Numerical solution of obstacle and parabolic obstacle problems based on piecewise linear systems

Piecewise Linear Systems (PLSs) are linear systems whose coefficient matrix is a piecewise constant function of the solution itself. Their general formulation has been introduced in [1] and their application toflowsin porous media has already been studied in [2]. Here we consider another important application of such kind of systems, that is the numerical solution of obstacle and parabolic obstacle problems. The discrete formulation of such problems is expressed as a linear complementarity problem and it is then formulated as a specific PLS for the elliptic case and as a finite sequence of such systems for the parabolic case. A semi-iterative Newton-type method is proposed for the solution of the obtained PLSs and it is possible to prove that monotonic convergence in afinitenumber of steps is guaranteed. Some numerical results are presented to show the effectiveness of the proposed approach.