Stabilised FEM for Degenerate Convex Minimisation Problems under Weak Regularity Assumptions

Wolfgang Boiger , Carsten Carstensen

MSC 2000

65K10 Optimization and variational techniques

65N12 Stability and convergence of numerical methods

Abstract
The discretisation of degenerate convex minimisation problems experiences numerical difficulties with a singular or nearly singular Hessian matrix. Some discrete analog of the surface energy in microstrucures is added to the energy functional to define a stabilisation technique. This paper proves (a) strong convergence of the stress even without any smoothness assumption for a class of stabilised degenerate convex minimisation problems. Given the limitted a priori error control in those cases, the sharp a posteriori error control is of even higher relevance. This paper derives (b) guaranteed a posteriori error control via some equilibration technique which does not rely on the strict Galerkin orthogonality of the unperturbed problem. In the presence of L2 control in the original minimisation problem, some realistic model scenario with piecewise smooth exact solution allows for strong convergence of the gradients plus refined a posteriori error estimates. This paper presents (c) an improved a posteriori error control in this interface problem and so narrows the efficiency reliability gap. Numerical experiments illustrate the theoretical convergence rates for uniform and adaptive mesh-refinements and the improved a posteriori error control for four benchmark examples in the computational microstructures.