@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} } @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 {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} } @article {https://doi.org/10.1002/nme.4854, title = {Hourglass stabilization and the virtual element method}, journal = {International Journal for Numerical Methods in Engineering}, volume = {102}, number = {3-4}, year = {2015}, pages = {404-436}, abstract = {SummaryIn this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of -continuous VEMs on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation and present a virtual element approach for the isoparametric four-node quadrilateral and eight-node hexahedral elements. In addition, we show quantitative comparisons of the consistency and stabilization matrices in the VEM with those in the hourglass control method of Belytschko and coworkers. Numerical examples in two and three dimensions are presented for different stabilization parameters, which reveals that the method satisfies the patch test and delivers optimal rates of convergence in the L2 norm and the H1 seminorm for Poisson problems on quadrilateral, hexahedral, and arbitrary polygonal meshes. Copyright {\textcopyright} 2015 John Wiley \& Sons, Ltd.}, keywords = {consistency matrix, hourglass control, polygonal and polyhedral finite elements, stabilization matrix, underintegration, virtual element method}, doi = {https://doi.org/10.1002/nme.4854}, url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.4854}, author = {Andrea Cangiani and Manzini, G. and Russo, A. and Sukumar, N.} } @article {MR3340083, title = {Hourglass stabilization and the virtual element method}, journal = {Internat. J. Numer. Methods Engrg.}, volume = {102}, number = {3-4}, year = {2015}, pages = {404{\textendash}436}, issn = {0029-5981}, doi = {10.1002/nme.4854}, url = {https://doi.org/10.1002/nme.4854}, author = {Andrea Cangiani and Manzini, G. and Russo, A. and Sukumar, N.} } @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} }