Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case

Abstract

We develop a one-parameter family of hp-version discontinuous Galerkin finite element methods, parameterised by θ ∈ [−1, 1], for the numerical solution of quasilinear elliptic equations in divergence form on a bounded open set Ω ⊂ ℝd, d ≥ 2. In particular, we consider the analysis of the family for the equation −∇ ·{μ(x, |∇u|)∇u} = f(x) subject to mixed Dirichlet–Neumann boundary conditions on ∂ Ω. It is assumed that μ is a real-valued function, μ ∈ C(Ω̄ × [0, ∞)), and there exist positive constants mμ and Mμ such that mμ(t − s) ≤ μ(x, t)t − μ(x, s)s ≤ Mμ(t − s) for t ≥ s ≥ 0 and all x ∈ Ω̄. Using a result from the theory of monotone operators for any value of θ ∈ [−1, 1], the corresponding method is shown to have a unique solution uDG in the finite element space. If u ∈ C1(Ω) ∩ Hk(Ω), k ≥ 2, then with discontinuous piecewise polynomials of degree p ≥ 1, the error between u and uDG, measured in the broken H1(Ω)-norm, is 𝒪(hs−1/pk−3/2), where 1 ≤ s ≤ min {p + 1, k}.