Newton-Krylov-FAC methods for problems discretized on locally refined grids

Many problems in computational science and engineering are nonlinear and time-dependent. The solutions to these problems may include spatially localized features, such as boundary layers or sharp fronts, that require very fine grids to resolve. In many cases, it is impractical or prohibitively expensive to resolve these features with a globally fine grid, especially in three dimensions. Adaptive mesh refinement (AMR) is a dynamic gridding approach that employs a fine grid only where necessary to resolve such features. Numerous AMR codes exist for solving hyperbolic problems with explicit time stepping and some classes of linear elliptic problems. Researchers have paid much less attention to the development of AMR algorithms for the implicit solution of systems of nonlinear equations. Recent efforts encompassing a variety of applications demonstrate that Newton-Krylov methods are effective when combined with multigrid preconditioners. This suggests that hierarchical methods, such as the Fast Adaptive Composite grid (FAC) method of McCormick and Thomas, can provide effective preconditioning for problems discretized on locally refined grids. In this paper, we address algorithm and implementation issues for the use of Newton-Krylov-FAC methods on structured AMR grids. In our software infrastructure, we combine nonlinear solvers from KINSOL and PETSc with the SAMRAI AMR library, and include capabilities for implicit time stepping. We have obtained convergence rates independent of the number of grid refinement levels for simple, nonlinear, Poisson-like, problems. Additional efforts to employ this infrastructure in new applications are underway. Communicated by: G. Wittum