@article {doi:10.1142/S0218202521500172,
	title = {Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods},
	journal = {Mathematical Models and Methods in Applied Sciences},
	volume = {31},
	number = {04},
	year = {2021},
	pages = {711-751},
	abstract = {We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces between time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The a posteriori error estimates, for the error measured in the L2(H1) and L$\infty$(L2) norms, are derived using the elliptic reconstruction technique in an abstract framework designed to precisely encapsulate our notion of inconsistency and non-hierarchicality and requiring no particular compatibility between the computational meshes used on consecutive time-steps, thereby significantly relaxing this basic assumption underlying previous estimates.},
	doi = {10.1142/S0218202521500172},
	url = {https://doi.org/10.1142/S0218202521500172},
	author = {Andrea Cangiani and E.H. Georgoulis and Sutton, Oliver J.}
}
@article {MR4008583,
	title = {Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems},
	journal = {J. Comput. Appl. Math.},
	volume = {367},
	year = {2020},
	pages = {112397, 15},
	issn = {0377-0427},
	doi = {10.1016/j.cam.2019.112397},
	url = {https://doi.org/10.1016/j.cam.2019.112397},
	author = {Andrea Cangiani and E.H. Georgoulis and Sabawi, Younis A.}
}
@article {MR4056776,
	title = {\it A posteriori error analysis for implicit-explicit $hp$-discontinuous Galerkin timestepping methods for semilinear parabolic problems},
	journal = {J. Sci. Comput.},
	volume = {82},
	number = {2},
	year = {2020},
	pages = {Paper No. 26, 24},
	issn = {0885-7474},
	doi = {10.1007/s10915-020-01130-2},
	url = {https://doi.org/10.1007/s10915-020-01130-2},
	author = {Andrea Cangiani and E.H. Georgoulis and Sabawi, Mohammad}
}
@article {CANGIANI20193090,
	title = {hp-adaptive discontinuous Galerkin methods for non-stationary convection{\textendash}diffusion problems},
	journal = {Computers \& Mathematics with Applications},
	volume = {78},
	number = {9},
	year = {2019},
	note = {Applications of Partial Differential Equations in Science and Engineering},
	pages = {3090-3104},
	abstract = {An a posteriori error estimator for the error in the (L2(H1)+L$\infty$(L2))-type norm for an interior penalty discontinuous Galerkin (dG) spatial discretisation and backward Euler temporal discretisation of linear non-stationary convection{\textendash}diffusion initial/boundary value problems is derived, allowing for anisotropic elements. The proposed error estimator is used to drive an hp-space{\textendash}time adaptive algorithm wherein directional mesh refinement is employed to give rise to highly anisotropic elements able to accurately capture layers. The performance of the hp-space{\textendash}time adaptive algorithm is assessed via a number of standard test problems characterised by sharp and/or moving layers.},
	keywords = {A posteriori error estimation, Adaptive finite element methods, Anisotropic meshes, Discontinuous Galerkin, Unsteady convection{\textendash}diffusion},
	issn = {0898-1221},
	doi = {https://doi.org/10.1016/j.camwa.2019.04.002},
	url = {https://www.sciencedirect.com/science/article/pii/S0898122119302007},
	author = {Andrea Cangiani and E.H. Georgoulis and Stefano Giani and S. Metcalfe}
}
@article {10.1093/imanum/drz035,
	title = {Virtual element method for quasilinear elliptic problems},
	journal = {IMA Journal of Numerical Analysis},
	volume = {40},
	number = {4},
	year = {2019},
	month = {07},
	pages = {2450-2472},
