An Efficient Solver for Multi--Right-Hand-Side Linear Systems Based on the CCCG($\eta$) Method with Applications to Implicit Time-Dependent Partial Differential Equations

Abstract

The controlled Cholesky factorization (CCF) has been shown to be a robust preconditioner for the conjugate gradient (CG) method. In this scheme the amount of fill-in is defined in terms of a parameter $\eta$, the number of extra nonzero elements allowed per column. This parameter $\eta$ can be chosen in such a way that the preconditioning factor will require more or less storage than the coefficient matrix. The selection of nonzero elements kept for the preconditioner is made by value instead of by position, so the sparsity pattern of the original matrix is never considered by the proposed factorization. This preconditioned method is termed the CCCG($\eta$) method and is a generalization of a method developed by Jones and Plassmann [\textit{ACM Trans.\ Math.\ Software}, 21 (1995), pp.\ 5--17]. It is demonstrated how an optimum value of $\eta$ can be automatically determined when solving systems with multiple right-hand sides. Typically such systems occur when using time-stepping procedures. Numerical examples are presented for some time-dependent parabolic partial differential equations discretized using linear triangular finite elements on unstructured grids. Comparisons made between CCCG($\eta$), standard ICCG, and the Jones and Plassmann method show CCCG($\eta$) to be an efficient solver.