Volume 10, Issue 6
Nonconforming FEMs for the $p$-Laplace Problem

Adv. Appl. Math. Mech., 10 (2018), pp. 1365-1383.

Published online: 2018-09

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• Abstract

The $p$-Laplace problems in topology optimization eventually lead to a degenerate convex minimization problem $E(v):= ∫_ΩW(∇v)dx − ∫_Ωf vdx$ for $v∈W^{1,p}_0(Ω)$ with unique minimizer $u$ and stress $σ := DW(∇u)$. This paper proposes the discrete Raviart-Thomas mixed finite element method (dRT-MFEM) and establishes its equivalence with the Crouzeix-Raviart nonconforming finite element method (CR-NCFEM). The sharper quasi-norm a priori and a posteriori error estimates of this two methods are presented. Numerical experiments are provided to verify the analysis.

• Keywords

Adaptive finite element methods, nonconforming, $p$-Laplace problem, dual energy.

• AMS Subject Headings

65N12, 65N30, 65Y20

• Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-10-1365, author = {D. J. Liu , and A. Q. Li , and Chen , Z. R.}, title = {Nonconforming FEMs for the $p$-Laplace Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {6}, pages = {1365--1383}, abstract = {

The $p$-Laplace problems in topology optimization eventually lead to a degenerate convex minimization problem $E(v):= ∫_ΩW(∇v)dx − ∫_Ωf vdx$ for $v∈W^{1,p}_0(Ω)$ with unique minimizer $u$ and stress $σ := DW(∇u)$. This paper proposes the discrete Raviart-Thomas mixed finite element method (dRT-MFEM) and establishes its equivalence with the Crouzeix-Raviart nonconforming finite element method (CR-NCFEM). The sharper quasi-norm a priori and a posteriori error estimates of this two methods are presented. Numerical experiments are provided to verify the analysis.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0117}, url = {http://global-sci.org/intro/article_detail/aamm/12715.html} }
TY - JOUR T1 - Nonconforming FEMs for the $p$-Laplace Problem AU - D. J. Liu , AU - A. Q. Li , AU - Chen , Z. R. JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1365 EP - 1383 PY - 2018 DA - 2018/09 SN - 10 DO - http://doi.org/10.4208/aamm.OA-2018-0117 UR - https://global-sci.org/intro/article_detail/aamm/12715.html KW - Adaptive finite element methods, nonconforming, $p$-Laplace problem, dual energy. AB -

The $p$-Laplace problems in topology optimization eventually lead to a degenerate convex minimization problem $E(v):= ∫_ΩW(∇v)dx − ∫_Ωf vdx$ for $v∈W^{1,p}_0(Ω)$ with unique minimizer $u$ and stress $σ := DW(∇u)$. This paper proposes the discrete Raviart-Thomas mixed finite element method (dRT-MFEM) and establishes its equivalence with the Crouzeix-Raviart nonconforming finite element method (CR-NCFEM). The sharper quasi-norm a priori and a posteriori error estimates of this two methods are presented. Numerical experiments are provided to verify the analysis.

D. J. Liu, A. Q. Li & Z. R. Chen. (1970). Nonconforming FEMs for the $p$-Laplace Problem. Advances in Applied Mathematics and Mechanics. 10 (6). 1365-1383. doi:10.4208/aamm.OA-2018-0117
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