Error estimates for the approximation of semicoercive variational inequalities
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.
scientific article; zbMATH DE number 739431