REGULARIZATION METHODS FOR THE NUMERICAL SOLUTION OF THE DIVERGENCE EQUATION ▽·u =f

• 来源期刊: 计算数学（英文版）
• 关键词: Divergence equation Bounded solutions Regularization methods Augmented Lagrangian Uzawa algorithm Nonlinear variational problems
• 年，卷（期）: 2012,（4）
• 页码范围: 354-380
• 页数: 27页
• 语种: 英文
• DOI: 10.4208/jcm.1111-m3776

摘要

The problem of finding a L∞-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 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 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 L∞-bounded solutions.