An Interior Penalty Discontinuous Galerkin Method for a Class of Monotone Quasilinear Elliptic Problems
Year: 2015
International Journal of Numerical Analysis and Modeling, Vol. 12 (2015), Iss. 4 : pp. 750–777
Abstract
A family of interior penalty $hp$-discontinuous Galerkin methods is developed and analyzed for the numerical solution of the quasilinear elliptic equation $-\nabla \cdot (\rm{A}$ $(\nabla u)\nabla u)=f$ posed on the open bounded domain $\Omega\subset\mathbb{R}^d$, $d\geq2$. Subject to the assumption that the map $\rm{v}\mapsto \rm{A}(\rm{v})\rm{v}$, $\rm{v} \in \mathbb{R}^d$, is Lipschitz continuous and strongly monotone, it is proved that the proposed method is well-posed. A priori error estimates are presented of the error in the broken $H^1(\Omega)$-norm, exhibiting precisely the same $h$-optimal and mildly $p$-suboptimal convergence rates as obtained for the interior penalty approximation of linear elliptic problems. A priori estimates for linear functionals of the error and the $L^2(\Omega)$-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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2015-IJNAM-510
International Journal of Numerical Analysis and Modeling, Vol. 12 (2015), Iss. 4 : pp. 750–777
Published online: 2015-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 28
Keywords: $hp$-discontinuous Galerkin methods interior penalty methods second-order quasilinear elliptic problems.