APPROXIMATION AND NUMERICAL RESULTS FOR PHASE FIELD SYSTEM BY A FRACTIONAL STEP SCHEME

COSTICÃ MOROŞANU
(Iaşi)

We consider the phase field system

subject to the Dirichlet boundary conditions and initial conditions

where $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega$\mathrm{\Omega }$ is a bounded domain in ${\mathbf{R}}^n$${\mathbf{R}}^n$R^(n)\mathbf{R}^{n}${\mathbf{R}}^n$ with smooth boundary $\partial \mathrm{\Omega }$$\partial \mathrm{\Omega }$del Omega\partial \Omega$\partial \mathrm{\Omega }$, and $\phi ,u,\tau ,\xi ,l$$\phi ,u,\tau ,\xi ,l$varphi,u,tau,xi,l\varphi, u, \tau, \xi, l$\phi ,u,\tau ,\xi ,l$ and $k$$k$kk$k$ are as in [9], [11].

Setting

system (1.1)-(1.4) takes the form

Let $X=L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$$X=L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$X=L^(2)(Omega)xxL^(2)(Omega)X=L^{2}(\Omega) \times L^{2}(\Omega)$X=L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$. Then $X$$X$XX$X$ is a real Banach space with respect to the norm $\Vert \cdot \Vert$$\Vert \cdot \Vert$||*||\|\cdot\|$\Vert \cdot \Vert$ defined by

Define the operator $A:D(A)\subset X\to X$$A:D(A)\subset X\to X$A:D(A)sub X rarr XA: D(A) \subset X \rightarrow X$A:D(A)\subset X\to X$ by

and the operator $B:D(B)\subset X\to X:$$B:D(B)\subset X\to X:$B:D(B)sub X rarr X:B: D(B) \subset X \rightarrow X:$B:D(B)\subset X\to X:$

Thus, system (1.6)-(1.7) can be rewritten in the form

For others settings into the abstract framework of the phase-field equations (1.1)-(1.4) see, e.g., [6], [14].

The idea behind the Lie-Trotter scheme (known as the method of fractional step in numerical approximation of PDE's) is to decompose the original problem into several simpler problems.

Here we associate to system ( $S$$S$SS$S$ ) the following approximating scheme

where $0<\epsilon <\dots <M\epsilon =T$$0<\epsilon <\dots <M\epsilon =T$0 < epsi < dots < M epsi=T0<\varepsilon<\ldots<M \varepsilon=T$0<\epsilon <\dots <M\epsilon =T$ is a partition of the time-interval $[0,T],{\phi }_{\epsilon }^{+}(i\epsilon )$$[0,T],{\phi }_{\epsilon }^{+}(i\epsilon )$[0,T],varphi_(epsi)^(+)(i epsi)[0, T], \varphi_{\varepsilon}^{+}(i \varepsilon)$[0,T],{\phi }_{\epsilon }^{+}(i\epsilon )$ is the right limit of ${\phi }_{\epsilon }$${\phi }_{\epsilon }$varphi_(epsi)\varphi_{\varepsilon}${\phi }_{\epsilon }$ at is. We assume the following convention: ${\phi }_{\epsilon }^{+}(0)={\phi }_0,y_{\epsilon }(0)=y_0$${\phi }_{\epsilon }^{+}(0)={\phi }_0,y_{\epsilon }(0)=y_0$varphi_(epsi)^(+)(0)=varphi_(0),y_(epsi)(0)=y_(0)\varphi_{\varepsilon}^{+}(0)=\varphi_{0}, y_{\varepsilon}(0)=y_{0}${\phi }_{\epsilon }^{+}(0)={\phi }_0,y_{\epsilon }(0)=y_0$.

Recall that $J:X\to X^{\ast }$$J:X\to X^{\ast }$J:X rarrX^(**)J: X \rightarrow X^{*}$J:X\to X^{\ast }$ is the duality mapping of the space $X$$X$XX$X$ (see, for instance, [2]) and that $A\subset X\times X$$A\subset X\times X$A sub X xx XA \subset X \times X$A\subset X\times X$ is:
accretive, if for every pair $[x_1,y_1],[x_2,y_2]\in A$$\left[x_1,y_1\right],\left[x_2,y_2\right]\in A$[x_(1),y_(1)],[x_(2),y_(2)]in A\left[x_{1}, y_{1}\right],\left[x_{2}, y_{2}\right] \in A$[x_1,y_1],[x_2,y_2]\in A$, there exist $w\in J(x_1-x_2)$$w\in J\left(x_1-x_2\right)$w in J(x_(1)-x_(2))w \in J\left(x_{1}-x_{2}\right)$w\in J(x_1-x_2)$ such that

or, equivalently,

$m$$m$mm$m$-accretive, if it is accretive and $R(I+A)=X$$R(I+A)=X$R(I+A)=XR(I+A)=X$R(I+A)=X$,
$\omega$$\omega$omega\omega$\omega$-accretive, if $A+\omega I$$A+\omega I$A+omega IA+\omega I$A+\omega I$ is accretive, where $\omega \in \mathbb{R}$$\omega \in \mathbb{R}$omega inR\omega \in \mathbb{R}$\omega \in \mathbb{R}$,
$\omega -m$$\omega -m$omega-m\omega-m$\omega -m$-accretive, if $A+\omega I$$A+\omega I$A+omega IA+\omega I$A+\omega I$ is $m$$m$mm$m$-accretive,
where $<,>$$<,>$< , ><,>$<,>$ is the pairing between $X$$X$XX$X$ and $X^{\ast }$$X^{\ast }$X^(**)X^{*}$X^{\ast }$ (the dual space of $X$$X$XX$X$ ), $I$$I$II$I$ is the identity operator in $X,R(A)$$X,R(A)$X,R(A)X, R(A)$X,R(A)$ is the range of $A$$A$AA$A$.

