Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
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Adaptive in-time local grid refinement techniques use multilevel local discretizations designed to achieve local accuracy. The changing nature of the matrix structure of the linear systems arising from the multilevel local discretizations requires flexible approximate factorizations that focus on local components and coordinate their interaction. The solution of these composite grid systems with Krylov solvers is considered. The selection of an adequate preconditioner is crucial. The incomplete Cholesky (IC) factorization of the composite matrix and the inexact BEPS preconditioner are two such potential preconditioners. The inexact BEPS preconditioner can be constructed and applied with significantly more flexibility than the IC factorizations of the composite multilevel grid matrix.An extension of the IC factorizations for matrices arising from discretizations of self-adjoint PDEs on multilevel composite grids is proposed. The resulting factorization is referred to as the incomplete multilevel Cholesky (IMC) factorization. The IMC factorization is spectrally equivalent to the IC factorization of the matrix of the composite grid system.IMC factorization can be constructed with the same flexibility as the inexact BEPS preconditioner. The application of IMC factorization is achieved via a multilevel LLT-cycle consisting of a forward elimination pass proceeding downward on the grids from fine to coarse followed by an reverse-order upward back substitution pass.The application of multilevel factorization as a preconditioner requires roughly one-half as many operations as the inexact BEPS preconditioner. The numerical results illustrate the potential of the method.