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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board Member: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold

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Online
ISSN
1569-3953
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Volume 16, Issue 3 (Jan 2008)

Issues

Numerical solution of the Dirichlet problem for a Pucci equation in dimension two. Application to homogenization

L. A. Caffarelli / R. Glowinski
Published Online: 2008-11-24 | DOI: https://doi.org/10.1515/JNUM.2008.009

Abstract

The main goal of this article is two fold: (i) To discuss a methodology for the numerical solution of the Dirichlet problem for a Pucci equation in dimension two. (ii) Use the ensuing algorithms to investigate the homogenization properties of the solutions when a coefficient in the Pucci equation oscillates periodically or randomly in space. The solution methodology relies on the combination of a least-squares formulation of the Pucci equation in an appropriate Hilbert space with operator-splitting techniques and mixed finite element approximations. The results of numerical experiments suggest second order accuracy when globally continuous piecewise affine space approximations are used; they also show that the solution of the problem under consideration can be reduced to a sequence of discrete Poisson–Dirichlet problems coupled with one-dimensional optimization problems (one per grid point).

Keywords:: Fully nonlinear elliptic equations; nonlinear least squares; finite element approximation; homogenization

About the article

Received: 2008-03-06

Published Online: 2008-11-24

Published in Print: 2008-11-01


Citation Information: Journal of Numerical Mathematics, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/JNUM.2008.009.

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