# fixed point iteration on DD method

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.

• My, $f(u)=u^3$ or $u^2$ – 420 Jun 7 '20 at 16:06
• 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 $F(u)= - \Delta^2 u + f(u)$ – Maxim Umansky Jun 7 '20 at 23:52
• yes numerically, because of nonlinearity of $f$, I have a problem of getting convergence factor, and linearize it using iteration technique. – 420 Jun 8 '20 at 5:59