During his outstanding career, Olivier Pironneau has addressed the solution of a large variety of problems from the Natural Sciences, Engineering and Finance to name a few, an evidence of his activity being the many articles and books he has written. It is the opinion of these authors, and former collaborators of O. Pironneau (cf. [DGP91]), that this chapter is well-suited to a volume honoring him. Indeed, the two pillars of the solution methodology that we are going to describe are: (1) a nonlinear least squares formulation in an appropriate Hilbert space, and (2) a mixed finite element approximation, reminiscent of the one used in [DGP91] and [GP79] for solving the Stokes and Navier-Stokes equations in their stream function-vorticity formulation; the contributions of O. Pironneau on the two above topics are well-known world wide. Last but not least, we will show that the solution method discussed here can be viewed as a solution method for a non-standard variant of the incompressible Navier-Stokes equations, an area where O. Pironneau has many outstanding and celebrated contributions (cf. [Pir89], for example).
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References
Th. Aubin. Nonlinear Analysis on Manifolds, Monge–Ampère Equations. Springer-Verlag, Berlin, 1982.
Th. Aubin. Some Nonlinear Problems in Riemanian Geometry. Springer-Verlag, Berlin, 1998.
J.-D. Benamou and Y. Brenier. A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math., 84(3):375–393, 2000.
X. Cabré. Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete Contin. Dyn. Syst., 8(2):331–359, 2002.
L. A. Caffarelli and X. Cabré. Fully Nonlinear Elliptic Equations. American Mathematical Society, Providence, RI, 1995.
R. Courant and D. Hilbert. Methods of Mathematical Physics, Vol. II. Wiley Interscience, New York, 1989.
M. G. Crandall, H. Ishii, and P.-L. Lions. User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.), 27(1):1–67, 1992.
L. A. Caffarelli, S. A. Kochenkgin, and V. I. Oliker. On the numerical solution of reflector design with given far field scattering data. In L. A. Caffarelli and M. Milman, editors, Monge-Ampère Equation: Application to Geometry and Optimization, pages 13–32. American Mathematical Society, Providence, RI, 1999.
E. J. Dean and R. Glowinski. Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach. C. R. Math. Acad. Sci. Paris, 336(9):779–784, 2003.
E. J. Dean and R. Glowinski. Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: a least-squares approach. C. R. Math. Acad. Sci. Paris, 339(12):887–892, 2004.
E. J. Dean and R. Glowinski. On the numerical solution of a two-dimensional Pucci’s equations with Dirichlet boundary conditions: a least-squares approach. C. R. Math. Acad. Sci. Paris, 341(6):375–380, 2005.
E. J. Dean and R. Glowinski. An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge–Ampère equation in two dimensions. Electron. Trans. Numer. Anal., 22:71–96, 2006.
E. J. Dean and R. Glowinski. Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type. Comput. Methods Appl. Mech. Engrg., 195(13–16):1344–1386, 2006.
E. J. Dean, R. Glowinski, and O. Pironneau. Iterative solution of the stream function-vorticity formulation of the Stokes problem. Applications to the numerical simulation of incompressible viscous flow. Comput. Methods Appl. Mech. Engrg., 87(2–3):117–155, 1991.
J. E. Dennis and R. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia, PA, 1996.
R. Glowinski. Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, New York, 1984.
R. Glowinski. Finite element methods for incompressible viscous flow. In P. G. Ciarlet and J.-L. Lions, editors, Handbook of Numerical Analysis, Vol. IX, pages 3–1176. North-Holland, Amsterdam, 2003.
R. Glowinski and O. Pironneau. Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Rev., 17(2):167–212, 1979.
D. Gilbarg and N. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, 2001.
R. Jansen. The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rational Mech. Anal., 101:1–27, 1988.
V. I. Oliker and L. D. Prussner. On the numerical solution of the equation (∂ 2 z∕∂x 2)(∂ 2 z∕∂y 2) − ((∂ 2 z∕∂x∂y))2 = f and its discretization, I. Numer. Math., 54(3):271–293, 1988.
O. Pironneau. Finite Element Methods for Fluids. Wiley, Chichester, 1989.
J. I. E. Urbas. Regularity of generalized solutions of Monge–Ampère equations. Math. Z., 197(3):365–393, 1988.
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Dean, E.J., Glowinski, R. (2008). On the Numerical Solution of the Elliptic Monge—Ampère Equation in Dimension Two: A Least-Squares Approach. In: Glowinski, R., Neittaanmäki, P. (eds) Partial Differential Equations. Computational Methods in Applied Sciences, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8758-5_3
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