Finite element approximation of the 𝑝-Laplacian

John W. Barrett, W. B. Liu · Mathematics of Computation · 1993

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 \[ − ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) = f in Ω ⊂ R 2 , u = g on ∂ Ω . - abla \cdot (| abla u{|^{p - 2}} abla u) = f\quad {\text {in}}\;\Omega \subset {\mathbb {R}^2},\quad u = g\quad {\text {on}}\;\partial \Omega . \] 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

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