SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1979-2003, August 2024.
Abstract. We introduce a new overlapping domain decomposition method (DDM) to solve fully nonlinear elliptic partial differential equations (PDEs) approximated with monotone schemes. While DDMs have been extensively studied for linear problems, their application to fully nonlinear PDEs 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. We also provide a specific implementation for the Monge–Ampère equation. 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.

中文翻译：

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Monge-Ampère 方程的域分解方法


SIAM 数值分析杂志，第 62 卷，第 4 期，第 1979-2003 页，2024 年 8 月。

抽象的。我们引入了一种新的重叠域分解方法（DDM）来求解用单调方案近似的完全非线性椭圆偏微分方程（PDE）。虽然 DDM 已针对线性问题进行了广泛研究，但其在完全非线性偏微分方程中的应用在文献中仍然有限。为了解决这一差距，我们使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。我们还提供了 Monge-Ampère 方程的具体实现。进行了几次数值测试来验证收敛定理。这些数值实验涉及不同规律的例子。计算实验表明该方法高效、鲁棒，并且需要相对较少的迭代来收敛。结果揭示了 DDM 方法的巨大潜力，可以为使用现有求解方法在计算上难以解决的大规模问题提供高效且可并行的求解器。