	abstract = {A virtual element method for the quasilinear equation \\$-\\textrm\{div\} (\{\\boldsymbol \{\k a}ppa \}(u)\\operatorname\{grad\} u)=f\\$ using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal-order a priori error estimates in the \\$H^1\\$- and \\$L^2\\$-norm are proven. In addition, the convergence of fixed-point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes.},
	issn = {0272-4979},
	doi = {10.1093/imanum/drz035},
	url = {https://doi.org/10.1093/imanum/drz035},
	author = {Andrea Cangiani and Chatzipantelidis, P and Diwan, G and E.H. Georgoulis}
}
@article {MR3834681,
	title = {Adaptive discontinuous Galerkin methods for elliptic interface problems},
	journal = {Math. Comp.},
	volume = {87},
	number = {314},
	year = {2018},
	pages = {2675{\textendash}2707},
	issn = {0025-5718},
	doi = {10.1090/mcom/3322},
	url = {https://doi.org/10.1090/mcom/3322},
	author = {Andrea Cangiani and E.H. Georgoulis and Sabawi, Younis A.}
}
@article {doi:10.1098/rspa.2017.0608,
	title = {Revealing new dynamical patterns in a reaction\&\#x2013;diffusion model with cyclic competition via a novel computational framework},
	journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
	volume = {474},
	number = {2213},
	year = {2018},
	pages = {20170608},
	abstract = {Understanding how patterns and travelling waves form in chemical and biological reaction{\textendash}diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction{\textendash}diffusion models, such as Lotka{\textendash}Volterra competition models with spatial dependence. In this study, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterized by a high regularity in space, but are different from patterns previously known to exist in reaction{\textendash}diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel a posteriori error estimate which is proved to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns in both two and three spatial dimensions.},
	doi = {10.1098/rspa.2017.0608},
	url = {https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.0608},
	author = {Andrea Cangiani and E.H. Georgoulis and Morozov, A. Yu. and Sutton, O. J.}
}
@book {MR3729265,
	title = {$hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes},
	series = {SpringerBriefs in Mathematics},
	year = {2017},
	pages = {viii+131},
	publisher = {Springer, Cham},
	organization = {Springer, Cham},
	isbn = {978-3-319-67671-5; 978-3-319-67673-9},
	author = {Andrea Cangiani and Dong, Zhaonan and E.H. Georgoulis and Houston, Paul}
}
@article {MR3672375,
	title = {$hp$-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes},
	journal = {SIAM J. Sci. Comput.},
	volume = {39},
	number = {4},
	year = {2017},
	pages = {A1251{\textendash}A1279},
	issn = {1064-8275},
	doi = {10.1137/16M1073285},
	url = {https://doi.org/10.1137/16M1073285},
	author = {Andrea Cangiani and Dong, Zhaonan and E.H. Georgoulis}
}
@article {MR3719046,
	title = {A posteriori error estimates for the virtual element method},
	journal = {Numer. Math.},
	volume = {137},
	number = {4},
	year = {2017},
	pages = {857{\textendash}893},
	issn = {0029-599X},
	doi = {10.1007/s00211-017-0891-9},
	url = {https://doi.org/10.1007/s00211-017-0891-9},
	author = {Andrea Cangiani and E.H. Georgoulis and Pryer, Tristan and Sutton, Oliver J.}
}
@article {MR3584583,
	title = {Adaptivity and blow-up detection for nonlinear evolution problems},
	journal = {SIAM J. Sci. Comput.},
	volume = {38},
	number = {6},
	year = {2016},
	pages = {A3833{\textendash}A3856},
	issn = {1064-8275},
	doi = {10.1137/16M106073X},
	url = {https://doi.org/10.1137/16M106073X},
	author = {Andrea Cangiani and E.H. Georgoulis and Kyza, Irene and Metcalfe, Stephen}
}
@article {MR3478183,
