An interior penalty discontinuous Galerkin method for a class of monotone quasilinear elliptic problems

Abstract

A family of interior penalty $hp$-discontinuous Galerkin methods is developed and analyzed for the numerical solution of the quasilinear elliptic equation $-\nabla{} \cdot (\mathbf{A}(\nabla{u}) \nabla{u} = f$ posed on the open bounded domain $Ω\subset \mathbb{R}^d$, $d \geq 2$. Subject to the assumption that the map $\mathbf{v} \mapsto \mathbf{A}(\mathbf{v}) \mathbf{v}$, $\mathbf{v} \in \mathbb{R}^d$, is Lipschitz continuous and strongly monotone, it is proved that the proposed method is well-posed. \emph{A priori} error estimates are presented of the error in the broken $H^1(Ω)$-norm, exhibiting precisely the same $h$-optimal and mildly $p$-suboptimal convergence rates as obtained for the interior penalty approximation of linear elliptic problems. \emph{A priori} estimates for linear functionals of the error and the $L^2(Ω)$-norm of the error are also established and shown to be $h$-optimal for a particular member of the proposed family of methods. The analysis is completed under fairly weak conditions on the approximation space, allowing for non-affine and curved elements with multilevel hanging nodes. The theoretical results are verified by numerical experiments.