hp-Discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems

Abstract

In this paper, we have analyzed a one parameter family of hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems −∇⋅a(u,∇u)+f(u,∇u)=0 with Dirichlet boundary conditions. These methods depend on the values of the parameter θ∈[−1,1] , where θ = + 1 corresponds to the nonsymmetric and θ = −1 corresponds to the symmetric interior penalty methods when a(u,∇u)=∇u and f(u,∇u) = −f, that is, for the Poisson problem. The error estimate in the broken H 1 norm, which is optimal in h (mesh size) and suboptimal in p (degree of approximation) is derived using piecewise polynomials of degree p ≥ 2, when the solution u∈H5/2(Ω) . In the case of linear elliptic problems also, this estimate is optimal in h and suboptimal in p. Further, optimal error estimate in the L 2 norm when θ = −1 is derived. Numerical experiments are presented to illustrate the theoretical results.