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The problem of finding a L∞-bounded two-dimensional vector field whose divergence is given in L 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∞-norm. To solve this problem from calculus of variations, we use a method relying on a wellchosen 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∞-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 Lregularization 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∞-bounded solutions.

Regularization Methods for the Numerical Solution of the Divergence Equation $∇· u = f$