收敛的半拉格朗日方法用于无结构网格上的Monge-Ampère方程
Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids
AI总结：
本文提出并分析了用于一般三角网格上非线性椭圆Monge-Ampère方程的收敛半拉格朗日方法，通过建立等价的Hamilton-Jacobi-Bellman形式，消除了凸性约束，并利用半拉格朗日方案求解Monge-Ampère方程。
摘要：
本文致力于开发和分析用于一般三角网格上的非线性椭圆Monge-Ampère方程的收敛半拉格朗日方法。通过建立Monge-Ampère方程的等价（在粘性意义下）的Hamilton-Jacobi-Bellman形式，显著的好处是去除了可接受空间中的凸性约束，因为凸性成为新形式的内置属性。此外，这种方法允许利用Hamilton-Jacobi-Bellman方程的丰富数值方法，如半拉格朗日方案，来求解Monge-Ampère类型方程。证明所考虑的数值方法是单调的、点一致一致的和均匀稳定的。因此，其解均匀收敛到Monge-AmpèreDirichlet问题的唯一凸粘性解。利用一个超线性收敛的Howard算法（一种牛顿型方法）作为非线性求解器，以利用方案的单调性。还进行了数值实验，以评估所提出的数值方法和非线性求解器的性能。
英文摘要：
This paper is concerned with developing and analyzing convergent semi-Lagrangian methods for the fully nonlinear elliptic Monge-Ampère equation on general triangular grids. This is done by establishing an equivalent (in the viscosity sense) Hamilton-Jacobi-Bellman formulation of the Monge-Ampère equation. A significant benefit of the reformulation is the removal of the convexity constraint from the admissible space as convexity becomes a built-in property of the new formulation. Moreover, this new approach allows one to tap the wealthy numerical methods, such as semi-Lagrangian schemes, for Hamilton-Jacobi-Bellman equations to solve Monge-Ampère type equations. It is proved that the considered numerical methods are monotone, pointwise consistent and uniformly stable. Consequently, its solutions converge uniformly to the unique convex viscosity solution of the Monge-Ampère Dirichlet problem. A super-linearly convergent Howard's algorithm, which is a Newton type method, is utilized as the nonlinear solver to take advantage of the monotonicity of the scheme. Numerical experiments are also presented to gauge the performance of the proposed numerical method and the nonlinear solver.
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