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 [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.

翻译：我们采用三变量样条函数对三维椭圆型Monge-Ampère方程的Dirichlet问题进行数值求解。主要利用文献[SIAM J. Numerical Analysis, 2405-2434,2022]中提出的样条配置法对迭代泊松方程进行数值求解，并通过平均化算法确保迭代收敛。在充分条件下建立了收敛速率，并给出数值证据以展示数值收敛阶。随后通过大量计算结果证明该方法具有优异性能。特别地，我们测试了凸域和非凸域上多个已知的凸解与非凸解，并与若干现有数值方法进行对比，充分验证了所提方法的高效性与有效性。