A combined Newton and load increment method for solving nonlinear equations
O. Axelsson

Severely nonlinear problems can only be solved by some homotopy continuation method. An example of a homotopy, method is the continuous Newton method which, however, must be discretized which leads to the damped step version of Newtons method.

The standard Newton iteration method for solving systems of nonlinear equations F(x) = 0 must be modified in order to get global convergence, i.e. convergence from any initial point. Furthermore, the applicability of the method is also restricted in as much as it assumes a nonsingular and everywhere differentiable mapping F(·). Such issues are of particular importance in various problems in large scale Scientific Computation. The control of steplengths is the damped step Newton method can lead to many small steps and slow convergence.

A new method in the form of a coupled Newton and load increment method is presented and shown to have a global convergence already from the start and second order of accuracy with respect to the load increment step and with less restrictive regularity assumptions. An adaptive control of the method is also presented.

Preconditioning of systems arising from finite element discretizations of phase-field models
M. Neytcheva     (Joint work with O. Axelsson, P. Boyanova and M. Do-Quang)

Flows dominated by capillarity and wetting are important in many processes in nature and are of increasing interest, in particular, in microfluidic applications. Accurate and computationally efficient numerical simulations of such flows remain a challenging task and put extra demands on the numerical solution methods used.

In this talk we consider preconditioned iterative solution methods to solve the algebraic systems of equations arising from finite element discretizations of multiphase flow problems, modeled using the phase-field model, coupled with the Navier-Stokes equation. The model allows us to simulate the motion of a free surface in the presence of surface tension and surface chemistry energy.

We focus on the phase-field model, described by the Cahn-Hilliard equation. The problem is time-dependent and nonlinear. We consider the task to solve the linear systems with the arising Jacobian matrices by a preconditioned iterative solution method. When discretized, the Cahn-Hilliard equation gives raise to large scale algebraic systems of equations with matrices of a particular two-by-two block form. The block structure is utilized when constructing preconditioners to be used in an iterative procedure. Interestingly enough, the preconditioning techniques are applicable to solving systems with symmetric complex matrices as well as for problems arising from some optimization problems with a constraint given by a partial differential equation.

We illustrate the performance of the preconditioners with several numerical experiments.