Two-scale methods for convex envelopes

Li, Wenbo; Nochetto, Ricardo

doi:10.1090/mcom/3521

Two-scale methods for convex envelopes
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by Wenbo Li and Ricardo H. Nochetto HTML | PDF

Math. Comp. 91 (2022), 111-139 Request permission

Abstract:

We develop two-scale methods for computing the convex envelope of a continuous function over a convex domain in any dimension. This hinges on a fully nonlinear obstacle formulation (see A. M. Oberman [Proc. Amer. Math. Soc. 135 (2007), pp. 1689–1694]). We prove convergence and error estimates in the max norm. The proof utilizes a discrete comparison principle, a discrete barrier argument to deal with Dirichlet boundary values, and the property of flatness in one direction within the non-contact set. Our error analysis extends to a modified version of the finite difference wide stencil method provided by Oberman [Math. Models Methods Appl. Sci. 18 (2008), pp. 759–780].

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