We investigate the use of piecewise linear systems, whose coecient 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 eective semi-iterative Newton-type methods to nd 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 nite number of steps. Numerical examples are presented to demonstrate the eectiveness of the proposed methods.
Iterative solution of Piecewise Linear Systems for the numerical solution of obstacle problems / L.Brugnano; A.Sestini. - In: JOURNAL OF NUMERICAL ANALYSIS,INDUSTRIAL AND APPLIED MATHEMATICS. - ISSN 1790-8140. - STAMPA. - 6:(2011), pp. 67-82.
Iterative solution of Piecewise Linear Systems for the numerical solution of obstacle problems
BRUGNANO, LUIGI;SESTINI, ALESSANDRA
2011
Abstract
We investigate the use of piecewise linear systems, whose coecient 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 eective semi-iterative Newton-type methods to nd 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 nite number of steps. Numerical examples are presented to demonstrate the eectiveness of the proposed methods.File | Dimensione | Formato | |
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