	title = {Discontinuous Galerkin methods for fast reactive mass transfer through semi-permeable membranes},
	journal = {Appl. Numer. Math.},
	volume = {104},
	year = {2016},
	pages = {3{\textendash}14},
	issn = {0168-9274},
	doi = {10.1016/j.apnum.2014.06.007},
	url = {https://doi.org/10.1016/j.apnum.2014.06.007},
	author = {Andrea Cangiani and E.H. Georgoulis and Jensen, Max}
}
@article {MR3507270,
	title = {$hp$-version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes},
	journal = {ESAIM Math. Model. Numer. Anal.},
	volume = {50},
	number = {3},
	year = {2016},
	pages = {699{\textendash}725},
	issn = {0764-583X},
	doi = {10.1051/m2an/2015059},
	url = {https://doi.org/10.1051/m2an/2015059},
	author = {Andrea Cangiani and Dong, Zhaonan and E.H. Georgoulis and Houston, Paul}
}
@inbook {MR3585793,
	title = {Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains},
	booktitle = {Building bridges: connections and challenges in modern approaches to numerical partial differential equations},
	series = {Lect. Notes Comput. Sci. Eng.},
	volume = {114},
	year = {2016},
	pages = {279{\textendash}308},
	publisher = {Springer, [Cham]},
	organization = {Springer, [Cham]},
	author = {Antonietti, Paola F. and Andrea Cangiani and Collis, Joe and Dong, Zhaonan and E.H. Georgoulis and Stefano Giani and Houston, Paul}
}
@article {MR3269437,
	title = {Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems},
	journal = {IMA J. Numer. Anal.},
	volume = {34},
	number = {4},
	year = {2014},
	pages = {1578{\textendash}1597},
	issn = {0272-4979},
	doi = {10.1093/imanum/drt052},
	url = {https://doi.org/10.1093/imanum/drt052},
	author = {Andrea Cangiani and E.H. Georgoulis and Metcalfe, Stephen}
}
@article {MR3211116,
	title = {$hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes},
	journal = {Math. Models Methods Appl. Sci.},
	volume = {24},
	number = {10},
	year = {2014},
	pages = {2009{\textendash}2041},
	issn = {0218-2025},
	doi = {10.1142/S0218202514500146},
	url = {https://doi.org/10.1142/S0218202514500146},
	author = {Andrea Cangiani and E.H. Georgoulis and Houston, Paul}
}
@article {MR3218334,
	title = {On local super-penalization of interior penalty discontinuous Galerkin methods},
	journal = {Int. J. Numer. Anal. Model.},
	volume = {11},
	number = {3},
	year = {2014},
	pages = {478{\textendash}495},
	issn = {1705-5105},
	author = {Andrea Cangiani and Chapman, John and E.H. Georgoulis and Jensen, Max}
}
@article {MR3121762,
	title = {Discontinuous Galerkin methods for mass transfer through semipermeable membranes},
	journal = {SIAM J. Numer. Anal.},
	volume = {51},
	number = {5},
	year = {2013},
	pages = {2911{\textendash}2934},
	issn = {0036-1429},
	doi = {10.1137/120890429},
	url = {https://doi.org/10.1137/120890429},
	author = {Andrea Cangiani and E.H. Georgoulis and Jensen, Max}
}
@inbook {MR3307565,
	title = {Implementation of the continuous-discontinuous Galerkin finite element method},
	booktitle = {Numerical mathematics and advanced applications 2011},
	year = {2013},
	pages = {315{\textendash}322},
	publisher = {Springer, Heidelberg},
	organization = {Springer, Heidelberg},
	author = {Andrea Cangiani and Chapman, J. and E.H. Georgoulis and Jensen, M.}
}
@article {MR3108752,
	title = {On the stability of continuous-discontinuous Galerkin methods for advection-diffusion-reaction problems},
	journal = {J. Sci. Comput.},
	volume = {57},
	number = {2},
	year = {2013},
	pages = {313{\textendash}330},
	issn = {0885-7474},
	doi = {10.1007/s10915-013-9707-y},
	url = {https://doi.org/10.1007/s10915-013-9707-y},
	author = {Andrea Cangiani and Chapman, John and E.H. Georgoulis and Jensen, Max}
}
