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

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 SOLVE → ESTIMATE -> MARK → REFINE and employs the bulk criterion. The convergence in L P' (Ω; R m×n ) relies on new sharp strict convexity estimates of degenerate convex minimization problems with J(v):= ∫ Ω W(Dν)dx-∫ Ω fνdx for ν ∈ V:= W 0 1,p (Ω; R m ). The class of minimization problems includes strong convex problems and allows applications in an optimal design task, Hencky elastoplasticity or relaxation of two-well problems allowing for microstructures.

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