Regularization methods for the numerical solution of the divergence equation ∇ · u = f

Caboussat, Alexandre ; Glowinski, Roland

In: Journal of computational mathematics, 2012, vol. 30, no. 4, p. 354-380

The problem of finding a L1-bounded two-dimensional vector field whose divergence is given in L2 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 L1-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... Plus

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    Summary
    The problem of finding a L1-bounded two-dimensional vector field whose divergence is given in L2 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 L1-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 L1-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 L2- 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 L1-bounded solutions. Mathematics subject classification: 65N30, 65K10, 65J20, 49K20, 90C47. Key words: Divergence equation, Bounded solutions, Regularization methods, Augmented Lagrangian, Uzawa algorithm, Nonlinear variational problems.