v+\alpha_{2})\delta|\epsilon_{h}^{n}|_{h}^{2}+c\delta h^{2}(\frac{v}{\alpha_{1}}\Vert u^{n}\Vert_{2}^{2}+\frac{1}{\alpha_{2}}\Vert p^{n}\Vert_{1}^{2})$

.

(57)

Combining (56) and (57) with (53), we have, for any positive numbers$\delta,$$\alpha_{0},$$\alpha_{1}$and$\alpha_{2}$and $n=1$ ,–, $Nr$ ,$\overline{D}_{[be]}(\frac{1}{2}\Vert e_{h}^{n}\Vert_{0}^{2})+v\Vert D(e_{h}^{n})\Vert_{0}^{2}+\delta|\epsilon_{h}^{n}|_{h}^{2}$$\leq$

cto II$e_{h}^{n} \Vert_{0}^{2}+c_{1}\Vert e_{h}^{n-1}\Vert_{0}^{2}+(\alpha_{1}v+\alpha_{2})\delta|\epsilon_{h}^{n}|_{h}^{2}+c\delta h^{2}(\frac{v}{\alpha_{1}}$I$u^{n} \Vert_{2}^{2}+\frac{1}{\alpha_{2}}$I$p^{n}\Vert_{1}^{2})$$+ \frac{c_{2}}{\alpha_{0}}\{\Delta t(\Vert u\Vert_{H^{2}(r^{n-1},r^{n};\text{乙^{}2})}^{2}+\Vert u\Vert_{H^{1}(t^{n-1},t^{n},H^{1})}^{2}+\Vert u\Vert_{\text{乙^{}2}(t^{n-1},t^{n};H^{2})}^{2})+\frac{h^{2}}{\Delta t}\Vert(u,p)\Vert_{H^{1}(t^{n-1},t^{n};H^{2}\cross H^{2})}^{2}\},$

and, then, for$\alpha_{0}=1/(4\Delta t_{0}),$$\alpha_{1}=1/(4v)$$\overline{D}_{\Delta!}(\frac{1}{2}\Vert e_{h}^{n}\Vert_{0}^{2})+v\Vert D(e_{h}^{n})\Vert_{0}^{2}+\frac{\delta}{2}|\epsilon_{h}^{n}|_{h}^{2}$

and $\alpha_{2}=1/4$ ,

(58)

.

60
$\leq\frac{1}{4\Delta t_{0}}\Vert e_{h}^{n}\Vert_{0}^{2}+c_{1}\Vert e_{h}^{n-1}$

No$+c\delta h^{2}(v^{2}\Vert u^{n}\Vert_{2}^{2}+\Vert p^{n}\Vert_{1}^{2})$.

$+c_{2} \{\Delta t(\Vert u\Vert_{H^{2}(t^{n-1},t^{n};L^{2})}^{2}+\Vert u\Vert_{H^{1}(t^{n-1},t^{n};H^{1})}^{2}+\Vert u\Vert_{\text{乙^{}2}(t^{n-1},t^{n};H^{2})}^{2})+\frac{h^{2}}{\Delta t}\Vert(u,p)\Vert_{H^{1}(t^{n-1},t^{n};H^{2}\cross H^{2})}^{2}\}$(59)

By the discrete Gronwall inequality, it holds that$\Vert e_{h}\Vert_{l^{\infty}(L^{2})}+\sqrt{v}\Vert D(e_{h})\Vert_{l^{2}(L^{2})^{+}}\sqrt{\delta}|\epsilon_{h}|_{l^{2}(M_{h})}$$\leq c_{2}\{\Vert e_{h}^{0}\Vert_{0}+\Delta t(\Vert u\Vert_{H^{2}(0,T;\text{乙^{}2})}+\Vert u\Vert_{H^{1}(0,T,H^{1})}+\Vert u\Vert_{\text{乙^{}2}(0,T;H^{2})})$$+h(\sqrt{\delta}(v\Vert u\Vert_{l^{2}(H^{2})}+\Vert p\Vert_{l^{2}(H^{1})})+\Vert(u,p)\Vert_{H^{1}(0,T;H^{2}\cross H^{2})})\}$$\leq c_{2}$$\{$$(\Vert u\Vert_{H^{2}(0,T;L^{2})}+\Vert u\Vert_{H^{1}(0,T;H^{1})}+\Vert u\Vert_{\text{乙^{}2}(0,T;H^{2})})$

,

(60)

$+h(\Vert u^{0}\Vert_{2}+\sqrt{\delta}(v\Vert u\Vert_{l^{2}(H^{2})}+\Vert p\Vert_{l^{2}(H^{1})})+\Vert(u,p)\Vert_{H^{1}(0,T;H^{2}\cross H^{2})})\}$

whiCh implies (48).

口

4 Conclusions
We have introduced a characteristics finite element scheme for a linearized Navier-Stokes equations. The
scheme employs Pl $lP1$ element, and the coefficient matrix appearing in the scheme is symmetric. Theseadvantages reduces computational time and cost by half. Therefore, the scheme is useful especially forthree dimensional computation. We have shown stability and convergence results with the optimal -error
estimate for the velocity. Although the system of the equations is linear, the analysis is useful even in the
nonlinear case. We note that the convergence of the pressure can be proved under some assumptions.
$L^{2}$

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