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Iterative solution of piecewise linear systems for the numerical solution of obstacle problems. (English) Zbl 1432.65079
Summary: 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.

MSC:
65K10 Numerical optimization and variational techniques
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C53 Methods of quasi-Newton type
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