NAKANE, KAZUAKI

OKAMURA, TOSHIHIRO

OMATA, SEIRO

Kyoto University Research Information Repository

TitleNUMERICAL ANALYSIS FOR THE DISCRETE MORSESEMIFLOW(Variational Problems and Related Topics)Author(s)NAKANE, KAZUAKI; OKAMURA, TOSHIHIRO; OMATA,SEIROCitation数理解析研究所講究録 (1996), 951: 1-11Issue Date1996-05URL http://hdl.handle.net/2433/60369RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityNUMERICAL ANALYSIS FOR THE DISCRETE MORSE SEMIFLOW$.+\mathcal{X}7_{\backslash }$, $4^{13}\ovalbox{\tt\small REJECT}_{q}^{\neg}$’$\Re,\dagger_{\sim 7}$$\theta \mathrm{A}\overline{3}\angle\vdash\iota,\backslash$ $\iota\uparrow\backslash l_{\mathrm{K}}^{O}’\backslash$,$1^{-_{\llcorner}}\tilde{x}^{\grave{\tau}_{\backslash }}t]$KAZUAKI NAKANE TOSHIHIRO OKAMURA SEIRO OMATADepartment of Mathematics, Faculty of Science, Kanazawa University1. Definition of discrete Morse semiflowsWe treat numerical analysis for an approximate solution of a heat flow related to aminimizing problem of a functional (1.1) below. Our method leads to a new (general)treatment of a numerical analysis for heat flows (See [4]). Moreover, by use of a limitfunction of this flow, we can obtain a solution of an elliptic problem with better regularity.Let $\Omega(\subset \mathrm{R}^{n})$ be a bounded domain with a smooth boundary $(n\geq 1)$ . Consider aminimizing problem of the functional:$I(u):= \int_{\Omega}(|\nabla u|^{2}+f(u))dX$ , in $\mathcal{K}=W_{\varphi}^{1,2}(\Omega;\mathrm{R}N)$ , (1.1)where $N\geq 1$ and $\mathcal{K}$ is an admissible function space whose elements are square summable upto the first (weak) derivative and satisfy boundary data, $\varphi$ , in the sense of $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$ . Moreover$f$ is a nonnegative function. In this paper, $f$ will be chosen in the following:$f(u):= \frac{1}{\delta}(|u|^{2}-1)^{2}$ Ginzburg-Landau type $(n=N=2)$ (1.2)$f(u):=xu>\mathrm{o}(X)$ Free boundary problem $(n=2, N=1)$ . (1.3)We introduce the notion of the discrete Morse semiflow. For many types of energy func-tional, we can consider the following type of time semidiscretized functional. Let $h$ be apositive number which tends to zero later. Consider the energy functionals:$J_{m}(u)=f_{\Omega} \frac{|u-u_{m-1}|2}{h}dx+I(u)$, $(m=1,2, \cdots)$ . (1.4)We will determine the sequence $\{u_{m}\}$ offunctions in $\mathcal{K}$ inductively. Firstly, for any $u_{0}\in \mathcal{K}$with $I(u_{0})<\infty$ , we define $u_{1}$ , as the minimizer of $J_{1}$ in $\mathcal{K}$ . The next function $u_{2}\in \mathcal{K}$ is theminimizer of $J_{2}$ in $\mathcal{K}$ , and so on.Since the minimizers $\{u_{m}\}$ depend on the positive constant $h$ , we should write $\{u_{m}^{h}\}$and also $J_{m}=J_{m}^{h}$ . However we use the notation $\{u_{m}\}$ and $J_{m}$ unless any confusions occur.The function $u_{m}$ satisfies the following Euler-Lagrange equation when $f$ is differentiable:$\int_{\Omega}(2\frac{u_{m}-u_{m-1}}{h}\phi+2\nabla u_{m}\nabla\phi+f’(u_{m})\phi)dx=0$ (1.5)951 1996 1-11 1for any $\phi\in C_{0}^{\infty}(\Omega;\mathrm{R}N)$ . The equation (1.5) is the time-semidiscretization of the heatequation which possibly describes the Morse semiflow for $I(u)$ . Hence, we call $\{u_{m}\}$ thediscrete Morse semiflow.This approach to determine the family $\{u_{m}\}$ by the minimality of the auxiliary functional$J_{m}$ is proposed by Kikuchi in [1] and related result can be found in [2,3,4].2. UniquenessIn this section, we consider the uniqueness of a discrete Morse semiflow.Theorem 2.1. Let $u$ and $v$ be minimiz$\mathrm{e}\mathrm{r}s$ of (1.4). If $f^{\prime/}(u)$ is bounded then the secon $d$variation formula becomes positive $\mathrm{w}h$en we choose $h$ small enou$gh$ .Proof. We have$\frac{d^{2}}{d\epsilon^{2}}J_{m}(u+\epsilon\phi)|_{\epsilon=0}=\int_{\Omega}((\frac{2}{h}+f^{\prime;}(u))\phi^{2}+2|\nabla\phi|^{2)dX}$ . (2.1)If we choose $h$ small enough, we can make (2.1) positive. It implies the functional $J_{m}$ isuniquly determined. $\square$Now, we shall mention for each case $(1,2)$ and (1.3). For the case (1.2), by the choiseof testing function $v=u/|u|$ , we can show the minimizers are bounded. And for the case(1.3), if we choose suitable approximation function $\chi_{u>0}^{\epsilon}$ (see (7.1) for definition), we canmake $\chi_{u>}^{\epsilon’’}0$ is bounded. Therefore the both of the cases, the discreate Morse semiflows areuniquely determined.Remark 2.1. In the case (1.3), by using maximam principle, we have each minimizer $u_{m}$ isbounded.By use of this theorem, it becomes very easy for us to make a good algorithm for seekinga minimizer.3. Convergence theory for $harrow 0$The essential estimate on this flow is based on the following property:$J_{m}^{h}(u_{m}^{h}) \equiv\int_{\Omega}\frac{|u_{m}^{h}-u_{m-1}^{h}|^{2}}{h}dX+I(u^{h}m)\leq J_{m}^{h}(u_{m-1}h)\equiv I(u_{m-1}^{h})$ ,and therefore we have$\int_{\Omega}\frac{|u_{m}^{h}-u_{m-1}h|^{2}}{h}dx\leq I(u_{m-1}^{h})-I(u_{m})h$ . (3.1)Summing up from $m=1$ to $M$ , we have the estimate:$I(u_{M}^{h})+ \sum_{m=1}^{M}\int_{\Omega}\frac{|u_{m}^{h}-u^{h}m-1|^{2}}{h}d_{X}\leq I(u_{0})$ . (3.2)This estimate is a basic estimate of this flow, from which many properties are obtained.Before showing the convergence theory, firstly, we define the approximate solution of theheat equation related to (1.2).2Definition 3.1. We define functions $\overline{u}^{h}$ and $u^{h}$ on $\Omega\cross(0, \infty)$ by$\overline{u}^{h}(x,t)=u_{m}(hx)$ ,$u^{h}(x,t)= \frac{t-(m-1)h}{h}u_{m}^{hh}(x)+\frac{mh-t}{h}um-1(x)$ ,for $(x,t)\in\Omega\cross((m-1)h, mh]$ .It is easy to see that the functions above satisfy the following relations in a weak sense:$\frac{\partial u^{h}(x,t)}{\partial t}=\Delta\overline{u}^{h}(x,t)-\frac{1}{2}f’(\overline{u}^{h})$ in $\Omega \mathrm{x}\bigcup_{m=2}((\infty m-1)h, mh)$ ,$\overline{u}^{h}(x,t)=u^{h}(x,t)=u_{0}(x)$ on $\partial\Omega$ ,$u^{h}(x,0)=u0(X)$ in $\Omega$ .Here, we investigate the convergence theory when $h$ tends to zero. By use of (3.2), wecan easily obtain the following results.Theorem 3.2. The following $\mathrm{n}o\mathrm{f}\mathrm{m}s$ are uniformly bounded with respect to $h$ :$|| \frac{\partial u^{h}}{\partial t}||_{L^{2}}((0,\infty)\mathrm{X}\Omega)$, $||\nabla\overline{u}^{h}||L^{\infty}((0,\infty);L2(\Omega))$ , $||\nabla u^{h}||_{L}\infty((0,\infty);L^{2}(\Omega))$ ,$||u^{h}||L\infty((0,\infty);L2(\Omega))$ , $||\overline{u}^{h}||L\infty((0,\infty);L2(\Omega))$ , $||u^{h}||W^{1,2}((0,\tau)\mathrm{X}\Omega)$ , (for all $T>0$ ).Theorem 3.3. Th$\mathrm{e}\mathrm{r}e$ exists a subsequence, $s\mathrm{u}ch$ that$\overline{u^{h}.