A scheme with accuracy \(O(h^ 2)\) for determining a free boundary in one-dimensional problem with an obstacle (Q1842496)

A grid approximation for a one-dimensional problem with an obstacle inside the domain is given. The author considers the solution with respect to \(u \in K\) of the variational inequality \(a(u,v - u) \geq (f,v - u)\) \(\forall v \in K\) where \(a(u,v) = \int_ \Omega (pu'v' + quv) dx\), \((f,v) = \int_ \Omega fvdx\), \(\Omega = (0,1)\), \(K = \{v \in W^ 1_ 2 (\Omega) : v \geq 0\) in \(\Omega\), \(v(0) = \mu_ 0\), \(v(1) = \mu_ 1\}\), \(\mu_ 0, \mu_ 1 \in \mathbb{R}_ +\). The main result of the paper is the construction of a finite element method solution \(u_ h\) based on linear elements.