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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1996-1997 Mathematics in High Performance Computing