A framework for grid solvers

The design and implementation of an object-oriented framework for grid solvers is presented. Unlike conventional sparse solvers, a grid solver solves the sparse linear system embedded in a computational grid without creating the sparse matrix explicitly at any stage of the solution process. This allows it to take advantage of additional information about the problem that is available within the grid data structure. Such an approach leads to an optimal parallel algorithm for matrix reordering by cartesian nested dissection. It also facilitates the efficient computation of the elimination tree and factorization of the sparse system by the multi-frontal method in a high-performance computing environment. In addition, it enables the construction of a structural preconditioner that is highly effective in accelerating the rates of convergence of iterative methods. The problem of non-universality of grid data structures is addressed by designing and implementing the grid solver as an object-oriented framework that can be used by a wide variety of applications through instantiation and customization. Such a framework can encompass diverse grid representations in different spatial dimensions without any loss of functionality.