We use trivariate spline functions for the numerical solution of the Dirichlet problem of the 3D elliptic Monge-Amp\'ere equation. Mainly we use the spline collocation method introduced in \cite{LL21} 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.
翻译:3D 椭圆蒙古- Amp\' ere 等式的 Dirichlet 问题的数字解决方案,我们使用三变量样条函数。 我们主要使用在\ cite{LLL21} 中引入的样条合用法来从数字上解答迭接 Poisson 方程式, 并使用平均算法来确保迭代的趋同。 我们还将在足够的条件下确定趋同率, 并提供一些数字证据来显示数字率。 然后我们提出许多计算结果, 以证明这个方法效果很好。 特别是, 我们测试了许多已知的 convex 方块和非convex 域的二次曲线共用法以及非convex 方程式, 并将它们与一些现有的数字方法进行比较, 以显示我们方法的效率和有效性。