Hybridizable discontinuous Galerkin methods for the two-dimensional Monge-Ampère equation (Q6592266)

The authors consider two hybridizable discontinuous Galerkin (HDG) methods that are appropriate for the numerical solution of the Monge-Ampère equation in two dimensions. The paper is split into two parts. In the first part (Section 2), the Monge-Ampère equation is written as a first-order nonlinear system of equations, which is solved in two different ways that are both related to HDG methods. The first HDG method is derived by solving the Monge-Ampère equation using Newton's method directly on the first-order system. The second HDG method decouples the computation of the Hessian from the remaining variables and it is based on the solution of a of Poisson equation for a sequence of right hand side vectors, until convergence to a fixed-point solution of the Monge-Ampère equation is achieved. Several interesting numerical examples are presented. Finally, in Section 3, the authors consider the use of transport theory to r-adaptive mesh generation. This approach requires the solution of the Monge-Ampère equation with nonlinear Neumann boundary condition, which is solved by extending the HDG methods described in Section 2. Several numerical examples to demonstrate the performance of the HDG methods for r-adaptive mesh generation are also presented.