Multigrid Method for Nonsmooth Problems
|Title||Multigrid Method for Nonsmooth Problems|
|Year of Publication||2015|
Multigrid methods have been shown to be an efficient tool for solv- ing partial differential equations. In this paper, the idea of a multi- grid method for nonsmooth problems is presented based on techniques from piecewise linear differentiation. In detail, the original nonsmooth problem is approximated by a sequence of piecewise linear models, which can be written in abs-normal form by using additional switch- ing variables. In certain cases, one can exploit the structure of the piecewise linearization and formulate an efficient modulus fixed-point iteration for these switching variables. Moreover, using the idea of multigrid methods, one can find a solution of the modulus fixed-point equation for the switching variables on a coarse discretization, which then serves as an initial guess for the next finer level. Here, the impor- tant aspect is the right choice for the prolongation operator in order to avoid undesirable smoothing effects as it will be shown. Numerical results indicate (almost) mesh-independent behavior of the resulting method if done in the right way.