We consider the numerical construction of minimal Lagrangian graphs, which is related to recent applications in materials science, molecular engineering, and theoretical physics. It is known that this problem can be formulated as an eigenvalue problem for a fully nonlinear elliptic partial differential equation. We introduce and implement a two-step generalized finite difference method, which we prove converges to the solution of the eigenvalue problem. Numerical experiments validate this approach in a range of challenging settings. We further discuss the generalization of this new framework to Monge-Ampere type equations arising in optimal transport. This approach holds great promise for applications where the data does not naturally satisfy the mass balance condition, and for the design of numerical methods with improved stability properties.
翻译:我们考虑了与材料科学、分子工程和理论物理学最近应用有关的最小Lagrangian图形的数值构造,众所周知,这个问题可以被描述为完全非线性椭圆部分差异方程式的精华值问题。我们引入并实施了两步通用的有限差异法,我们证明这种方法与解决电子价值问题的方法一致。数字实验在一系列具有挑战性的环境中验证了这一方法。我们进一步讨论了在最佳运输中将这一新框架概括到蒙古-安珀尔型方程式的问题。在数据不能自然满足质量平衡条件的情况下,这一方法对于设计具有更好稳定性的数字方法来说很有希望。