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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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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Additional Information
  • Wenbo Li
  • Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
  • ORCID: 0000-0002-6678-6857
  • Email: wli50@utk.edu
  • Ricardo H. Nochetto
  • Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
  • MR Author ID: 131850
  • ORCID: 0000-0002-6678-6857
  • Email: rhn@umd.edu
  • Received by editor(s): December 29, 2018
  • Received by editor(s) in revised form: February 25, 2019, and December 24, 2019
  • Published electronically: October 13, 2021
  • Additional Notes: Both authors were partially supported by the NSF Grant DMS -1411808. The first author was also partially supported by the Patrick and Marguerite Sung Fellowship in Mathematics
  • © Copyright 2021 American Mathematical Society
  • Journal: Math. Comp. 91 (2022), 111-139
  • MSC (2020): Primary 65N06, 65N12, 65N15, 65N30; Secondary 35J70, 35J87
  • DOI: https://doi.org/10.1090/mcom/3521
  • MathSciNet review: 4350534