Degenerate variational problems often result from a relaxation technique in effective numerical simulation of nonconvex minimisation problems. The relaxed energy density is the convex envelope of the original one and so convex but not strictly convex. Hence strong convergence of straightforward finite element approximations cannot be expected but is relevant in many applications. This paper establishes a modified discretization by stabilisation and proves its convergence in strong norms.
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S. Bartels, Carsten Carstensen, P. Plechac, Andreas Prohl, Convergence for stabilisation of degenerately convex minimisation problems. Interfaces Free Bound. 6 (2004), no. 2, pp. 253–269DOI 10.4171/IFB/99