Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems

A wide range of problems occuring in engineering and industry is characterized by the presence of a free (i.e. a priori unknown) boundary where the underlying physical situation is changing in a discontinuous way. Mathematically, such phenomena can be often reformulated as variational inequalities or related non-smooth minimzation problems.These research notes are devoted to a new and promising way of constructing fast solvers for the corresponding discretized problems providing globally convergent iterative schemes with (asymptotic) multigrid convergence speed. The presentation covers: physical modelling - existence and uniqueness results - finite element approximation - monotone mutligrid methods - a posteriori error estimation - adaptive mesh refinement.The numerical properties of the resulting adaptive multilevel algorithm are illustrated by typical applications, such as semiconductor device simulation or continuous casting. "... The presentation is very good motivated, short but very clear so that the book is good readable. It can be recommended for all who are teaching or researching in the field of numerical methods." ZAA Zeitschrift für Analysis und ihre Anwendung, Leipzig