Study on the convergence criterion of linear iteration in inexact Newton methods

Newton-like iteration methods are usually used for implicit time stepping to solve large scale nonlinear systems. Each of the nonlinear iterative step requires solving large linear equations composed of a Jacobian matrix of the nonlinear system, and the error of solving such linear equations can have a significant impact on the convergence of the nonlinear system. However, the convergence criterion of the linear iteration is less studied when the Jacobian matrix error exists in the Newton method. Aiming at the above problem, this work first presents the Newton iterative formula with both the Jacobian matrix error and the linear iteration error, and verifies the significant influence of the Jacobian matrix error on the iteration through numerical tests. Then, two different types of linear iterative convergence criteria are tested numerically, and the produced over-solving problem is investigated at the presence of Jacobian matrix error. Finally, a new convergence criterion of the linear iteration is developed based on two of the previous convergence criteria. Numerical tests suggest that the proposed method is effective to relieve over-solving problems, thus improves the computational efficiency.