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I have to solve the the problem $u_t+\Delta^2u=f(u)$, where $f(u)$ is non-linear, using domain-decomposition method.

My approach is first using fixed point iteration on mixed form i.e to say $u^{k+1}_t+\Delta w^{k+1}=f(u^k)$ and $w^{k+1}=\Delta u^{k+1}$, then using usual domain decomposition procedure.

Another approach: Use fixed point iteration on the sub-problem i.e to say first use domain decomposition then for each sub-problem use fixed point iteration.

1) which approach is good?

2) Is the selection of the operator $T=$$\begin{bmatrix}(\partial_t)^{-1}+(\Delta)^{-1}\\(\Delta)^{-1}+I\end{bmatrix}$ is correct for first approach.

Thanks in advance.

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    $\begingroup$ Neither approach guarantees convergence for a general RHS function f(u), if it is strongly nonlinear you'd have to go fully implicit. $\endgroup$
    – Maxim Umansky
    Jun 7, 2020 at 15:34
  • $\begingroup$ My, $f(u)=u^3$ or $u^2$ $\endgroup$
    – 420
    Jun 7, 2020 at 16:06
  • $\begingroup$ What do you mean by solving this problem? Integrating it numerically in time to obtain $u(x,t)$? Since you have the equation of the form $u_t = F(u)$ that should be enough to integrate the equations in time with some standard time-integrating software. Domain decomposition may be used for evaluating the right-hand side function $F(u)= - \Delta^2 u + f(u)$ $\endgroup$
    – Maxim Umansky
    Jun 7, 2020 at 23:52
  • $\begingroup$ yes numerically, because of nonlinearity of $f$, I have a problem of getting convergence factor, and linearize it using iteration technique. $\endgroup$
    – 420
    Jun 8, 2020 at 5:59

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