Preconditioned conjugate gradient methods for the solution of Love’s integral equation with very small parameter

Abstract

In this paper, we propose efficient numerical methods for the solution of the following Love’s integral equation f ( x ) + 1 π ∫ − 1 1 c ( x − y ) 2 + c 2 f ( y ) d y = 1 , x ∈ [ − 1 , 1 ] , where c > 0 is a very small parameter. We introduce a new unknown function h ( x ) = f ( x ) − 0 . 5 as in Lin et al. (2013), and then apply a composite Gauss–Legendre quadrature to the resulting integral equation as in Pastore (2011). The coefficient matrix of corresponding linear system is a nonsymmetric block matrix with Toeplitz blocks. We transform the nonsymmetric linear system into a symmetric linear system and introduce a preconditioner which is a block matrix with circulant blocks. Spectral properties of relevant matrices are analyzed and numerical results are presented to illustrate the efficiency of the proposed methods.