Convergence of adaptive FEM for a class of degenerate convex minimization problems

Abstract

A class of degenerate convex minimization problems allows for some adaptive finite element method (AFEM) to compute strongly converging stress approximations. The algorithm AFEM consists of successive loops of the form \begin{equation*} \texttt{SOLVE} \rightarrow \texttt{ESTIMATE} \rightarrow \texttt{MARK} \rightarrow \texttt{REFINE} \end{equation*} and employs the bulk criterion. The convergence in $L^{p'}(\Omega; \R^{m \times n})$ relies on new sharp strict convexity estimates of degenerate convex minimization problems with \[ \J(v):=\int_{\Omega}W(Dv)\,dx-\int_{\Omega}fv\,dx\quad\mbox{for } v\in V:=W^{1,p}_0(\Omega;\R^m). \] The class of minimization problems includes strong convex problems and allows applications in an optimal design task, Hencky elastoplasticity, or relaxation of 2-well problems allowing for microstructures.