We will describe numerical approximation of the relaxation for certain non-convex variational problems which arise, e.g., in mathematical modelling of phase transitions, or in optimal shape design.
Although these relaxed energy functionals are convex, they
are not strictly convex. Therefore error estimates for finite
element approximation are more difficult to obtain than for
uniformly convex problems. A priori and a posteriori error estimates
will be presented together with error indicators proposed for
adaptive mesh refinement. Approximation of generalised solutions
to the original non-convex problem will be also discussed. Using
some standard test problems from optimal shape design we compare
the proposed error indicators with some heuristic mesh-refinement
strategies used in engineering applications.
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