一种用于带有传输边界的Monge-Ampère方程的有限元方法
A finite element method for the Monge-Ampère equation with transport boundary conditions
AI总结：
本文提出了一种基于Galerkin方法求解带有传输边界的Monge-Ampère方程的有限元方法，通过牛顿-拉夫森迭代线性化问题，采用非变分有限元方法得到最优收敛率，并通过数值测试展示其在光学和网格运动中的应用潜力。
摘要：
我们针对最优质量运输、几何光学和计算网格或网格运动技术中出现的带有传输边界的Monge-Ampère方程，采用Galerkin类型方法进行数值求解。该 fully nonlinear elliptic 问题可通过牛顿-拉夫森迭代线性化，导致非发散形式椭圆方程的斜导数边界值问题。我们通过采用非变分有限元方法进行离散化，经经验观察得到最优收敛率，前提是使用恢复技术近似未知函数的梯度和Hessian。我们提供了广泛的数值测试，以说明我们方法的优势及其在光学和网格运动中的潜在应用。
英文摘要：
We address the numerical solution via Galerkin type methods of the Monge-Ampère equation with transport boundary conditions arising in optimal mass transport, geometric optics and computational mesh or grid movement techniques. This fully nonlinear elliptic problem admits a linearisation via a Newton-Raphson iteration, which leads to an oblique derivative boundary value problem for elliptic equations in nondivergence form. We discretise these by employing the nonvariational finite element method, which lead to empirically observed optimal convergence rates, provided recovery techinques are used to approximate the gradient and the Hessian of the unknown functions. We provide extensive numerical testing to illustrate the strengths of our approach and the potential applications in optics and mesh movement.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
