CONVERGENCE ANALYSIS OF OVERLAPPING DOMAIN DECOMPOSITION PRECONDITIONERS FOR NONLINEAR PROBLEMS

Numerical simulations of nonlinear partial differential equations often involve solving large nonlinear systems, for which Newton's method is widely employed due to its fast convergence near the solution. However, its performance can deteriorate in the presence of strong nonlinearities or poor initial guesses. Nonlinear overlapping domain decomposition methods, such as RASPEN and substructured RASPEN (SRASPEN), have proven effective in addressing these challenges. Because SRASPEN reduces the problem size by restricting computations to a substructure, it does not update the solution outside the substructure, so that no natural initial guesses for the nonlinear local solution exists that might lead to additional inner subdomain nonlinear iterations or even prevent the local solvers to converge. In this study, we analyze the convergence of RASPEN. We show how domain decomposition improves the convergence rate of the Newton's method by highlighting the key role of the substructure on the global error contraction. Moreover, our analysis provides insight into an inexpensive modification to SRASPEN that mitigates the lack of iterations outside the substructure. The proposed variant significantly reduces computational cost while improving overall efficiency compared to existing techniques in the literature. Numerical experiments confirm the computational performance and robustness of the improved SRASPEN, establishing it as a reliable approach for solving large-scale nonlinear systems.