Convergence of a two-stage Richardson process for nonlinear equations
One considers a nonlinear equation \(A(x)=b\), where \(A: R^ n\to R^ n\) is differentiable and can be decomposed as \(A=M-N\), and the equations \(M(y)=c\) can be easily solved. The iterative process is of the form \(x_{k+1}=(1-\omega)x_{k-1}+(\alpha z_ k+x_ k)\) with \(\alpha,\omega >0\) and \(Mz_ k=r_ k+q_ k\), \(r_ k=b-A(x_ k)\), \(\| q_ k\| \leq \delta \| r_ k\|\), \(0\leq \delta <1\). As a numerical example, one solves the equation obtained by approximating the differential equation \(-\Delta u+u^ 3=f\) with nonhomogeneous Dirichlet conditions on the boundary of the unit square, using finite differences.
two-stage Richardson iterative processes
scientific article; zbMATH DE number 3963104