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 $-\nabla \cdot {\rm a} (u, \nabla u) + f (u, \nabla u) = 0$ with Dirichlet boundary conditions. These methods depend on the values of the parameter $\theta\in[-1,1]$ , where θ =  + 1 corresponds to the nonsymmetric and θ = −1 corresponds to the symmetric interior penalty methods when ${\rm a}(u,\nabla u)={\nabla}u$ 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 $u\in H^{5/2}(\Omega)$ . 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.