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Finite element approximations of the three dimensionalMonge-Ampère equation

Published online by Cambridge University Press:  13 February 2012

Susanne Cecelia Brenner  and
Michael Neilan
Susanne Cecelia Brenner
Affiliation:
Department of Mathematics and Center for Computation & Technology, Louisiana State University, Baton Rouge, 70803 LA, USA. [email protected] ; Supported in part by the National Science Foundation under Grant Numbers DMS-07-13835 and DMS-10-16332. ,
Michael Neilan
Affiliation:
Department of Mathematics, University of Pittsburgh, 15260 PA USA; [email protected] ,
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Abstract

In this paper, we construct and analyze finite element methods for the three dimensionalMonge-Ampère equation. We derive methods using the Lagrange finite element space such thatthe resulting discrete linearizations are symmetric and stable. With this in hand, we thenprove the well-posedness of the method, as well as derive quasi-optimal error estimates.We also present some numerical experiments that back up the theoretical findings.

Keywords

Monge-Ampère equationthree dimensionsfinite element methodconvergence analysis
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 5 , September 2012 , pp. 979 - 1001
Copyright
© EDP Sciences, SMAI, 2012

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References

Barles, G. and Souganidis, P.E., Convergence of approximation schemes for fully nonlinear second order equtions. Asymptotic Anal. 4 (1991) 271–283. Google Scholar
Bernardi, C., Optimal finite element interpolation on curved domains. SIAM J. Numer. Anal. 26 (1989) 1212–1240. Google Scholar
Böhmer, K., On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 46 (2008) 1212–1249. Google Scholar
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3th edition. Springer (2008).
Brenner, S.C., Gudi, T., Neilan, M. and Sung, L.-Y., 𝒞0 penalty methods for the fully nonlinear Monge-Ampère equation. Math. Comput. 80 (2011) 1979–1995. Google Scholar
Caffarelli, L.A. and Gutiérrez, C.E., Properties of the solutions of the linearized Monge-Ampère equation. Amer. J. Math. 119 (1997) 423–465. Google Scholar
L.A. Caffarelli and M. Milman, Monge-Ampère Equation : Applications to Geometry and Optimization. Amer. Math. Soc. Providence, RI (1999).
Caffarelli, L.A., Nirenberg, L. and Spruck, J., The Dirichlet problem for nonlinear second-order elliptic equations I. Monge-Ampère equation. Comm. Pure Appl. Math. 37 (1984) 369–402. Google Scholar
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
Crandall, M.G., Ishii, H. and Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67. Google Scholar
Dean, E.J. and Glowinski, R., Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1344–1386. Google Scholar
Delzanno, G.L., Chacón, L., Finn, J.M., Chung, Y. and Lapenta, G., An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization. J. Comput. Phys. 227 (2008) 9841–9864. Google Scholar
L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics. Providence, RI. Amer. Math. Soc. 19 (1998).
Feng, X. and Neilan, M., Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations. J. Sci. Comput. 38 (2009) 74–98. Google Scholar
Feng, X. and Neilan, M., Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal. 47 (2009) 1226–1250. Google Scholar
Froese, B.D. and Oberman, A.M., Convergent finite difference solvers for viscosity solutions of the ellptic Monge-Ampère equation in dimensions two and higher. SIAM J. Numer. Anal. 49 (2011) 1692–1714. Google Scholar
Froese, B.D. and Oberman, A.M., Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation. J. Comput. Phys. 230 (2011) 818–834. Google Scholar
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (2001).
P. Grisvard, Elliptic Problems on Nonsmooth Domains. Pitman Publishing Inc. (1985).
C.E. Gutiérrez, The Monge-Ampère Equation, Progress in Nonlinear Differential Equations and Their Applications 44. Birkhauser, Boston, MA (2001).
T. Muir, A Treatise on the Theory of Determinants. Dover Publications Inc., New York (1960).
Neilan, M., A nonconforming Morley finite element method for the fully nonlinear Monge-Ampère equation. Numer. Math. 115 (2010) 371–394. Google Scholar
M. Neilan, A unified analysis of some finite element methods for the Monge-Ampère equation. Submitted.
Nitsche, J.A., Über ein Variationspirinzip zur Lösung Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unteworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15. Google Scholar
Oberman, A.M., 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 (2008) 221–238. Google Scholar
Sorensen, D.C. and Glowinski, R., A quadratically constrained minimization problem arising from PDE of Monge-Ampère type. Numer. Algorithm 53 (2010) 53–66. Google Scholar
N.S. Trudinger and X.-J. Wang, The Monge-Ampère equation and its geometric applications, Handbook of Geometric Analysis I. International Press (2008) 467–524.
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics. Providence, RI. Amer. Math. Soc. 58 (2003).
Ženíšek, A., Polynomial approximation on tetrahedrons in the finite element method. J. Approx. Theory 7 (1973) 334–351. Google Scholar
Zheligovsky, V., Podvigina, O. and Frisch, U., The Monge-Ampère equation : various forms and numerical solutions. J. Comput. Phys. 229 (2010) 5043–5061. Google Scholar