}arrow u$ wealily star in $L^{\infty}.((0, \infty);L^{2}(\Omega))$ , (3.3)$\nabla\overline{u}^{h}arrow\nabla u$$\mathrm{w}\mathrm{e}\mathrm{a}\mathit{1}\iota^{r}l_{f}$ star in $L^{\infty}((0, \infty);L^{2}(\Omega))$, (3.4)$u^{h}arrow u$ weakly in $W^{1,2}((0, \tau)\cross\Omega)$ , (3.5)$u^{h}arrow u$ strongly in $L^{2}((0,T)\cross\Omega)$ , (3.6)$\overline{u}^{h}arrow u$$s$trongly in $L^{2}((0,T)\cross\Omega)$ , (3.7)By use of above estimates, we have:Theorem 3.4. Functions $\overline{u}^{h}$ and $u^{h}co\mathrm{n}$verge to the $s$ame function $u$ in the following sense:For any $T$ ,$\overline{u}^{h}arrow u$ weakly in $L^{2}(\Omega\cross(0,T))$ ,$u^{h}arrow u$ $\mathrm{w}e\mathrm{a}l\iota’l\mathrm{y}$ in $W^{1,2}(\Omega\cross(0,T))$ ,and strongly in $L^{2}(\Omega\cross(0,T))$ .Proof. From (3.2), we have the estimate$\int_{\Omega}|\nabla\overline{u}^{h}(x,Mh)|^{2}dx+\int_{\Omega}f(\overline{u}^{h}(x, Mh))d_{X}+\int_{0}^{Mh}\int_{\Omega}|\frac{\partial u^{h}}{\partial t}(x,t)|^{2}dXdt\leq I(u_{0})$.3It implies that $\{\overline{u}^{h}\}_{h>}0$ and $\{u^{h}\}_{h>}0$ are bounded sets in $L^{2}(\Omega\cross(0, T))$ and $W^{1,2}(\Omega\cross(0, T))$respectively for any $T>0$ . Therefore we can extract a subsequence $\{h_{j}\}$ such that $h_{j}\downarrow 0$and $\overline{u}^{h_{\mathrm{j}}}arrow u$ weakly in $L^{2}(\Omega\cross(0, T)),$ $uh\mathrm{j}arrow v$ weakly in $W^{1,2}(\Omega \mathrm{x}(0,T))$ and strongly in$L^{2}(\Omega \mathrm{x}(0, T))$ as $jarrow\infty$ . It follows from $|u^{h}- \overline{u}^{h}|\leq h|\frac{\partial u^{h}}{\partial t}|$ that$\int_{0}^{T}\int_{\Omega}|u^{h}-\overline{u}|h2d_{Xd}t\leq h^{2}\int_{0}^{\infty}\int_{\Omega}|\frac{\partial u^{h}}{\partial t}|^{2}dXdt\leq h^{2}I(u_{0})arrow 0$ as $h\downarrow 0$ ,which shows $u=v$ .By use of Theorem 3.4, we can show that the limit function satisfies the following equation:Here, in general, we assume the following:Assumption 3.5. There is a constant $M>0_{s\mathrm{u}C}h$ that$||u^{h}||L\infty((0,\tau)\cross\Omega)\leq M$ and $||\overline{u}^{h}||L^{\infty}((0,T)\mathrm{x}\Omega)\leq M$ (for all $T>0$ ) hold.Theorem 3.6. If $f’$ is continuous and $u^{h}$ an $\mathrm{d}\overline{u}^{h}$ satisfies $Ass$ umption 3.3, then the limitfunction $u$ belongs to $V_{2}^{\circ}((0,T)\mathrm{X}\Omega)$ an $\mathrm{d}$ satisfies$\int_{\Omega}u_{0}\eta(x, 0)dX=\int_{0}^{T}\int_{\Omega}D_{t}u\eta d_{Xd}t+\int_{0}^{T}\int_{\Omega}DuD\eta dxdt+\int_{0}^{T}\int_{\Omega}f/(u)\eta d_{X}dt$ (3.8)for all $\eta\in W_{2}^{1,1}((\mathrm{o}\mathrm{o}, T)\cross\Omega))$ with $\eta(x, T)=0$ , where $V_{2}((\mathrm{o},\tau)\mathrm{X}\circ\Omega)=\{u\in L^{2}(Q_{T}),$ $u_{x}\in$$L^{2}(Q_{T});|u|_{Q_{T}}= \mathrm{e}\mathrm{S}\mathrm{s}\sup_{0\leq t\leq\tau}||u(X,t)||L2(\Omega)+||u_{x}||_{L^{2}}(Q_{T})<\infty\}$. We call the function $u$ a weaksolution.Proof. Obviously, approximate solutions satisfy$\int_{\Omega}u_{0}\eta(x, 0)dX=\int_{0}^{T}\int_{\Omega}D_{t}u^{h}\eta dxdt+\int_{0}^{T}\int_{\Omega}D\overline{u}^{h}D\eta d_{Xdt}+\int_{0}^{T}\int_{\Omega}f’(\overline{u}^{h})\eta dXdt$.Thus Theorem 3.4 guarantees that approximate solutions converge to $u$ which satisfy (3.8).