Finite element approximation of the 𝑝-Laplacian

In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given p ∈ ( 1 , ∞ ) p ∈ (1,∞ ) , f , and g , find u such that \[ − ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) = f in Ω ⊂ R 2 , u = g on ∂ Ω . - ∇ · (|∇ u{|p - 2}∇ u) = f {in}\;Ω ⊂ {R^2}, u = g {on}\;∂ Ω . \] The finite element approximation is defined over Ω h {Ω ^h} , a union of regular triangles, yielding a polygonal approximation to Ω Ω . For sufficiently regular solutions u , achievable for a subclass of data f , g , and Ω Ω , we prove optimal error bounds for this approximation in the norm W 1 , q ( Ω