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On the updating and assembly of the Hessian matrix in finite element methods. (English) Zbl 0673.65068
This paper is concerned with the iterative solution of systems of nonlinear equations arising in finite element approximations of some nonlinear problems such as an electromagnetic field equation, the torsion of an elastic bar, the full potential equation in aerodynamics. The authors discuss Newton methods involving the assembly of the Hessian matrix. In general, such methods as the damped inexact Newton procedure are expensive to use because the Hessian matrix has to be updated too many times. On the other hand, it is shown that, when the Hessian matrix has a special structure, updating it is not more expensive than updating the gradient vector. Effective methods of evaluation, factorization and preconditioning of a stiffness matrix are given. Several numerical examples are presented.
Reviewer: V.V.Kobkov

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65H10 Numerical computation of solutions to systems of equations
35Q99 Partial differential equations of mathematical physics and other areas of application
35J65 Nonlinear boundary value problems for linear elliptic equations
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