Quadratic finite element approximations of the Monge-Ampère equation
The paper deals with the Dirichlet boundary value problem for the fully nonlinear Monge-Ampère equation \[ \text{det}\big(D^2u\big)=f \; \text{in} \; \Omega, \] \[ u=g \; \text{on} \; \partial \Omega, \] where \(\Omega \in \mathbb{R}^2\) is a smooth strictly convex domain, \(f>0\), \(g\) are given functions, and \(\text{det}\big(D^2u\big)\) denotes the determinant of the Hessian matrix \(D^2u\) of the unknown function. The author considers the finite element method for this problem, shows the convergence of quadratic finite element approximations, obtains quasi-optimal error estimates in a \(W^{2,p}\)-type norm, localizes pointwise error estimates, and the convergence of Newton's method. Numerical experiments are presented which confirm the theoretical results.
scientific article; zbMATH DE number 6134443
scientific article