Another convenient way to define the accretiveness is obtained using $[✓\cdot {]}_S-$$[✓\cdot {]}_S-$[✓*]_(S)-[\checkmark \cdot]_{S}-$[✓\cdot {]}_S-$ the directional derivative of the norm

i.e, (ii) can be equivalently written as
(ii')

(see also [2], [15], [16]). Recall that if $X$$X$XX$X$ is a real Hilbert space then $[x,y{]}_S=⟨x,y⟩\mathrm{\forall }x,y\in X$$[x,y{]}_S=⟨x,y⟩\mathrm{\forall }x,y\in X$[x,y]_(S)=(:x,y:)AA x,y in X[x, y]_{S}=\langle x, y\rangle \forall x, y \in X$[x,y{]}_S=⟨x,y⟩\mathrm{\forall }x,y\in X$ (see [16], Remark 1.4.1).

It is well known that, under certain hypotheses on $A$$A$AA$A$, the Cauchy problem

has a generalized solution $v\in C([0.\mathrm{\infty }),X)$$v\in C([0.\mathrm{\infty }),X)$v in C([0.oo),X)v \in C([0 . \infty), X)$v\in C([0.\mathrm{\infty }),X)$ given by the exponential formula

for every $v_0\in \overline{D(A)}$$v_0\in \overline{D(A)}$v_(0)in bar(D(A))v_{0} \in \overline{D(A)}$v_0\in \overline{D(A)}$ (a classical result of Crandall-Ligget, sec, e.g., [2]).
This is the sense in which we will treat the problems (1.6)-(1.9) and (1.10)-(1.13).

Let us recall the following result due to Barbu and Iannelli ([5]).
THEOREM 2.1. Let $Y$$Y$YY$Y$ be a real Banach space, let $C$$C$CC$C$ be a closed subset of $Y$$Y$YY$Y$ and $K=C\cap D(A)$$K=C\cap D(A)$K=C nn D(A)K=C \cap D(A)$K=C\cap D(A)$ be a convex subset of $Y$$Y$YY$Y$ such that
(H1) $A$$A$AA$A$ is $\omega$$\omega$omega\omega$\omega$-accretive and $R(I+\lambda A)\supset \overline{D(A)},(\mathrm{\forall })\lambda \in (0,{\lambda }_0)$$R(I+\lambda A)\supset \overline{D(A)},(\mathrm{\forall })\lambda \in \left(0,{\lambda }_0\right)$R(I+lambda A)sup bar(D(A)),(AA)lambda in(0,lambda_(0))R(I+\lambda A) \supset \overline{D(A)},(\forall) \lambda \in\left(0, \lambda_{0}\right)$R(I+\lambda A)\supset \overline{D(A)},(\mathrm{\forall })\lambda \in (0,{\lambda }_0)$;
(H2) $B$$B$BB$B$ is a continuous $\omega$$\omega$omega\omega$\omega$-accretive operator on $C$$C$CC$C$ such that

(H3) $R(I+\lambda (A+B))\supset K,(I+\lambda A{)}^{-1}K\subset K,(I+\lambda B{)}^{-1}K\subset K$$R(I+\lambda (A+B))\supset K,(I+\lambda A{)}^{-1}K\subset K,(I+\lambda B{)}^{-1}K\subset K$R(I+lambda(A+B))sup K,(I+lambda A)^(-1)K sub K,(I+lambda B)^(-1)K sub KR(I+\lambda(A+B)) \supset K,(I+\lambda A)^{-1} K \subset K,(I+\lambda B)^{-1} K \subset K$R(I+\lambda (A+B))\supset K,(I+\lambda A{)}^{-1}K\subset K,(I+\lambda B{)}^{-1}K\subset K$;
(H4) For every $[x,y]\in A$$[x,y]\in A$[x,y]in A[x, y] \in A$[x,y]\in A$, there exists $\left\{x_h\right\}\subset Y$$\left\{x_h\right\}\subset Y${x_(h)}sub Y\left\{x_{h}\right\} \subset Y$\left\{x_h\right\}\subset Y$ such that

Then, for every $y_0\in K$$y_0\in K$y_(0)in Ky_{0} \in K$y_0\in K$, we have

and the limit is uniform on bounded $t$$t$tt$t$ intervals.
Here, by $y(t)$$y(t)$y(t)y(t)$y(t)$, we have denoted the generalized solution to the Cauchy problem:

and by $y_{\epsilon }(t)$$y_{\epsilon }(t)$y_(epsi)(t)y_{\varepsilon}(t)$y_{\epsilon }(t)$ the solution of the corresponding approximative scheme.
This result is not applicable to the problem (1.10)-(1.13) because we cannot find a subset $C$$C$CC$C$ as in (H2) and such that the operator $B$$B$BB$B$ be continuous on $C$$C$CC$C$.

