Finite element approximation of the 𝑝-Laplacian

Abstract

In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given p ∈ ( 1 , ∞ ) p \in (1,\infty ) , f, and g, find u such that \[ − ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) = f in Ω ⊂ R 2 , u = g on ∂ Ω . - \nabla \cdot (|\nabla u{|^{p - 2}}\nabla u) = f\quad {\text {in}}\;\Omega \subset {\mathbb {R}^2},\quad u = g\quad {\text {on}}\;\partial \Omega . \] The finite element approximation is defined over Ω h {\Omega ^h} , a union of regular triangles, yielding a polygonal approximation to Ω \Omega . For sufficiently regular solutions u, achievable for a subclass of data f, g, and Ω \Omega , we prove optimal error bounds for this approximation in the norm W 1 , q ( Ω h ) , q = p {W^{1,q}}({\Omega ^h}),q = p for p > 2 p > 2 and q ∈ [ 1 , 2 ] q \in [1,2] for p > 2 p > 2 , under the additional assumption that Ω h ⊆ Ω {\Omega ^h} \subseteq \Omega . Numerical results demonstrating these bounds are also presented.