This Small Business Innovative Research Phase I project is testing and development of robust algebraic multigrid software for solving unstructured-mesh problems, designed to run in both serial and parallel environments, and which efficiently incorporates local refinement strategies in the solution process. Many commercial and government codes, including those in hydrodynamics, heat conduction and multigroup thermal radiation transport, require solution of second-order elliptic partial differential equations, and often employ unstructured discretizations in order to resolve complex geometries and to facilitate adaptive refinement. A common solution strategy is to set up an outer iteration (e.g., Newton's or Newton-like methods, preconditioned conjugate gradient, implicit time-stepping) in which each step requires the solution of discretized second-order elliptic partial differential equations. Current solution methods are limited, with convergence degrading with problem size, time steps, and highly resolved local refinement levels, so that solution of realistic problems is prohibitively expensive. To counter these difficulties, algebraic multigrid (AMG) uses the principles of standard multigrid methods, but with an automated preprocessing phase in which the various multigrid components are constructed. This leads to a very efficient and robust solution method for such problems which is ideally suited for unstructured grids since all of the processing is based on the matrices, thus eliminating the usual multigrid requirement that the user supply coarser discretization levels, which would be prohibitive or impossible for most unstructured mesh applications. In Phase I, the emphasis will be on testing existing AMG codes on problems of interest, determining efficient ways to incorporate local refinement strategies in the solution proven, and extending multigrid parallel implementation strategies to AMG. Robust algebraic multigrid software that is easily combined with existi ng codes has the potential to greatly enhance the range of problems (in size and complexity) that can be efficiently solved. Unstructured meshed at used widely in codes for bow the public and private sectors in many applications. and these codes would benefit greatly by the availability of robust AMG solvers