Therefore, we will replace the operator $B$$B$BB$B$ with another one having all the properties required by Theorem 2.1 and we will show that the approximate solution $(\genfrac{}{}{0ex}{}{y_{\epsilon }}{{\phi }_{\epsilon }})$$(\genfrac{}{}{0ex}{}{y_{\epsilon }}{{\phi }_{\epsilon }})$((y_(epsi))/(varphi_(epsi)))\binom{y_{\varepsilon}}{\varphi_{\varepsilon}}$(\genfrac{}{}{0ex}{}{y_{\epsilon }}{{\phi }_{\epsilon }})$ corresponding to this new operator is in fact an approximate solution corresponding to $B$$B$BB$B$ (see Remark 3.1, below). Namely, we consider the operator $B_r$$B_r$B_(r)B_{r}$B_r$. defined by (see also Figure 1)

where

Substituting in $(S)$$(S)$(S)(S)$(S)$ and in (2.12) $B(\genfrac{}{}{0ex}{}{y_{\epsilon }(t)}{{\phi }_{\epsilon }(t)})$$B(\genfrac{}{}{0ex}{}{y_{\epsilon }(t)}{{\phi }_{\epsilon }(t)})$B((y_(epsi)(t))/(varphi_(epsi)(t)))B\binom{y_{\varepsilon}(t)}{\varphi_{\varepsilon}(t)}$B(\genfrac{}{}{0ex}{}{y_{\epsilon }(t)}{{\phi }_{\epsilon }(t)})$ by $B_r(\genfrac{}{}{0ex}{}{y_{\epsilon }(t)}{{\phi }_{\epsilon }(t)})$$B_r(\genfrac{}{}{0ex}{}{y_{\epsilon }(t)}{{\phi }_{\epsilon }(t)})$B_(r)((y_(epsi)(t))/(varphi_(epsi)(t)))B_{r}\binom{y_{\varepsilon}(t)}{\varphi_{\varepsilon}(t)}$B_r(\genfrac{}{}{0ex}{}{y_{\epsilon }(t)}{{\phi }_{\epsilon }(t)})$ we obtain:

We associate to system ( $S^{\prime }$$S^{\prime }$S^(')S^{\prime}$S^{\prime }$ ) the approximating scheme (1.10), (1.11), (2.1) and (1.13).

Now we prove
Proposition 2.1. If $k{l}^2<16{\xi }^2/\tau$$k{l}^2<16{\xi }^2/\tau$kl^(2) < 16xi^(2)//tauk l^{2}<16 \xi^{2} / \tau$k{l}^2<16{\xi }^2/\tau$ and $l>1/2a$$l>1/2a$l > 1//2al>1 / 2 a$l>1/2a$ then, the operators $A,B_r$$A,B_r$A,B_(r)A, B_{r}$A,B_r$. and $Y=C=K=L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$$Y=C=K=L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$Y=C=K=L^(2)(Omega)xxL^(2)(Omega)Y=C=K=L^{2}(\Omega) \times L^{2}(\Omega)$Y=C=K=L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$ satisfy all the hypotheses of Theorem 2.1.

We shall prove first the following lemma.
Lemma 2.1. If $k{l}^2<16{\xi }^2/\tau$$k{l}^2<16{\xi }^2/\tau$kl^(2) < 16xi^(2)//tauk l^{2}<16 \xi^{2} / \tau$k{l}^2<16{\xi }^2/\tau$ and $l>1/2a$$l>1/2a$l > 1//2al>1 / 2 a$l>1/2a$ then $A$$A$AA$A$ is 0 -accretive and satisfies the range condition

Proof. Using the definition (ii') we must show that, for every $(\genfrac{}{}{0ex}{}{y}{\phi })\in X,(A$$(\genfrac{}{}{0ex}{}{y}{\phi })\in X,(A$((y)/( varphi))in X,(A\binom{y}{\varphi} \in X,(A$(\genfrac{}{}{0ex}{}{y}{\phi })\in X,(A$ is linear and $J$$J$JJ$J$ is univalued, i.e., $J(\genfrac{}{}{0ex}{}{y}{\phi })=(\genfrac{}{}{0ex}{}{y}{\phi })=w)$$\left.J(\genfrac{}{}{0ex}{}{y}{\phi })=(\genfrac{}{}{0ex}{}{y}{\phi })=w\right)${:J((y)/( varphi))=((y)/( varphi))=w)\left.J\binom{y}{\varphi}=\binom{y}{\varphi}=w\right)$J(\genfrac{}{}{0ex}{}{y}{\phi })=(\genfrac{}{}{0ex}{}{y}{\phi })=w)$

Using Green formula and Cauchy-Schwarz's inequality, we get ( $X$$X$XX$X$ is real Hilbert space)