We introduce two hybridizable discontinuous Galerkin (HDG) methods for numerically solving the Monge-Ampere equation. The first HDG method is devised to solve the nonlinear elliptic Monge-Ampere equation by using Newton's method. The second HDG method is devised to solve a sequence of the Poisson equation until convergence to a fixed-point solution of the Monge-Ampere equation is reached. Numerical examples are presented to demonstrate the convergence and accuracy of the HDG methods. Furthermore, the HDG methods are applied to r-adaptive mesh generation by redistributing a given scalar density function via the optimal transport theory. This r-adaptivity methodology leads to the Monge-Ampere equation with a nonlinear Neumann boundary condition arising from the optimal transport of the density function to conform the resulting high-order mesh to the boundary. Hence, we extend the HDG methods to treat the nonlinear Neumann boundary condition. Numerical experiments are presented to illustrate the generation of r-adaptive high-order meshes on planar and curved domains.

翻译：我们提出了两种可杂交间断伽辽金（HDG）方法，用于数值求解蒙日-安培方程。第一种HDG方法通过使用牛顿法求解非线性椭圆型蒙日-安培方程。第二种HDG方法通过求解一系列泊松方程，直至收敛到蒙日-安培方程的定点解。数值算例展示了HDG方法的收敛性和精度。此外，我们将HDG方法应用于基于最优输运理论的r-自适应网格生成，通过重新分布给定的标量密度函数实现。该r-自适应方法通过密度函数的最优输运导出具有非线性诺伊曼边界条件的蒙日-安培方程，使生成的高阶网格贴合边界。因此，我们扩展了HDG方法以处理非线性诺伊曼边界条件。数值实验展示了在平面和曲面区域上生成r-自适应高阶网格的效果。