An inexact Newton method for systems arising from the finite element method

In this paper, we introduce an efficient and robust technique for approximating the Jacobian matrix for a nonlinear system of algebraic equations which arises from the finite element discretization of a system of nonlinear partial differential equations.

It is demonstrated that when an iterative solver, such as preconditioned GMRES, is used to solve the linear systems of equations that result from the application of Newton's method, this approach is generally more efficient than using matrix-free techniques: the price paid being the extra memory requirement for storing the sparse Jacobian. The advantages of this approach over attempting to calculate the Jacobian exactly or of using other approximations are also discussed. A numerical example is included which is based upon the solution of a 2-d compressible viscous flow problem.