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Volume 30, Issue 4
Regularization Methods for the Numerical Solution of the Divergence Equation $∇· u = f$

Alexandre Caboussat & Roland Glowinski

J. Comp. Math., 30 (2012), pp. 354-380.

Published online: 2012-08

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  • Abstract

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 $L^∞$-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 $L^∞$-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 $L^∞$-bounded solutions.

  • AMS Subject Headings

65N30, 65K10, 65J20, 49K20, 90C47.

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COPYRIGHT: © Global Science Press

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@Article{JCM-30-354, author = {}, 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 = {

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 $L^∞$-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 $L^∞$-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 $L^∞$-bounded solutions.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1111-m3776}, url = {http://global-sci.org/intro/article_detail/jcm/8436.html} }
TY - JOUR T1 - Regularization Methods for the Numerical Solution of the Divergence Equation $∇· u = f$ JO - Journal of Computational Mathematics VL - 4 SP - 354 EP - 380 PY - 2012 DA - 2012/08 SN - 30 DO - http://doi.org/10.4208/jcm.1111-m3776 UR - https://global-sci.org/intro/article_detail/jcm/8436.html KW - Divergence equation, Bounded solutions, Regularization methods, Augmented Lagrangian, Uzawa algorithm, Nonlinear variational problems. AB -

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 $L^∞$-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 $L^∞$-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 $L^∞$-bounded solutions.

Alexandre Caboussat & Roland Glowinski. (1970). Regularization Methods for the Numerical Solution of the Divergence Equation $∇· u = f$. Journal of Computational Mathematics. 30 (4). 354-380. doi:10.4208/jcm.1111-m3776
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