A direct algorithm in some free boundary problems

Abstract

Abstract. In this paper we propose a new algorithm for the well known elliptic obstacle problem and for parabolic variational inequalities like one and two phase Stefan problem and of obstacle type. Our approach enters the category of fixed domain methods and solves just linear elliptic or parabolic equations and their discretization at each iteration. We prove stability and convergence properties. The approximating coincidence set is explicitly computed and it converges in the Hausdorff-Pompeiu sense to the searched geometry. In the numerical examples, the algorithm has a very fast convergence and the obtained solutions (including the free boundaries) are accurate.

A Penalization Method for the Elliptic Bilateral Obstacle Problem

“…In [18], we have tested numerically with positive results the stability of a similar algorithm when f the right-hand side in (1) is perturbed. Test 2.…”

“…Moreover, we need just a scalar penalization parameter in our method. A similar strategy was employed in [18] for the elliptic unilateral obstacle problem and for parabolic variational inequalities. Our approach is inspired from shape optimization techniques, but no shape optimization problem is used here although this is a known method in free boundary problems, [2].…”

Optimization of a plate with holes

“…The unknowns to be found are the position, the shape, the size, the number of the holes defining the optimal plate and the given thickness is assumed constant. The main tools that we use is the fictitious domain approach Neittaanmäki, Pennanen, Tiba [13], Neittaanmäki, Tiba [15], Halanay, Murea, Tiba [6], Murea, Tiba [12] and the control variational method, Barboteu, Sofonea, Tiba [2], Sofonea, Tiba [18], Neittaanmäki, Sprekels, Tiba [14] and their references.…”

Approximation of a simply supported plate with obstacle

“…In the setting of variational inequalities and free boundary problems, applications of such ideas have been recently discussed in Halanay, Murea and Tiba (2013), (2016), Murea and Tiba (2016) with reference to elliptic or parabolic obstacle problems, one or two phase Stefan problems, fluid-solid contact models. As examples of applications that may be handled via the fixed domain methods, we mention as well the electrochemical machining process Neittaanmaki, Sprekels and Tiba (2006) or various contact problems related to the deformation of the elastic membranes with obstacle, melting or solidification processes.…”

“…As examples of applications that may be handled via the fixed domain methods, we mention as well the electrochemical machining process Neittaanmaki, Sprekels and Tiba (2006) or various contact problems related to the deformation of the elastic membranes with obstacle, melting or solidification processes. Applications to financial models (the two-asset American options) are also investigated in Murea and Tiba (2016).…”