论文标题
用于3D Monge-ampère方程的数值溶液的微粒样条搭配方法
Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère Equation
论文作者
论文摘要
我们使用三个椭圆形蒙奇 - 安培方程的dirichlet问题的数值解。主要是我们使用[Siam J. J.数值分析，2405-2434,2022]中引入的样条搭配方法来求解迭代泊松方程，并使用平均算法来确保迭代的收敛性。我们还应在充分条件下建立收敛速度，并提供一些数值证据以显示数值率。然后，我们提出许多计算结果，以证明这种方法效果很好。特别是，我们测试了许多已知的凸解决方案以及凸和非凸域上的非凸溶液，并将它们与几种现有的数值方法进行了比较，以显示我们方法的效率和有效性。
We use trivariate spline functions for the numerical solution of the Dirichlet problem of the 3D elliptic Monge-Ampére equation. Mainly we use the spline collocation method introduced in [SIAM J. Numerical Analysis, 2405-2434,2022] to numerically solve iterative Poisson equations and use an averaged algorithm to ensure the convergence of the iterations. We shall also establish the rate of convergence under a sufficient condition and provide some numerical evidence to show the numerical rates. Then we present many computational results to demonstrate that this approach works very well. In particular, we tested many known convex solutions as well as nonconvex solutions over convex and nonconvex domains and compared them with several existing numerical methods to show the efficiency and effectiveness of our approach.
