English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A family of interior penalty $hp$-discontinuous Galerkin methods is developedand analyzed for the numerical solution of the quasilinear elliptic equation$-\nabla{} \cdot (\mathbf{A}(\nabla{u}) \nabla{u} = f$ posed on the openbounded domain $\Omega \subset \mathbb{R}^d$, $d \geq 2$. Subject to theassumption 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} errorestimates are presented of the error in the broken $H^1(\Omega)$-norm,exhibiting precisely the same $h$-optimal and mildly $p$-suboptimal convergencerates as obtained for the interior penalty approximation of linear ellipticproblems. \emph{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. Theanalysis is completed under fairly weak conditions on the approximation space,allowing for non-affine and curved elements with multilevel hanging nodes. Thetheoretical results are verified by numerical experiments.

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