TY - JOUR

T1 - Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined mesh

AU - Li, Buyang

N1 - Funding Information:
Received by the editor May 16, 2021, and, in revised form, November 12, 2021, and November 30, 2021. 2020 Mathematics Subject Classification. Primary 65M12, 65M15; Secondary 65L06. Key words and phrases. Finite element methods, Neumann problem, maximum-norm stability, nonconvex polygon, graded mesh, corner singularity. This work was partially supported by a grant from the Research Grants Council of Hong Kong (GRF Project No. PolyU15300519), and an internal grant at The Hong Kong Polytechnic University (PolyU Project ID: P0031035, Work Programme: ZZKQ).
Publisher Copyright:
© 2022. American Mathematical Society

PY - 2022/7

Y1 - 2022/7

N2 - 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.

AB - 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.

KW - Corner singularity

KW - Finite element methods

KW - Graded mesh

KW - Maximum-norm stability

KW - Neumann problem

KW - Nonconvex polygon

UR - http://www.scopus.com/inward/record.url?scp=85131912539&partnerID=8YFLogxK

U2 - https://doi.org/10.1090/mcom/3724

DO - https://doi.org/10.1090/mcom/3724

M3 - Journal article

VL - 91

SP - 1533

EP - 1585

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 336

ER -