@Article{JCM-30-354,
author = {Alexandre Caboussat and Roland Glowinski},
title = {Regularization Methods for the Numerical Solution of the Divergence Equation $∇· u = f$},
journal = {Journal of Computational Mathematics},
year = {2012},
volume = {30},
number = {4},
pages = {354--380},
abstract = {<p style="text-align: justify;">The problem of finding a $L^∞$-bounded two-dimensional vector field whose divergence is given in $L^2$ is discussed from the numerical viewpoint. A systematic way to find such a vector field is to introduce a non-smooth variational problem involving a $L^∞$<span style="text-align: justify;"></span>-norm. To solve this problem from calculus of variations, we use a method relying on a well-chosen augmented Lagrangian functional and on a mixed finite element approximation. An Uzawa algorithm allows to decouple the differential operators from the nonlinearities introduced by the $L^∞$<span style="text-align: justify;"></span>-norm, and leads to the solution of a sequence of Stokes-like systems and of an infinite family of local nonlinear problems. A simpler method, based on a $L^2$-regularization is also considered. Numerical experiments are performed, making use of appropriate numerical integration techniques when non-smooth data are considered; they allow to compare the merits of the two approaches discussed in this article and to show the ability of the related methods at capturing $L^∞$<span style="text-align: justify;"></span>-bounded solutions.</p>},
issn = {1991-7139},
doi = {https://doi.org/10.4208/jcm.1111-m3776},
url = {http://global-sci.org/intro/article_detail/jcm/8436.html}
}