Nonconforming FEMs for the $p$-Laplace Problem

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.