%0 Journal Article %J Mathematical Models and Methods in Applied Sciences %D 2021 %T Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods %A Andrea Cangiani %A E.H. Georgoulis %A Sutton, Oliver J. %X 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∞(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. %B Mathematical Models and Methods in Applied Sciences %V 31 %P 711-751 %G eng %U https://doi.org/10.1142/S0218202521500172 %R 10.1142/S0218202521500172 %0 Journal Article %J J. Comput. Appl. Math. %D 2020 %T Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems %A Andrea Cangiani %A E.H. Georgoulis %A Sabawi, Younis A. %B J. Comput. Appl. Math. %V 367 %P 112397, 15 %G eng %U https://doi.org/10.1016/j.cam.2019.112397 %R 10.1016/j.cam.2019.112397 %0 Journal Article %J J. Sci. Comput. %D 2020 %T \it A posteriori error analysis for implicit-explicit $hp$-discontinuous Galerkin timestepping methods for semilinear parabolic problems %A Andrea Cangiani %A E.H. Georgoulis %A Sabawi, Mohammad %B J. Sci. Comput. %V 82 %P Paper No. 26, 24 %G eng %U https://doi.org/10.1007/s10915-020-01130-2 %R 10.1007/s10915-020-01130-2 %0 Journal Article %J Computers & Mathematics with Applications %D 2019 %T hp-adaptive discontinuous Galerkin methods for non-stationary convection–diffusion problems %A Andrea Cangiani %A E.H. Georgoulis %A Stefano Giani %A S. Metcalfe %K A posteriori error estimation %K Adaptive finite element methods %K Anisotropic meshes %K Discontinuous Galerkin %K Unsteady convection–diffusion %X An a posteriori error estimator for the error in the (L2(H1)+L∞(L2))-type norm for an interior penalty discontinuous Galerkin (dG) spatial discretisation and backward Euler temporal discretisation of linear non-stationary convection–diffusion initial/boundary value problems is derived, allowing for anisotropic elements. The proposed error estimator is used to drive an hp-space–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–time adaptive algorithm is assessed via a number of standard test problems characterised by sharp and/or moving layers. %B Computers & Mathematics with Applications %V 78 %P 3090-3104 %G eng %U https://www.sciencedirect.com/science/article/pii/S0898122119302007 %R https://doi.org/10.1016/j.camwa.2019.04.002 %0 Journal Article %J IMA Journal of Numerical Analysis %D 2019 %T Virtual element method for quasilinear elliptic problems %A Andrea Cangiani %A Chatzipantelidis, P %A Diwan, G %A E.H. Georgoulis %X A virtual element method for the quasilinear equation \\$-\\textrm\{div\} (\{\\boldsymbol \ąppa \}(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. %B IMA Journal of Numerical Analysis %V 40 %P 2450-2472 %8 07 %G eng %U https://doi.org/10.1093/imanum/drz035 %R 10.1093/imanum/drz035 %0 Journal Article %J Math. Comp. %D 2018 %T Adaptive discontinuous Galerkin methods for elliptic interface problems %A Andrea Cangiani %A E.H. Georgoulis %A Sabawi, Younis A. %B Math. Comp. %V 87 %P 2675–2707 %G eng %U https://doi.org/10.1090/mcom/3322 %R 10.1090/mcom/3322 %0 Journal Article %J Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences %D 2018 %T Revealing new dynamical patterns in a reaction&\#x2013;diffusion model with cyclic competition via a novel computational framework %A Andrea Cangiani %A E.H. Georgoulis %A Morozov, A. Yu. %A Sutton, O. J. %X Understanding how patterns and travelling waves form in chemical and biological reaction–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–diffusion models, such as Lotka–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–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. %B Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences %V 474 %P 20170608 %G eng %U https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.0608 %R 10.1098/rspa.2017.0608 %0 Book Section %B Generalized barycentric coordinates in computer graphics and computational mechanics %D 2018 %T Virtual element methods for elliptic problems on polygonal meshes %A Andrea Cangiani %A Sutton, Oliver J. %A Gyrya, Vitaliy %A Manzini, Gianmarco %B Generalized barycentric coordinates in computer