Newton method is a widely used iteration method in solving nonlinear algebraic equations. In this method, a linear algebraic equations need to be solved in every step. The coefficient matrix of the algebraic equations is so-called Jacobian matrix, which needs to be determined at every step. For a complex non-linear system, usually no explicit form of Jacobian matrix can be found. Several methods are introduced to obtain an approximated matrix, which are classified as Jacobian-free method. The finite difference method is used to approximate the derivatives in Jacobian matrix, and a small parameter is needed in this process. Some problems may arise because of the interaction of this parameter and round-off errors. In the present work, we show that this kind of Newton method may encounter difficulties in solving non-linear partial differential equation (PDE) on fine mesh. To avoid this problem, the continuous Newton method is presented, which is a modification of classical Newton method for non-linear PDE.
Identification of Flexural Rigidity in a Kirchhoff Plates Model Using a Convex Objective and Continuous Newton Method
