Iterative Solution of Piecewise Linear Systems for the Numerical Solution of Obstacle Problems

Luigi Brugnano and Alessandra Sestini
Dipartimento di Matematica “U. Dini” Viale Morgagni 67/A, 50134 Firenze, Italy
Dedicated to Prof. D. Trigiante on the occasion of his 65th birthday.
Received 16 December, 2009; accepted in revised form 24 September, 2011
Abstract: We investigate the use of piecewise linear systems, whose coefficient matrix is a
piecewise constant function of the solution itself. Such systems arise, for example, from
the numerical solution of linear complementarity problems and in the numerical solution
of free-surface problems. In particular, we here study their application to the numerical
solution of both the (linear) parabolic obstacle problem and the obstacle problem. We
propose a class of effective semi-iterative Newton-type methods to find the exact solution
of such piecewise linear systems. We prove that the semi-iterative Newton-type methods
have a global monotonic convergence property, i.e., the iterates converge monotonically
to the exact solution in a finite number of steps. Numerical examples are presented to
demonstrate the effectiveness of the proposed methods.


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