graphics and computational mechanics %I CRC Press, Boca Raton, FL %P 263–279 %G eng %0 Journal Article %J IMA J. Numer. Anal. %D 2017 %T Conforming and nonconforming virtual element methods for elliptic problems %A Andrea Cangiani %A Manzini, Gianmarco %A Sutton, Oliver J. %B IMA J. Numer. Anal. %V 37 %P 1317–1354 %G eng %U https://doi.org/10.1093/imanum/drw036 %R 10.1093/imanum/drw036 %0 Book %B SpringerBriefs in Mathematics %D 2017 %T $hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes %A Andrea Cangiani %A Dong, Zhaonan %A E.H. Georgoulis %A Houston, Paul %B SpringerBriefs in Mathematics %I Springer, Cham %P viii+131 %@ 978-3-319-67671-5; 978-3-319-67673-9 %G eng %0 Journal Article %J SIAM J. Sci. Comput. %D 2017 %T $hp$-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes %A Andrea Cangiani %A Dong, Zhaonan %A E.H. Georgoulis %B SIAM J. Sci. Comput. %V 39 %P A1251–A1279 %G eng %U https://doi.org/10.1137/16M1073285 %R 10.1137/16M1073285 %0 Journal Article %J Numer. Math. %D 2017 %T A posteriori error estimates for the virtual element method %A Andrea Cangiani %A E.H. Georgoulis %A Pryer, Tristan %A Sutton, Oliver J. %B Numer. Math. %V 137 %P 857–893 %G eng %U https://doi.org/10.1007/s00211-017-0891-9 %R 10.1007/s00211-017-0891-9 %0 Journal Article %J SIAM J. Sci. Comput. %D 2016 %T Adaptivity and blow-up detection for nonlinear evolution problems %A Andrea Cangiani %A E.H. Georgoulis %A Kyza, Irene %A Metcalfe, Stephen %B SIAM J. Sci. Comput. %V 38 %P A3833–A3856 %G eng %U https://doi.org/10.1137/16M106073X %R 10.1137/16M106073X %0 Journal Article %J Appl. Numer. Math. %D 2016 %T Discontinuous Galerkin methods for fast reactive mass transfer through semi-permeable membranes %A Andrea Cangiani %A E.H. Georgoulis %A Jensen, Max %B Appl. Numer. Math. %V 104 %P 3–14 %G eng %U https://doi.org/10.1016/j.apnum.2014.06.007 %R 10.1016/j.apnum.2014.06.007 %0 Journal Article %J ESAIM Math. Model. Numer. Anal. %D 2016 %T $hp$-version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes %A Andrea Cangiani %A Dong, Zhaonan %A E.H. Georgoulis %A Houston, Paul %B ESAIM Math. Model. Numer. Anal. %V 50 %P 699–725 %G eng %U https://doi.org/10.1051/m2an/2015059 %R 10.1051/m2an/2015059 %0 Journal Article %J SIAM J. Numer. Anal. %D 2016 %T The nonconforming virtual element method for the Stokes equations %A Andrea Cangiani %A Gyrya, Vitaliy %A Manzini, Gianmarco %B SIAM J. Numer. Anal. %V 54 %P 3411–3435 %G eng %U https://doi.org/10.1137/15M1049531 %R 10.1137/15M1049531 %0 Book Section %B Building bridges: connections and challenges in modern approaches to numerical partial differential equations %D 2016 %T Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains %A Antonietti, Paola F. %A Andrea Cangiani %A Collis, Joe %A Dong, Zhaonan %A E.H. Georgoulis %A Stefano Giani %A Houston, Paul %B Building bridges: connections and challenges in modern approaches to numerical partial differential equations %S Lect. Notes Comput. Sci. Eng. %I Springer, [Cham] %V 114 %P 279–308 %G eng %0 Journal Article %J Internat. J. Numer. Methods Engrg. %D 2015 %T Hourglass stabilization and the virtual element method %A Andrea Cangiani %A Manzini, G. %A Russo, A. %A Sukumar, N. %B Internat. J. Numer. Methods Engrg. %V 102 %P 404–436 %G eng %U https://doi.org/10.1002/nme.4854 %R 10.1002/nme.4854 %0 Journal Article %J International Journal for Numerical Methods in Engineering %D 2015 %T Hourglass stabilization and the virtual element method %A Andrea Cangiani %A Manzini, G. %A Russo, A. %A Sukumar, N. %K consistency matrix %K hourglass control %K polygonal and polyhedral finite elements %K stabilization matrix %K underintegration %K virtual element method %X 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 © 2015 John Wiley & Sons, Ltd. %B International Journal for Numerical Methods in Engineering %V 102 %P 404-436 %G eng %U https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.4854 %R https://doi.org/10.1002/nme.4854 %0 Journal Article %J IMA J. Numer. Anal. %D 2014 %T Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems %A Andrea Cangiani %A