$\square$4. Convergence theory for $marrow\infty$In this section we investigate the asymptotic behavior of the discrete Morse semiflow$\{u_{m}\}$ as $marrow\infty$ . From the inequality (3.2), we easily have the following:Lemma 4.1. We $h\mathrm{a}\mathrm{v}e$ the $L^{2}(\Omega)$-decay of differences $u_{m}-u_{m-1}$ as $marrow\infty$ , i.e., $||u_{m}-$$u_{m-1}||L^{2}(\Omega)arrow 0$ as $marrow\infty$ .Now we show the existence of limit $u_{\infty}$ of a subsequence.4Theorem 4.2. For any subsequence $\{u_{m_{i}}\}\subset\{u_{m}\}$ , there exists a subsequence $\{u_{m_{\mathrm{j}_{\nu}}}\}$$\subset\{u_{m_{\mathrm{j}}}\}$ and a function $u_{\infty}$ on $\Omega s\mathrm{u}ch$ that$u_{m_{\mathrm{j}_{\nu}}}arrow u_{\infty}$ wealrly in $W^{1,2}(\Omega)$ , (4.1)$u_{m_{\mathrm{j}\nu}}arrow u_{\infty}$ in $L^{2}(\Omega)$ , (4.2)as $\nuarrow\infty$ . Moreo$\mathrm{v}er$, we $h\mathrm{a}\mathrm{v}e$$u_{\infty}=u_{0}$ on $\partial\Omega$ in the sense of trace. (4.3)Proof. Since $\{J_{m}(u_{m})\}$ is a non-increasing sequence, $\{u_{m}\}$ is weakly compact in $W^{1,2}(\Omega)$ .Therefore (4.1) and (4.2) hold by use of the weak compactness argument and Rellich’stheorem. The boundary condition (4.3) follows from (4.1). $\square$Theorem 4.3. The $li$mit $u_{\infty}$ is a minimizer of the functional$J_{\infty}(u)= \int_{\Omega}(\frac{|u-u_{\infty}|^{2}}{h}+|\nabla u|2f+(u))dx$in $\mathcal{K}$ , hence, $u_{\infty}$ satisfies$- \int_{\Omega}(2\nabla u_{\infty}\nabla\phi+f’(u_{\infty})\phi)dx=0$ for any $\phi\in C_{0}^{\infty}(\Omega)$ .By use of a general theory, we can easily obtain the result, thus the proof is omitted.5. On the numerical methodWe mention here a minimizing algorithm used in this paper. Our method is based onthe finite elements method. We proceed discretization into finite elements. Firstly, split $\Omega$into $Q$ small finite elements (triangle) with $P$ nodes inside of $\Omega$ . Secondly, approximate acomparison function $u$ by a piecewise linear function,$\tilde{u}=\{\tilde{u}_{1},\tilde{u}_{2}, \cdots,\tilde{u}_{N}\}$$\tilde{u}^{i}=A_{1}^{i}x_{1}+\cdots+A_{j}^{i}x_{j}+\cdots+A_{n}^{i}x_{n}+C_{j}$,in a finite elements, which coincides with the given data at each nodal point. $\{A_{j}^{i}\}$ , and $C_{j}$are uniquely determined by the value at each nodal point. By this approximation, we canregard the elements of $\mathrm{R}^{NP}$ as the approximate comparison function. Thirdly, calculatethe gradient of $J_{m}(u)$ in $\mathrm{R}^{NP}$ and find a minimum point along the line with the direction$\nabla J_{m}(u)$ . Repeat this step until satisfies the given terminate conditions.As the usual finite elements method, we calculate the value of integral by summing upthe values of each element. For this purpose, we use a volume coordinate to calculate thesecond term of (1.1) and the first term of (1.2).56. Numerical examples (Ginzburg-Landau type problem)We choose $f(u)= \frac{1}{\delta}(|u|^{2}-1)^{2}$ . We treat the case when $n=N=2,$ $\Omega=\mathrm{B}^{2}$ in (1.2). Weare interested in the behavior of zeros and vortices.Let $p_{i}(i=2,3),$ $f_{z}$ be functions $\mathrm{B}^{2}-\succ \mathrm{R}^{2}$ such that, for $(x, y)\in \mathrm{B}^{2}$ ,$p_{2}(x,y)=( \frac{x^{2}-y^{2}}{\sqrt{x^{2}+y^{2}}},$ $\frac{2xy}{\sqrt{x^{2}+y^{2}}})$$p_{3}(x, y)=( \frac{4x^{3}-3x}{\sqrt{x^{2}+y^{2}}},$ $\frac{3y-4y^{3}}{\sqrt{x^{2}+y^{2}}})$andfor $z=(Z_{1}, z_{2})\in \mathrm{B}^{z}$ .We calculated examples in the following conditions:In Ex 1, we choose zero with degree 2. In this case, a zero with degree 2 splits into two zeroswith degree 1. In Ex 2, we choose degree 3 boundary condition and initial data which has azero with degree 3. In this case, in the final state, a zero splits into three zeros with degree1.7. Numerical example (Free boundary problem)We choose$f(u)=\chi_{u}>0=\xi\{$1 $;\epsilon<u$(7.1)$0$ $;u\leq 0$ ,with $|| \nabla\chi_{u>0}^{\epsilon}||_{\infty}\leq\frac{2}{\epsilon}||\nabla u||_{\infty}$ and $\chi^{\epsilon}u>0\in C^{2}(\mathrm{R})$ . We treat the case $n=2,$ $N=1$ and$\Omega=\mathrm{B}^{2}$ .The Ex 3 says that the graph peel off from $(x, y)$-plane as $tarrow\infty$ . On the other hand, inthe Ex 4, the graph of $u$ does not completely peeling off from the $(x, y)$-plane when $tarrow\infty$ .6$88\zeta \mathrm{q}$$\sim \mathrm{S}$$\alpha\sim$$-I$$.\mathfrak{Q}\approx||$$0\sim$$\mathrm{V}$$\Phi\sim$$\overline{\mathrm{a}\sim}$.$\underline{\mathfrak{Q}||}$7$8\aleph \mathrm{b}\mathrm{l}$$\sim \mathrm{S}$$\ltimes\sim \mathrm{Q}$,$\mathfrak{D}\approx||$$\circ 0$$\infty \mathrm{a}$$\frac{\sim}{\mathrm{a}}$$\underline{\mathfrak{D}||}$8Example 3$h=\mathit{0}.\mathit{0}\mathit{0}\mathit{5}$, epsilon$=\mathit{0}.\theta \mathit{5},$ $u_{-}^{-}l.\theta$ on the boudary9Example 4$h=\mathit{0}.\theta \mathit{0}\mathit{5}$, epsilon $=\theta.\theta \mathit{5},$ $u=\mathit{0}.\mathit{3}\mathit{5}$ on the boudary$t=0.\mathrm{O}$ $t=0.l$$t=\mathit{0}.\mathit{5}$ $t=\mathit{6}.\mathit{5}$$t=\mathit{6}.\mathit{5}$10References[1] N. Kikuchi, “An approach to the construction of Morse flows for variational function-als”, in “Nematics –Mathematical and Physical Aspects”, ed: J. -M. Coron, J. -M.Ghidaglia, F. H\’elein, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 332, KluwerAcad. Publ., Dordrecht- Boston- London (1991), 195-198.[2] T. Nagasawa-S. Omata, “Discrete Morse semifiows of a functional with free boundary”,Adv. Math. Sci. Appl. 2 (1993), 147-187.[3] T. Nagasawa- S. Omata, “Discrete Morse semiflows and their convergence of a func-tional with free boundary”, in Nonlinear Partial Differential Equations - Proceedingsof the International Conference Zhejiang University, June 1992, ed: Dong Guangchangand Lin Fanghua, International Academic Publishers, Beijing (1993), 205-213.[4] S. Omata-T.Okamura-K.Nakane, “Numercal analysis for the discrete Morse semiflowrelated to the Ginzburg Landau functinal”, to appear in Nonlinear World.11

NUMERICAL ANALYSIS FOR THE DISCRETE MORSE SEMIFLOW(Variational Problems and Related Topics)

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NUMERICAL ANALYSIS FOR THE DISCRETE MORSE SEMIFLOW(Variational Problems and Related Topics)

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