## 65H20 Global methods, including homotopy approaches [See also 58C30, 90C30]

### Refine

#### Keywords

- Affine invariant Newton methods (1)
- Euler discretization (1)
- adaptive trust region methods (1)
- contractivity of ordinary differential equations (1)
- global Newton methods (1)
- inexact Newton methods (1)
- lar (1)
- linearly implicit (1)
- nonlinear partial differential equa (1)
- pseudo--transient continuation (1)

#### Institute

- Computational Medicine (1)
- ZIB Allgemein (1)

The paper deals with three different Newton algorithms that have recently been worked out in the general frame of affine invariance. Of particular interest is their performance in the numerical solution of discretized boundary value problems (BVPs) for nonlinear partial differential equations (PDEs). Exact Newton methods, where the arising linear systems are solved by direct elimination, and inexact Newton methods, where an inner iteration is used instead, are synoptically presented, both in affine invariant convergence theory and in numerical experiments. The three types of algorithms are: (a) affine covariant (formerly just called affine invariant) Newton algorithms, oriented toward the iterative errors, (b) affine contravariant Newton algorithms, based on iterative residual norms, and (c) affine conjugate Newton algorithms for convex optimization problems and discrete nonlinear elliptic PDEs.

Pseudo--transient continuation methods are quite popular for the numerical solution of steady state problems, typically in PDEs. They are based on an embedding into a time dependent initial value problem. In the presence of dynamical invariants the Jacobian matrix of the nonlinear equation system is bound to be singular. The paper presents a convergence analysis which takes this property into account -- in contrast to known approaches. On the basis of the new analysis adaptive algorithms are suggested in detail. These include a variant with Jacobian approximations as well as inexact pseudo--transient continuation, both of which play an important role in discretized PDEs. Numerical experiments are left to future work.