A fast multiscale solver for modified Hammerstein equations

This paper presents a fast solver, called the multilevel augmentation method, for modified nonlinear Hammerstein equations. When we utilize the method to solve a large scale problem, most of components of the solution can be computed directly, and lower frequency components can be obtained by solving a fixed low-order algebraic nonlinear system. The advantage of using the algorithm to modified equations is that it leads to reduce the cost of numerical integrations greatly. The optimal error estimate of the method is established and the nearly linear computational complexity is proved. Finally, numerical examples are presented to confirm the theoretical results and illustrate the efficiency of the method.