Iterative Parallel Methods for Boundary Value Problems

A bordered almost block diagonal system (BABD) results from discretizing and linearizing ordinary diierential equation (ODE) boundary value problems (BVPs) with non-separated boundary conditions (BCs) by either spline collocation, nite diierences, or multiple shooting. After internal condensation, if necessary, this BABD system reduces to a standard-nite diierence BABD structure. This system can be solved either using a \direct" divide-and-conquer approach or an iterative scheme such as preconditioned conjugate gradients (PCG). Preconditioners approximating the inverse of the nite diierence operator are eeective and can be computed and applied eeciently in a parallel environment. We present theoretical computational costs, comparing direct and iterative methods, and numerical results computed on a Sequent Symmetry shared memory computer. These demonstrate that the PCG method can outperform the divide-and-conquer approach on systems with many processors when approximating large diierential systems. Also, the PCG method \scales up" better than the implemented divide-and-conquer method.