Fast Multilevel Augmentation Methods for Solving Hammerstein Equations

• Authors: Zhongying Chen;Bin Wu;Yuesheng Xu
• Venue: SIAM Journal on Numerical Analysis
• Year: 2009

Fast multilevel augmentation methods based on projections for solving Hammerstein equations are developed. Each of these methods requires availability of a multilevel decomposition of the solution space and a projection from the solution space onto a finite dimensional subspace. A subspace at a level is obtained from the subspace at the previous level by adding a difference subspace. Accordingly, the projection equation at the present level is obtained by augmenting the projection equation at the previous level. Using this idea recursively, we solve the nonlinear equation (by an iteration method) only at an initial lower resolution level while obtaining its solution at a higher resolution level. With the help of the multiscale analysis, we separate the procedure of solving the nonlinear operator equation at a high level into two major components: (1) solving the nonlinear equation only at an initial lower level, and (2) compensating the error by solving a linear system at the high level. We prove that the proposed methods require only linear (up to a logarithmic factor) computational complexity and have the optimal convergence order. A relationship between the proposed method and the multigrid method is discussed. Two specific fast methods based on the Galerkin projection and the collocation projection are developed. Numerical results are presented to confirm the theoretical estimates, with comparisons to the two-grid method.