Dirichlet–Neumann waveform relaxation methods for parabolic and hyperbolic problems in multiple subdomains

Abstract

In this paper, a new waveform relaxation variant of the Dirichlet–Neumann algorithm is introduced for general parabolic problems as well as for the second-order wave equation for decompositions with multiple subdomains. The method is based on a non-overlapping decomposition of the domain in space, and the iteration involves subdomain solves in space-time with transmission conditions of Dirichlet and Neumann type to exchange information between neighboring subdomains. Regarding the convergence of the algorithm, two main results are obtained when the time window is finite: for the heat equation, the method converges superlinearly, whereas for the wave equation, it converges after a finite number of iterations. The analysis is based on Fourier–Laplace transforms and detailed kernel estimates, which reveals the precise dependence of the convergence on the size of the subdomains and the time window length. Numerical experiments are presented to illustrate the performance of the algorithm and to compare its convergence behaviour with classical and optimized Schwarz Waveform Relaxation methods. Experiments involving heterogeneous coefficients and non-matching time grids, which are not covered by the theory, are also presented.

Appendix

For equal subdomains, with subdomain size h, we can deduce a better estimate:

$$\begin{aligned} {\displaystyle \max _{1\le i\le 2m}}\parallel w_{i}^{k}\parallel _{L^{\infty }(0,T)}\le \left( \min \left\{ 2m-1,Q(h,\nu ,T)\right\} \right) ^{k}{{\,\mathrm{erfc}\,}}\left( \frac{kh}{2\sqrt{\nu T}}\right) {\displaystyle \max _{1\le i\le 2m}}\parallel w_{i}^{0}\parallel _{L^{\infty }(0,T)}, \end{aligned}$$

where $Q(h,\nu ,T):=2\,\mathrm{erfc}\left( \frac{h}{2\sqrt{\nu T}}\right) +\sum _{i=0}^{\infty }2^{i+1}\mathrm{erfc}\left( \frac{ih}{2\sqrt{\nu T}}\right)$.

Here is an outline of the proof: For equal-length subdomains we have $h_{i}=h$ for $i=1,\ldots ,2m+1$, so that $\gamma _{i}=\gamma ,\sigma _{i}=\sigma$ and $\gamma _{i,j}=1,\sigma _{i,j}=0$ for all i, j in (3.10). Therefore, we derive for $i=1$ the recurrence.

$$\begin{aligned} {\hat{v}}_{1}^{k} \!\!= & {} \!\!\frac{1}{\gamma }{\hat{v}}_{1}^{k-1}-\left( 1-\frac{1}{\gamma ^{2}} \right) {\hat{v}}_{2}^{k-1}-\cdots -\frac{1}{\gamma ^{m-3}} \left( 1-\frac{1}{\gamma ^{2}}\right) {\hat{v}}_{m-1}^{k-1} -\frac{1}{\gamma ^{m-2}}{\hat{v}}_{m}^{k-1}+\frac{1}{\gamma ^{m-1}} {\hat{v}}_{m+1}^{k-1}\nonumber \\ \!\!=: & {} \!\! \sum _{i=1}^{m+1}{\hat{q}}_{i}{\hat{v}}_{i}^{k-1}. \end{aligned}$$

(A.1)

Note that the estimate from Theorem 2.1 also holds in this case with $h_{\max }=h_{m+1}=h$:

$$\begin{aligned} {\displaystyle \max _{1\le i\le 2m}}\parallel w_{i}^{k}\parallel _{L^{\infty }(0,T)}\le \left( 2m-1\right) ^{k}{{\,\mathrm{erfc}\,}}\left( \frac{kh}{2\sqrt{\nu T}}\right) {\displaystyle \max _{1\le i\le 2m}}\parallel w_{i}^{0}\parallel _{L^{\infty }(0,T)}. \end{aligned}$$

(A.2)

We now present a different estimate starting from (A.1). By back transforming into the time domain we obtain $v_{1}^{k}(t)=\sum _{i=1}^{m+1}\left( q_{i}*v_{i}^{k-1}\right) (t).$ Part 3 of Result 1 yields

$$\begin{aligned} \Vert v_{1}^{k}\Vert _{L^{\infty }(0,T)}\le & {} \sum _{i=1}^{m+1}\Vert v_{i}^{k-1} \Vert _{L^{\infty }(0,T)}\int _{0}^{T}\!|q_{i}(\tau )|d\tau \nonumber \\\le & {} {\displaystyle \max _{1\le i\le m+1}}\parallel v_{i}^{k-1}\parallel _{L^{\infty }(0,T)}\sum _{i=1}^{m+1}\int _{0}^{T} \!|q_{i}(\tau )|d\tau . \end{aligned}$$

(A.3)

Set $f_{k}(t):={\mathscr {L}}^{-1}\left( \frac{1}{\gamma ^{k}}\right)$. Since $\lim _{s\rightarrow 0+}\left| 1-\frac{1}{\gamma ^{2}}\right| \le 2$, we have from (A.3)

$$\begin{aligned} \sum _{i=1}^{m+1}\int _{0}^{T}|q_{i}(\tau )|d\tau \le \int _{0}^{T} \left( f_{1}(t)+2+2f_{1}(t)+\cdots +2f_{m-3}(t)+f_{m-2}(t)+f_{m-1}(t)\right) dt. \end{aligned}$$

Therefore using Lemma 3.3, we obtain

$$\begin{aligned} \sum _{i=1}^{m+1}\int _{0}^{T}|q_{i}(\tau )|d\tau \le 2\,\mathrm{erfc}\left( \frac{h}{2\sqrt{\nu T}}\right) +\sum _{i=0}^{m-1}2^{i+1}\mathrm{erfc}\left( \frac{ih}{2\sqrt{\nu T}}\right) \le Q(h,\nu ,T), \end{aligned}$$

where $Q(h,\nu ,T):=2\,\mathrm{erfc}\left( \frac{h}{2\sqrt{\nu T}}\right) +\sum _{i=0}^{\infty }2^{i+1}\mathrm{erfc}\left( \frac{ih}{2\sqrt{\nu T}}\right)$. So we get the second estimate as

$$\begin{aligned} {\displaystyle \max _{1\le i\le 2m}}\parallel w_{i}^{k}\parallel _{L^{\infty }(0,T)}\le Q^{k}\,\mathrm{erfc}\left( \frac{kh}{2\sqrt{\nu T}}\right) {\displaystyle \max _{1\le i\le 2m}}\parallel w_{i}^{0}\parallel _{L^{\infty }(0,T)}. \end{aligned}$$

(A.4)

The result follows combining the two estimates (A.2) and (A.4).

We compare the two estimates (A.2) and (A.4) in Fig. 16 for $\nu =1$. The region below the red curve is where the estimate (A.2) is more accurate than (A.4).

Fig. 16

Comparison of the two estimates (A.2) and (A.4) for $\nu =1$