Optimal Coarse Grid Size in Domain Decomposition (Journal of Computational Mathematics, 2021)
In most domain decomposition (DD) methods, a coarse grid solve is employed to provide the global coupling required to produce an $optimal$ method. The total cost of a method can depend sensitively on the choice of the coarse grid size $H$. In this paper, we give a simple analysis of this phenomenon for a model elliptic problem and a variant of Smith's vertex space domain decomposition method [11, 3]. We derive the optimal value $H_{opt}$, which asymptotically minimizes the total cost of method (number of floating point operations in the sequential case and execution time in the parallel case), for subdomain solvers with different complexities, Using the value of $H_{opt}$, we derive the overall complexity of the DD method, which can be significantly lower than that of the subdomain solver.