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2013 | OriginalPaper | Chapter

Applications of Nonvariational Finite Element Methods to Monge–Ampère Type Equations

Author : T. Pryer

Published in: Numerical Mathematics and Advanced Applications 2011

Publisher: Springer Berlin Heidelberg

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Abstract

The goal of this work is to illustrate the application of the nonvariational finite element method to a specific Monge–Ampère type nonlinear partial differential equation. The equation we consider is that of prescribed Gauss curvature however the method can be generalised to any Monge–Ampère operator.

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Literature
1.
go back to reference Néstor E. Aguilera and Pedro Morin. On convex functions and the finite element method. SIAM J. Numer. Anal., 47(4):3139–3157, 2009. Néstor E. Aguilera and Pedro Morin. On convex functions and the finite element method. SIAM J. Numer. Anal., 47(4):3139–3157, 2009.
3.
go back to reference Klaus Böhmer. On finite element methods for fully nonlinear elliptic equations of second order.SIAM J. Numer. Anal., 46(3):1212–1249, 2008. Klaus Böhmer. On finite element methods for fully nonlinear elliptic equations of second order.SIAM J. Numer. Anal., 46(3):1212–1249, 2008.
4.
go back to reference Susanne C. Brenner, Thirupathi Gudi, Michael Neilan, and Li-yeng Sung. C 0 penalty methods for the fully nonlinear Monge-Ampère equation. Math. Comp., 80(276):1979–1995, 2011. Susanne C. Brenner, Thirupathi Gudi, Michael Neilan, and Li-yeng Sung. C 0 penalty methods for the fully nonlinear Monge-Ampère equation. Math. Comp., 80(276):1979–1995, 2011.
5.
go back to reference Franco Brezzi and Michel Fortin. Mixed and hybrid finite element methods, volume 15 ofSpringer Series in Computational Mathematics. Springer-Verlag, New York, 1991. Franco Brezzi and Michel Fortin. Mixed and hybrid finite element methods, volume 15 ofSpringer Series in Computational Mathematics. Springer-Verlag, New York, 1991.
6.
go back to reference Luis A. Caffarelli and Xavier Cabré. Fully nonlinear elliptic equations, volume 43 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1995. Luis A. Caffarelli and Xavier Cabré. Fully nonlinear elliptic equations, volume 43 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1995.
7.
go back to reference Alexandre Ern and Jean-Luc Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004. Alexandre Ern and Jean-Luc Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.
8.
go back to reference Xiaobing Feng and Michael Neilan. Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal., 47(2):1226–1250, 2009. Xiaobing Feng and Michael Neilan. Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal., 47(2):1226–1250, 2009.
9.
go back to reference Xiaobing Feng and Michael Neilan. Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations.J. Sci. Comput., 38(1):74–98, 2009. Xiaobing Feng and Michael Neilan. Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations.J. Sci. Comput., 38(1):74–98, 2009.
10.
go back to reference Omar Lakkis and Tristan Pryer. A finite element method for second order nonvariational elliptic problems. SIAM J. Sci. Comput., 33(2):786–801, 2011. Omar Lakkis and Tristan Pryer. A finite element method for second order nonvariational elliptic problems. SIAM J. Sci. Comput., 33(2):786–801, 2011.
12.
go back to reference Anders Logg and Garth N. Wells. DOLFIN: automated finite element computing.ACM Trans. Math. Software, 37(2):Art. 20, 28, 2010. Anders Logg and Garth N. Wells. DOLFIN: automated finite element computing.ACM Trans. Math. Software, 37(2):Art. 20, 28, 2010.
13.
go back to reference Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete Contin. Dyn. Syst. Ser. B, 10(1):221–238, 2008. Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete Contin. Dyn. Syst. Ser. B, 10(1):221–238, 2008.
14.
go back to reference Tristan Pryer. Recovery methods for evolution and nonlinear problems.DPhil Thesis, University of Sussex, 2010. Tristan Pryer. Recovery methods for evolution and nonlinear problems.DPhil Thesis, University of Sussex, 2010.
Metadata
Title
Applications of Nonvariational Finite Element Methods to Monge–Ampère Type Equations
Author
T. Pryer
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-642-33134-3_47

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