GPCG–generalized preconditioned CG method and its use with non‐linear and non‐symmetric displacement decomposition preconditioners

The paper investigates a generalization of the preconditioned conjugate gradient method, which uses explicit orthogonalization of the search directions. This generalized preconditioned conjugate gradient (GPCG) method is suitable for solving the symmetric positive definite systems with preconditioners, which can be non‐symmetric or non‐linear . Such preconditioners can arise from computing of the pseudoresiduals by some additive or multiplicative space decomposition method with inexact solution of the auxiliary subproblems by inner iterations. The non‐linear and non‐symmetric preconditioners based on displacement decomposition for solving elasticity problems are described as an example of such preconditioners.