Schur complement preconditioned conjugate gradient methods for spline collocation equations

Abstract

We are interested in the efficient solution of linear second order Partial Differential Equation (PDE) problems on rectangular domains. The PDE discretisation scheme used is of Finite Element type and is based on quadratic splines and the collocation methodology. We integrate the Quadratic Spline Collocation (QSC) discretisation scheme with a Domain Decomposition (DD) technique. We develop DD motivated orderings of the QSC equations and unknowns and apply the Preconditioned Conjugate Gradient (PCG) method for the solution of the Schur Complement (SC) system. Our experiments show that the SC-PCG-QSC method in its sequential mode is very efficient compared to standard direct band solvers for the QSC equations. We have implemented the SC-PCG-QSC method on the iPSC/2 hypercube and present performance evaluation results for up to 32 processors configurations. We discuss a type of nearest neighbour communication scheme, in which the amount of data transfer per processor does not grow with the number of processors. The estimated efficiencies are at the order of 90%.