On the convergence and error estimates for waveform relaxation methods for neutral differential-functional systems

Marian Kwapisz

The Academy of Bydgoszcz, ul. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland

mkwapisz@math.univ.gda.pl

Contributed talk

We discuss the problem of delay-dependent error estimates for waveform relaxation methods applied to Volterra type systems of the form

$$w'(t) = \mathcal{F}(w,w')(t),\quad t\in I_a=[a,b],$$ (1)

$$w(t) = \phi(t),\quad t\in J_h=[h,a],\quad h<a.$$ (2)

Let $I_h=[h,b]$, $C_{\phi}(I_{h},\mathbb{R}^p)=\{x\in C(I_{h},\mathbb{R}^p):x(t)=\phi(t),\;t\in J_h\}$ and $\psi=\phi'$. Choosing a suitable integral operator $\mathcal{J}$ the problem (1)-(2) can be written as

$$z(t)=\mathcal{F}(\mathcal{J}z,z)(t),\quad t\in I_h.$$ (3)

Now, let $F$ be a splitting function such that for $z\in C_{\psi}(I_{h},\mathbb{R}^p)$

$$F(\mathcal{J}z,\mathcal{J}z,z,z)(t)=\mathcal{F}(\mathcal{J}z, z)(t),\quad t\in I_h.$$ (4)

We consider the following WR process

$$z_{k+1}(t)=F(\mathcal{J}z_{k+1},\mathcal{J}z_k,z_{k+1},z_k)(t),\quad t\in I_h,\quad k=0,1,\ldots.$$ (5)

We assume the Lipschitz condition (CONDITION L):
There exist nonnegative square matrices $L_1,L_2,K_1,K_2$ of dimension $p'$ and nondecreasing continuous functions $\alpha,\beta:I_a\rightarrow I_a$ satisfying the condition $a\le \alpha(t)\le t$ and $a\le \beta(t)\le t$ for $t\in I_a$ such that for any $q,u,\bar{q},\bar{u}\in C_{\phi}(I_{h},\mathbb{R}^p)$, $v,w,\bar{v},\bar{w}\in C_{\psi}(I_h,\mathbb{R}^p)$, $t\in I_a$, and a norm $\Vert\!\cdot\!\Vert:\mathbb{R}^p\rightarrow \mathbb{R}_+^{p'}$ $F$ satisfies the Lipschitz condition

$${\Vert F(q,u,v,w)(t)-F(\bar{q},\bar{u},\bar{v},\bar{w})(t)\Vert}$$ (6)

$$\le L_1\Vert q-\bar{q}\Vert _{t}+L_2\Vert u-\bar{u}\Vert _{\alpha(t)}+ K_1\Vert v-\bar{v}\Vert _{t}+K_2\Vert w-\bar{w}\Vert _{\beta(t)},$$

with the spectral radius $\rho(K_1)<1$ and nonreducible matrix $K=(I-K_1)^{-1}K_2$
and the condition (CONDITION D):
There exists $\delta>0$ such that: $\beta(t)=a$ for $t\in [a,a+\delta]$, $$\inf_{t\in [a+\delta,b]}(t-\beta(t))=\delta$$. We have the theorem:

Theorem  Assume that CONDITION L holds, $\alpha\in C^1[I_a,I_a]$, $\bar{\beta}(t)=\max{(\alpha(t),}$ ${\beta(t))}$, $t\in I_a$, and either $\rho(K)<1$ or CONDITION D holds. Then the WR process (5) is convergent with the following error estimate

$$\vert v_k(t)\vert _c\le \vert v_0(\bar{\beta}^k(t))\vert _... ...ho(K)^{k-i}\Psi_i(1)(t)\right],\\ k=0,1,\ldots,\; t\in I_a,$$ (7)

where $\Psi_i(1)$ is suitably defined sequence of functions. Moreover, if CONDITION D holds with $\beta$ replaced by $\bar{\beta}$ then there exists $k_0$ such that the error estimates are zeros for $k\ge k_0$, which means that the exact solution is obtained after a finite number of iterations.