An efficient approach for the numerical solution of the Monge-Ampère equation
The authors present an alternative method to compute the numerical solution of the elliptic Monge-Ampère equation \[ \rho _1\left( \nabla \Psi \left( \xi \right) \right) \det \nabla ^2\Psi \left( \xi \right) =\rho _0\left( \xi \right). \] The alternative aproach is based on finding a steady-state solution of the logarithmic parabolic Monge-Ampère equation \[ \frac{\partial \Psi }{\partial t} = \log \left( \frac{\rho _1\left( \nabla \Psi \left( \xi \right) \right) \det \nabla ^2\Psi \left( \xi \right) }{\rho _0\left( \xi \right) }\right), \] together with some suitable initial and boundary conditions. The authors study the problem of global existence, uniqueness and convergence of the solution of the fully nonlinear parabolic equation to the unique solution of the elliptic Monge-Ampère equation. Some numerical experiments illustrate the convergence and the regularity of the numerical solution.
logarithmic parabolic Monge-Ampère equation
scientific article; zbMATH DE number 5850232
scientific article