This is an example of the Monge-Ampere equation from "C0 penalty methods for the fully nonlinear Monge-Ampere equation, Mathematics of Computation, 80:1979-1995, 2011." by S.C. Brenner, T. Gudi, M. Neilan, and L.-Y. Sung.

So the Monge-Ampere equation is: $$\det(D^2u) = f \quad \textrm{in} \quad \Omega \quad (1.1)$$ $$ u = g \quad \textrm{on} \quad \partial{\Omega} \quad (1.1)$$ where $$\det(D^2u) = \frac{\partial^2{u}}{\partial{x_1^2}} \frac{\partial^2{u}}{\partial{x_2^2}} - (\frac{\partial^2{u}}{\partial{x_1x_2}})^2$$ This is given. The specific example says " we solve (2.7) for varying values of h and k, and choose our data such that the exact solution to the Monge-Ampere equation (1.1) is $u = 20e^{x_1^6/6+x_2}$. We take $Ω = (0,1)^2$, the unit square and set $σ = 100$." I'm understanding that u is the true solution. By the boundary of the domain you will get $g \rightarrow u \rightarrow f$ (Not that they map to each other I mean that, you will get them in that sequential order). But I'm still not exactly sure how to approach this to solve it numerically. The Monge-Ampere questions are generally approached with an elliptical PDE, which is what I'm assuming the $\Omega$ refers too, but it's the other parts that are confusing.

  • $\begingroup$ How about standard finite elements? Multiply by a test function, integrate by parts, linearize, etc. $\endgroup$ – knl Jul 13 '18 at 20:19

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