TY - JOUR
T1 - An Interior Penalty Discontinuous Galerkin Method for a Class of Monotone Quasilinear Elliptic Problems
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 750
EP - 777
PY - 2015
DA - 2015/12
SN - 12
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/510.html
KW - $hp$-discontinuous Galerkin methods, interior penalty methods, second-order quasilinear elliptic problems.
AB - <p style="text-align: justify;">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 <span style="text-align: justify;">$\Omega\subset\mathbb{R}^d$, $d\geq2$</span>. 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.</p>