The Pointwise Stabilities of Piecewise Linear Finite Element Method on Non-obtuse Tetrahedral Meshes of Nonconvex Polyhedra

Huadong Gao; Weifeng Qiu

doi:10.1007/s10915-021-01465-4

The Pointwise Stabilities of Piecewise Linear Finite Element Method on Non-obtuse Tetrahedral Meshes of Nonconvex Polyhedra

Huadong Gao
, Weifeng Qiu^*

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

2 Citations (Scopus)

Abstract

Let Ω be a Lipschitz polyhedral (can be nonconvex) domain in R³, and V_h denotes the finite element space of continuous piecewise linear polynomials. On non-obtuse quasi-uniform tetrahedral meshes, we prove that the finite element projection R_hu of u ∈ H¹(Ω) ∩ C (Ω¯) (with R_hu interpolating u at the boundary nodes) satisfies
‖R_hu‖L∞_(Ω) ≤ C| log h |‖u‖_L∞_(Ω).
If we further assume u ∈ W¹^,^∞(Ω), then
‖R_hu‖_W1,∞_(Ω) ≤ C| log h |‖u‖_W1,∞_(Ω).

Original language	English
Article number	53
Journal	Journal of Scientific Computing
Volume	87
Issue number	2
Online published	3 Apr 2021
DOIs	https://doi.org/10.1007/s10915-021-01465-4
Publication status	Published - May 2021

Research Keywords

Finite element method
Nonconvex polyhedra
The stability in L∞ and W1,∞

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10.1007/s10915-021-01465-4

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@article{303d6c635a334eb6b97308373a611f38,

title = "The Pointwise Stabilities of Piecewise Linear Finite Element Method on Non-obtuse Tetrahedral Meshes of Nonconvex Polyhedra",

abstract = "Let Ω be a Lipschitz polyhedral (can be nonconvex) domain in R3, and Vh denotes the finite element space of continuous piecewise linear polynomials. On non-obtuse quasi-uniform tetrahedral meshes, we prove that the finite element projection Rhu of u ∈ H1(Ω) ∩ C (Ω¯) (with Rhu interpolating u at the boundary nodes) satisfies ‖Rhu‖L∞(Ω) ≤ C| log h |‖u‖L∞(Ω). If we further assume u ∈ W1,∞(Ω), then ‖Rhu‖W1,∞(Ω) ≤ C| log h |‖u‖W1,∞(Ω). ",

keywords = "Finite element method, Nonconvex polyhedra, The stability in L∞ and W1,∞",

author = "Huadong Gao and Weifeng Qiu",

year = "2021",

month = may,

doi = "10.1007/s10915-021-01465-4",

language = "English",

volume = "87",

journal = "Journal of Scientific Computing",

issn = "0885-7474",

publisher = "Springer New York",

number = "2",

}

TY - JOUR

T1 - The Pointwise Stabilities of Piecewise Linear Finite Element Method on Non-obtuse Tetrahedral Meshes of Nonconvex Polyhedra

AU - Gao, Huadong

AU - Qiu, Weifeng

PY - 2021/5

Y1 - 2021/5

N2 - Let Ω be a Lipschitz polyhedral (can be nonconvex) domain in R3, and Vh denotes the finite element space of continuous piecewise linear polynomials. On non-obtuse quasi-uniform tetrahedral meshes, we prove that the finite element projection Rhu of u ∈ H1(Ω) ∩ C (Ω¯) (with Rhu interpolating u at the boundary nodes) satisfies ‖Rhu‖L∞(Ω) ≤ C| log h |‖u‖L∞(Ω). If we further assume u ∈ W1,∞(Ω), then ‖Rhu‖W1,∞(Ω) ≤ C| log h |‖u‖W1,∞(Ω).

AB - Let Ω be a Lipschitz polyhedral (can be nonconvex) domain in R3, and Vh denotes the finite element space of continuous piecewise linear polynomials. On non-obtuse quasi-uniform tetrahedral meshes, we prove that the finite element projection Rhu of u ∈ H1(Ω) ∩ C (Ω¯) (with Rhu interpolating u at the boundary nodes) satisfies ‖Rhu‖L∞(Ω) ≤ C| log h |‖u‖L∞(Ω). If we further assume u ∈ W1,∞(Ω), then ‖Rhu‖W1,∞(Ω) ≤ C| log h |‖u‖W1,∞(Ω).

KW - Finite element method

KW - Nonconvex polyhedra

KW - The stability in L∞ and W1,∞

UR - https://www.scopus.com/pages/publications/85103982378

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85103982378&origin=recordpage

U2 - 10.1007/s10915-021-01465-4

DO - 10.1007/s10915-021-01465-4

M3 - RGC 21 - Publication in refereed journal

SN - 0885-7474

VL - 87

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

IS - 2

M1 - 53

ER -

The Pointwise Stabilities of Piecewise Linear Finite Element Method on Non-obtuse Tetrahedral Meshes of Nonconvex Polyhedra

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GRF: Robust and Efficient Numerical Methods and Rigorous Analysis for Ginzburg-Landau Equations

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The Pointwise Stabilities of Piecewise Linear Finite Element Method on Non-obtuse Tetrahedral Meshes of Nonconvex Polyhedra

Abstract

Research Keywords

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GRF: Robust and Efficient Numerical Methods and Rigorous Analysis for Ginzburg-Landau Equations

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