Conjugate Gradient Methods for Spline Collocation Equations

Abstract

We study the parallel computation of linear second order elliptic Partial Differential Equation (PDE) problems in rectangular domains. We discuss the application of Conjugate Gradient (CG) and Preconditioned Conjugate Gradient (PCG) methods to the linear system arising from the discretisation of such problems using quadratic splines and the collocation discretisation methodology. Our experiments show that the number of iterations required for convergence of CG-QSC (Conjugate Gradient applied to Quadratic Spline Collocation equations) grows linearly with the square root of the number of equations. We implemented the CG and PCG methods for the solution of the Quadratic Spline Collocation (QSC) equations on the iPSC/2 hypercube and present performance evaluation results for up to 32 processors configurations. Our experiments show efficiencies of the order of 90%, for both the fixed and scaled speedups.