E.H. Georgoulis %A Metcalfe, Stephen %B IMA J. Numer. Anal. %V 34 %P 1578–1597 %G eng %U https://doi.org/10.1093/imanum/drt052 %R 10.1093/imanum/drt052 %0 Journal Article %J Math. Models Methods Appl. Sci. %D 2014 %T $hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes %A Andrea Cangiani %A E.H. Georgoulis %A Houston, Paul %B Math. Models Methods Appl. Sci. %V 24 %P 2009–2041 %G eng %U https://doi.org/10.1142/S0218202514500146 %R 10.1142/S0218202514500146 %0 Journal Article %J Int. J. Numer. Anal. Model. %D 2014 %T On local super-penalization of interior penalty discontinuous Galerkin methods %A Andrea Cangiani %A Chapman, John %A E.H. Georgoulis %A Jensen, Max %B Int. J. Numer. Anal. Model. %V 11 %P 478–495 %G eng %0 Journal Article %J Math. Models Methods Appl. Sci. %D 2013 %T Basic principles of virtual element methods %A Beirão da Veiga, L. %A Brezzi, F. %A Andrea Cangiani %A Manzini, G. %A Marini, L. D. %A Russo, A. %B Math. Models Methods Appl. Sci. %V 23 %P 199–214 %G eng %U https://doi.org/10.1142/S0218202512500492 %R 10.1142/S0218202512500492 %0 Journal Article %J SIAM J. Numer. Anal. %D 2013 %T Discontinuous Galerkin methods for mass transfer through semipermeable membranes %A Andrea Cangiani %A E.H. Georgoulis %A Jensen, Max %B SIAM J. Numer. Anal. %V 51 %P 2911–2934 %G eng %U https://doi.org/10.1137/120890429 %R 10.1137/120890429 %0 Book Section %B Numerical mathematics and advanced applications 2011 %D 2013 %T Implementation of the continuous-discontinuous Galerkin finite element method %A Andrea Cangiani %A Chapman, J. %A E.H. Georgoulis %A Jensen, M. %B Numerical mathematics and advanced applications 2011 %I Springer, Heidelberg %P 315–322 %G eng %0 Journal Article %J J. Sci. Comput. %D 2013 %T On the stability of continuous-discontinuous Galerkin methods for advection-diffusion-reaction problems %A Andrea Cangiani %A Chapman, John %A E.H. Georgoulis %A Jensen, Max %B J. Sci. Comput. %V 57 %P 313–330 %G eng %U https://doi.org/10.1007/s10915-013-9707-y %R 10.1007/s10915-013-9707-y %0 Journal Article %J Comput. Methods Appl. Mech. Engrg. %D 2011 %T Convergence of the mimetic finite difference method for eigenvalue problems in mixed form %A Andrea Cangiani %A Gardini, Francesca %A Manzini, Gianmarco %B Comput. Methods Appl. Mech. Engrg. %V 200 %P 1150–1160 %G eng %U https://doi.org/10.1016/j.cma.2010.06.011 %R 10.1016/j.cma.2010.06.011 %0 Journal Article %J J. Theoret. Biol. %D 2010 %T A spatial model of cellular molecular trafficking including active transport along microtubules %A Andrea Cangiani %A Natalini, R. %B J. Theoret. Biol. %V 267 %P 614–625 %G eng %U https://doi.org/10.1016/j.jtbi.2010.08.017 %R 10.1016/j.jtbi.2010.08.017 %0 Journal Article %J SIAM J. Numer. Anal. %D 2009 %T Convergence analysis of the mimetic finite difference method for elliptic problems %A Andrea Cangiani %A Manzini, Gianmarco %A Russo, Alessandro %B SIAM J. Numer. Anal. %V 47 %P 2612–2637 %G eng %U https://doi.org/10.1137/080717560 %R 10.1137/080717560 %0 Journal Article %J Comput. Methods Appl. Mech. Engrg. %D 2008 %T Flux reconstruction and solution post-processing in mimetic finite difference methods %A Andrea Cangiani %A Manzini, Gianmarco %B Comput. Methods Appl. Mech. Engrg. %V 197 %P 933–945 %G eng %U https://doi.org/10.1016/j.cma.2007.09.019 %R 10.1016/j.cma.2007.09.019 %0 Journal Article %J SIAM J. Numer. Anal. %D 2007 %T The residual-free-bubble finite element method on anisotropic partitions %A Andrea Cangiani %A Süli, Endre %B SIAM J. Numer. Anal. %V 45 %P 1654–1678 %G eng %U https://doi.org/10.1137/060658011 %R 10.1137/060658011 %0 Book Section %B Internat. J. Numer. Methods Fluids %D 2005 %T Enhanced residual-free bubble method for convection-diffusion problems %A Andrea Cangiani %A Süli, E. %B Internat. J. Numer. Methods Fluids %V 47 %P 1307–1313 %G eng %U https://doi.org/10.1002/fld.859 %R 10.1002/fld.859 %0 Journal Article %J Numer. Math. %D 2005 %T Enhanced RFB method %A Andrea Cangiani %A Süli, Endre %B Numer. Math. %V 101 %P 273–308 %G eng %U https://doi.org/10.1007/s00211-005-0620-7 %R 10.1007/s00211-005-0620-7