Splitting iteration method for simple singular points and simple bifurcation points (Q582014)

To avoid the large computational cost or loss of information caused by solving the linearized equation required in Newton's iteration, different independent blocks are introduced for singular points and the null vector of the derivative. Let E be a Hilbert space and G: \(E\to E\) be a \(C^ 3\)-mapping. Assme that the equation \(G(u)=0(*)\) has a solution \(u_ 0\in E\) satisfying hypotheses H1 and H2. H1) \(DG_ 0:=DG(u_ 0)\) is a Fredholm operator with index 0, and zero is a simple eigenvalue of it with \(\dim (N(DG_ 0))=1\). Then there exist \(\phi,\phi^*\in E\) such that \(N(DG_ 0)=Span[\phi]\), \(N(DG_ 0^*)=Span[\phi^*]\), \(<\phi,\phi^*>=<\phi^*,\phi^*>=1.\) H2) \(<\phi^*,D^ 2G_ 0\phi \phi >\neq 0\). Then \(u_ 0\) is a simple singular point of (*), which is to be found. Define \(E_ i:=(E\times {\mathbb{R}})^ 3\), \(x\in_ 1\), \(x:=(u,c,u_ 1,c_ 1,u_ 2,c_ 2)\), and \(C^ 2\)-continuous mappings \(H_ i: E_ 1\to E\times {\mathbb{R}}\) by \[ H_ 1(x):=\left( \begin{matrix} G(u)+cu_ 2/m\\ <u_ 2,DG(u)u_ 1>/m^ 3\end{matrix} \right),\quad H_ 2(x):=\left( \begin{matrix} DG(u)u_ 1+c_ 1u_ 2/m\\ [<u_ 2,u_ 1>-m^ 3]/m\end{matrix} \right), \] \[ H_ 3(x):=\left( \begin{matrix} DG(u)^*u_ 2+c_ 2u_ 2/m\\ [<u_ 2,u_ 2>- m^ 2]/2m\end{matrix} \right), \] where \(m>0\) is a normalization parameter. Let \(x^*:=(u_ 0,0,m^ 2\phi,0,m\phi^*,0)\in E_ 1\). Then \(x^*\) may be approximated by the followingsplitting iteration: Algorithm: For \(x^ 0\) in the neighbourhood of \(x^*\), \(k=1,2,...\), do \(x^{k+1}=F(x^ k):=(f_ 1(x^ k),f_ 2(x^ k),f_ 3(x^ k))\) where \(f_ i(x):=x_ i[D_{xi}H_ i(\tilde x)]^{-1}H_ i(x)\), \(x\in E_ 1\), and \(\tilde x:=(u,0,u_ 1,0,u_ 2,0)\in E_ 1\) is a restriction of x on \(E_ 1.\) Convergence with error proportional to \((1/2)^ k\) after k iterations is proved, and the precise rate depends on m. Faster convergence is obtained by using a Gauss-Seidel form of iteration in which the 3 components of \(x\in E_ 1\equiv (E\times {\mathbb{R}})^ 3\) are updated sequentially rather than in parallel. Precise complexities are given for the cases \(E={\mathbb{R}}^ n\) or is discrete. Now consider G to depend on a parameter \(\lambda\in {\mathbb{R}}\), and suppose that \((u_ 0,\lambda_ 0)\in E\times {\mathbb{R}}\) is a solution of \(G(u,\lambda)=0\). Conditions are given for \((u_ 0,\lambda_ 0)\) to be a simple bifurcation point, and the theory and algorithm are extended to find such a point. Three numerical examples are considered.