An iterative method for solving sparse linear algebraic systems with continuum solution dependent right-hand side for elliptic partial differential equations

Abstract

Krylov subspace iterative methods such as bi-conjugate gradients stabilized (BiCGStab) to approximately solve sparse linear algebraic systems are well known. However, there are certain instances in real-world engineering applications with underlying governing partial differential equation where the discretized right-hand side can only be exactly determined using the unavailable continuum solution. In such cases, an iterative method such as BiCGStab may not converge to a physically correct solution or may diverge completely. Such a method must be modified to accommodate inexact knowledge of the discrete right-hand side, using an updating scheme as the iteration proceeds. In this paper, we present such an updating strategy for physical problems governed by elliptic partial differential equations. This strategy must be performed in a numerically stable manner, which we also discuss. We present this as a modified BiCGStab iteration and investigate its effectiveness on both test problems, wherein it is shown to perform well and agrees with the analytical solutions, and on some more realistic problems arising in the study of Hele-Shaw flow, composite materials and power generation from wind farms.