Nonconforming FEMs for the $p$-Laplace Problem (Advances in Applied Mathematics and Mechanics, 2021)
The $p$-Laplace problems in topology optimization eventually lead to a degenerateconvex 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 discreteRaviart-Thomas mixed finite element method (dRT-MFEM) and establishes its equivalencewith the Crouzeix-Raviart nonconforming finite element method (CR-NCFEM).The sharper quasi-norm a priori and a posteriori error estimates of this two methodsare presented. Numerical experiments are provided to verify the analysis.