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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

$ \mathcal{C}^0$ penalty methods for the fully nonlinear Monge-Ampère equation


Authors: Susanne C. Brenner, Thirupathi Gudi, Michael Neilan and Li-yeng Sung
Journal: Math. Comp. 80 (2011), 1979-1995
MSC (2010): Primary 65N30, 35J60
Published electronically: March 9, 2011
MathSciNet review: 2813346
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Abstract: In this paper, we develop and analyze $ \mathcal{C}^0$ penalty methods for the fully nonlinear Monge-Ampère equation $ \det(D^2 u) = f$ in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.


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Additional Information

Susanne C. Brenner
Affiliation: Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803
Email: brenner@math.lsu.edu

Thirupathi Gudi
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore, 560012
Email: gudi@math.iisc.ernet.in

Michael Neilan
Affiliation: Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803
Email: neilan@math.lsu.edu

Li-yeng Sung
Affiliation: Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803
Email: sung@math.lsu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2011-02487-7
PII: S 0025-5718(2011)02487-7
Keywords: Monge-Ampère equation, fully nonlinear PDEs, finite element method, convergence analysis
Received by editor(s): May 9, 2010
Received by editor(s) in revised form: September 1, 2010
Published electronically: March 9, 2011
Additional Notes: The work of the first and fourth authors were supported in part by the National Science Foundation under Grants No. DMS-07-13835 and DMS-10-16332. The work of the third author was supported by the National Foundation under Grant No. DMS-09-02683
Article copyright: © Copyright 2011 American Mathematical Society