a posteriori error estimate
adaptive finite element methods
calculus of variations
convexification
degenerate convex problems
energy reduction
nonconvex minimisation
partial differential equation
relaxation
reliabilityefficiency gap
stabilisation
strong convergence
variational problem
Stabilised FEM for Degenerate Convex Minimisation Problems under Weak Regularity Assumptions
Wolfgang Boiger
Boiger
Wolfgang
Carsten Carstensen
Carstensen
Carsten
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 meshrefinements
and the improved a posteriori error control for
four benchmark examples in the computational microstructures.
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