Analysis of a damped nonlinear multilevel method
Let H be a real Hilbert space and n: \(H\to H'\) be a nonlinear operator which is monotone in the sense of Minty. Moreover n is assumed to be a gradient operator. Given a family of subspaces \(S_ h\), the corresponding operators \(n_ h: S_ h\to S_ h'\) define the discrete analoga of n. Specifically, the discretization of the nonlinear equation \(n(u)=f\) is considered. The uniform ellipticity of the Fréchet derivative \(Dn(u)(v,v)\geq \gamma \| v\|^ 2\) guarantees \(\| u_ 1-u_ 2\| \leq \gamma^{-1}\| f_ 1-f_ 2\|,\) if \(n(u_ i)=f_ i\). Therefore one has a generalization of Céa's lemma and convergence. The authors define a nonlinear multilevel algorithm with a fairly general class of nonlinear relaxations for the smoothing. An essential ingredient is a damping parameter which helps to control the multigrid iteration and global convergence can be proved. Numerical results show that damping may indeed be necessary in some cases to obtain convergence. Moreover they illustrate that the damped Newton nonlinear multigrid method is a very efficient method.
damped Newton nonlinear multigrid method
scientific article; zbMATH DE number 4102394