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 m(μ) and M(μ) such that m(μ)(t - s) ≤ μ(x, t)t - μ(x, s)s ≤ M(μ)(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 u(DG) in the finite element space. If u ∈ C 1(Ω) ∩ Hk(Ω), k ≥ 2, then with discontinuous piecewise polynomials of degree p ≥ 1, the error between u and u(DG), measured in the broken H1(Ω)-norm, is O(h s-1/pk-3/2), where 1 ≤ s ≤ min {p + 1, k}. © Institute of Mathematics and its Applications 2005; All rights reserved.