## 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 language | English |
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Pages (from-to) | 1533–1585 |

Number of pages | 53 |

Journal | Mathematics of Computation |

Volume | 91 |

Issue number | 336 |

DOIs | |

Publication status | Published - 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