Domain Decomposition Methods for the Monge-Amp\`ere equation

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.