Error estimates for the approximation of semicoercive variational inequalities (Q1347067)

The aim of the paper is to extend the well-known abstract error estimate obtained by \textit{R. S. Falk} [Math. Comput. 28, 963-971 (1974; Zbl 0297.65061)] to the case of semicoercive variational inequalities. The problem under consideration is: find \(u \in K\) such that \(a(u,v-u) \geq f(v-u)\) \((v\in K)\), where \(K\) is a closed convex subset of a real Hilbert space \((V,\| \cdot \|)\), \(f \in V'\) and the bilinear form \(a : V \times V \to \mathbb{R}\) is positive semidefinite on \(V\), i.e. \(a(v,v) \geq 0\) \((v\in V)\). Let \(A : V \to V'\) be defined by \(\langle Av, w\rangle = a(v,w)\) \((v,w \in V)\). The form \(a\) is supposed to satisfy Gårding's inequality, i.e. there is \(C > 0\) and a linear compact operator \(B_ A : V \to V\) such that \(a(v,v) \geq C(\| v\|^ 2 - \| B_ Av\|^ 2\) \((v \in V)\). Let \((V_ h)\) be a family of closed linear subspaces of \(V\) and \((K_ h)\) be a family of closed convex subsets with \(K_ h \subset V_ h\). Assume that there is a bounded linear operator \(\tau_ h : V \to V_ h\), \(\tau_ h \mid_{V_ h} = \text{id}\), satisfying \(\| \tau_ h v - v\| \to 0\) \((h \to 0)\) for each \(v \in V\). Let \(a_ h : V_ h \times V_ h \to \mathbb{R}\) be a family of uniformly bounded, positive semidefinite bilinear forms uniformly satisfying Gårding's inequality. Denote by \(u_ h \in K_ h\) the solution of the approximating variational inequality \(a_ h(u_ h,v_ h - u_ h) \geq f_ h(v_ h - u_ h)\) \((v_ h \in K_ h)\), where \(f_ h \in V'\), \(h > 0\). Then under some natural assumptions concerning \(A\), \(K\) and \(K_ h\) the following error estimate holds: \[ \begin{multlined} \| u - u_ h\|^ 2 \leq C_ 0 \left( \inf_{v_ h \in K_ h} (\| u - v_ h\|^ 2 + \langle Au - f, v_ h - u_ h\rangle_ +) + \sup_{v_ h \in V_ h, \| v_ h\| \leq 1} | (a - a_ h)(\tau_ h u, v_ h)|^ 2 + \right.\\ \left.+ \sup_{v_ h \in V_ h, \| v_ h\| \leq 1} | (f-f_ h)(v_ h)|^ 2 + \inf_{v \in K} \| u_ h - v\|^ 2\right).\end{multlined} \] The result is applied to the finite element approximation of Poisson's equation with Signorini boundary conditions and to the obstacle problem for the beam with no fixed boundary conditions.