Finite element methods for the displacement obstacle problem of clamped plates (Q2894507)

The authors consider the following displacement obstacle problem for a clamped Kirchhoff plate:NEWLINENEWLINEFind \(u \in K\) such thatNEWLINENEWLINE(1) \( u=\arg\min_{v \in K} G(v)\), where \(K=\{v \in H^2_0(\Omega):\Psi_1 \leq v \leq \Psi_2\) on \(\Omega \}\), \(G(v) = \frac{1}{2}a (\upsilon, v) - (f,\upsilon)\), \(a(w,\upsilon) = \int_\Omega D^2w: D^2 \upsilon dx\) and \((f, \upsilon) = \int_\Omega f \upsilon dx\), \(\Omega \subset R^2\) is a bounded convex polygonal domain, \(f \in L_2 (\Omega)\), \(\Psi_1, \Psi_2 \in C^2 (\Omega) \cap C (\overline{\Omega})\), \(\Psi_1 < \Psi_2\) on \(\overline{\Omega}\), and \(\Psi_1 <0<\Psi_2\) on \(\partial \Omega\), \(D^2w: D^2\upsilon = \Sigma^2_{i,j=1}w_{x_ix_j} v_{x_ix_j}\) is the Frobenius inner product between the Hessian matrices of \(w\) and \(v\).NEWLINENEWLINEThe obstacle problem (1) has a unique solution (uniquely determined by the variational inequality \(a (u,v-u) \geq (f, v-u) \quad \forall v \in K\)).NEWLINENEWLINEFor finite element methods for the obstacle problem, the authors introduce the piecewise Sobolev space \(H^3(\Omega, \tau_h) = \{ v \in L_2(\Omega): v_T = \upsilon|_T \in H^3 (T) \quad \forall T \in \tau_h\}\) (\(\tau_h\) is a regular triangulation of \(\Omega\) with mesh size \(h\)), and \(a_h(.,.)\) is a symmetric bilinear form on \(H^3(\Omega, \tau_h)\) such that \(a_h(w,v)=a(w,v) \forall v,w \in H^2_0(\Omega) \cap H^3(\Omega, \tau_h)\).NEWLINENEWLINEThe authors take advantage of the fact that \(H^2(\Omega)\) is compactly embedded in \(C(\overline{\Omega})\) to give a convergence analysis, that uses only the minimization (variational inequality formulations of the continuous and discrete problems). The authors introduce three types of finite element methods and present a general framework for the analysis of these methods. They study an auxiliary obstacle problem that plays a key role in the convergence analysis of the finite element methods. An extension of the \(C^2\) regularity to the two-obstacle problem is discussed. The paper contains an estimate for the Morley element, and provides a concise unified analysis of finite element methods for the biharmonic problem. The paper presents also a complete convergence analysis of finite element methods.