arrow
Volume 12, Issue 4
An Interior Penalty Discontinuous Galerkin Method for a Class of Monotone Quasilinear Elliptic Problems

Peter W. Fick

Int. J. Numer. Anal. Mod., 12 (2015), pp. 750-777.

Published online: 2015-12

Export citation
  • 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.

  • AMS Subject Headings

65N12, 65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{IJNAM-12-750, author = {}, title = {An Interior Penalty Discontinuous Galerkin Method for a Class of Monotone Quasilinear Elliptic Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2015}, volume = {12}, number = {4}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/510.html} }
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 -

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.

Peter W. Fick. (1970). An Interior Penalty Discontinuous Galerkin Method for a Class of Monotone Quasilinear Elliptic Problems. International Journal of Numerical Analysis and Modeling. 12 (4). 750-777. doi:
Copy to clipboard
The citation has been copied to your clipboard