We introduce a new overlapping Domain Decomposition Method (DDM) to solve the fully nonlinear Monge-Amp\`ere equation. While DDMs have been extensively studied for linear problems, their application to fully nonlinear partial differential equations (PDE) remains limited in the literature. To address this gap, we establish a proof of global convergence of these new iterative algorithms using a discrete comparison principle argument. Several numerical tests are performed to validate the convergence theorem. These numerical experiments involve examples of varying regularity. Computational experiments show that method is efficient, robust, and requires relatively few iterations to converge. The results reveal great potential for DDM methods to lead to highly efficient and parallelizable solvers for large-scale problems that are computationally intractable using existing solution methods.

翻译：我们提出了一种新的重叠域分解方法（DDM）来求解完全非线性的Monge-Ampère方程。尽管DDM在线性问题中已被广泛研究，但其在完全非线性偏微分方程（PDE）中的应用在文献中仍然有限。为弥补这一空白，我们利用离散比较原理论证，建立了这些新迭代算法的全局收敛性证明。通过多项数值测试验证了收敛定理，这些数值实验涵盖了不同正则性的例子。计算实验表明，该方法高效、鲁棒，且仅需相对较少的迭代次数即可收敛。结果揭示了DDM方法在开发用于大规模问题的高效可并行求解器方面具有巨大潜力，而这类问题使用现有求解方法在计算上难以处理。