Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined mesh

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Abstract

The Galerkin finite element solution uh of the Poisson equation −Δu = f under the Neumann boundary condition in a possibly nonconvex polygon Ω, with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability: (Formula Presented) where l h = ln(2+1/h) for piecewise linear elements and l h = 1 for higherorder elements. As a result of the maximum-norm stability, the following best approximation result holds: (Formula Presented) where I h denotes the Lagrange interpolation operator onto the finite element space. For a locally quasi-uniform triangulation sufficiently refined at the corners, the above best approximation property implies the following optimalorder error bound in the maximum norm: (Formula Presented) where r ≥ 1 is the degree of finite elements, k is any nonnegative integer no larger than r, and p ∈ [2,∞) can be arbitrarily large.

Original languageEnglish
Pages (from-to)1533–1585
Number of pages53
JournalMathematics of Computation
Volume91
Issue number336
DOIs
Publication statusPublished - Jul 2022

Keywords

  • Corner singularity
  • Finite element methods
  • Graded mesh
  • Maximum-norm stability
  • Neumann problem
  • Nonconvex polygon

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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