On the Use of Rigid Body Modes in the Deflated Preconditioned Conjugate Gradient Method

Abstract

Large discontinuities in material properties, such as those encountered in composite materials, lead to ill-conditioned systems of linear equations. These discontinuities give rise to small eigenvalues that may negatively affect the convergence of iterative solution methods such as the preconditioned conjugate gradient method. This paper considers the deflated preconditioned conjugate gradient method for solving such systems. Our deflation technique uses as the deflation space the rigid body modes of sets of elements with homogeneous material properties. We show that in the deflated spectrum the small eigenvalues are mapped to zero and no longer negatively affect the convergence. We justify our approach through mathematical analysis and show with numerical experiments on both academic and realistic test problems that the convergence of our DPCG method is independent of discontinuities in the material properties.