TY - JOUR T1 - Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods JF - Mathematical Models and Methods in Applied Sciences Y1 - 2021 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Sutton, Oliver J. AB - 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. VL - 31 UR - https://doi.org/10.1142/S0218202521500172 ER - TY - JOUR T1 - Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems JF - J. Comput. Appl. Math. Y1 - 2020 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Sabawi, Younis A. VL - 367 UR - https://doi.org/10.1016/j.cam.2019.112397 ER - TY - JOUR T1 - \it A posteriori error analysis for implicit-explicit $hp$-discontinuous Galerkin timestepping methods for semilinear parabolic problems JF - J. Sci. Comput. Y1 - 2020 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Sabawi, Mohammad VL - 82 UR - https://doi.org/10.1007/s10915-020-01130-2 ER - TY - JOUR T1 - hp-adaptive discontinuous Galerkin methods for non-stationary convection–diffusion problems JF - Computers & Mathematics with Applications Y1 - 2019 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Stefano Giani A1 - S. Metcalfe KW - A posteriori error estimation KW - Adaptive finite element methods KW - Anisotropic meshes KW - Discontinuous Galerkin KW - Unsteady convection–diffusion AB - 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. VL - 78 UR - https://www.sciencedirect.com/science/article/pii/S0898122119302007 N1 - Applications of Partial Differential Equations in Science and Engineering ER - TY - JOUR T1 - Virtual element method for quasilinear elliptic problems JF - IMA Journal of Numerical Analysis Y1 - 2019 A1 - Andrea Cangiani A1 - Chatzipantelidis, P A1 - Diwan, G A1 - E.H. Georgoulis AB - 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. VL - 40 UR - https://doi.org/10.1093/imanum/drz035 ER - TY - JOUR T1 - Adaptive discontinuous Galerkin methods for elliptic interface problems JF - Math. Comp. Y1 - 2018 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Sabawi, Younis A. VL - 87 UR - https://doi.org/10.1090/mcom/3322 ER - TY - JOUR T1 - Revealing new dynamical patterns in a reaction&\#x2013;diffusion model with cyclic competition via a novel computational framework JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Y1 - 2018 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Morozov, A. Yu. A1 - Sutton, O. J. AB - 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. VL - 474 UR - https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.0608 ER - TY - CHAP T1 - Virtual element methods for elliptic problems on polygonal meshes T2 - Generalized barycentric coordinates in computer graphics and computational mechanics Y1 - 2018 A1 - Andrea Cangiani A1 - Sutton, Oliver J. A1 - Gyrya, Vitaliy A1 - Manzini, Gianmarco JF - Generalized barycentric coordinates in computer graphics and computational mechanics PB - CRC Press, Boca Raton, FL ER - TY - JOUR T1 - Conforming and nonconforming virtual element methods for elliptic problems JF - IMA J. Numer. Anal. Y1 - 2017 A1 - Andrea Cangiani A1 - Manzini, Gianmarco A1 - Sutton, Oliver J. VL - 37 UR - https://doi.org/10.1093/imanum/drw036 ER - TY - BOOK T1 - $hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes T2 - SpringerBriefs in Mathematics Y1 - 2017 A1 - Andrea Cangiani A1 - Dong, Zhaonan A1 - E.H. Georgoulis A1 - Houston, Paul JF - SpringerBriefs in Mathematics PB - Springer, Cham SN - 978-3-319-67671-5; 978-3-319-67673-9 ER - TY - JOUR T1 - $hp$-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes JF - SIAM J. Sci. Comput. Y1 - 2017 A1 - Andrea Cangiani A1 - Dong, Zhaonan A1 - E.H. Georgoulis VL - 39 UR - https://doi.org/10.1137/16M1073285 ER - TY - JOUR T1 - A posteriori error estimates for the virtual element method JF - Numer. Math. Y1 - 2017 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Pryer, Tristan A1 - Sutton, Oliver J. VL - 137 UR - https://doi.org/10.1007/s00211-017-0891-9 ER - TY - JOUR T1 - Adaptivity and blow-up detection for nonlinear evolution problems JF - SIAM J. Sci. Comput. Y1 - 2016 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Kyza, Irene A1 - Metcalfe, Stephen VL - 38 UR - https://doi.org/10.1137/16M106073X ER - TY - JOUR T1 - Discontinuous Galerkin methods for fast reactive mass transfer through semi-permeable membranes JF - Appl. Numer. Math. Y1 - 2016 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Jensen, Max VL - 104 UR - https://doi.org/10.1016/j.apnum.2014.06.007 ER - TY - JOUR T1 - $hp$-version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes JF - ESAIM Math. Model. Numer. Anal. Y1 - 2016 A1 - Andrea Cangiani A1 - Dong, Zhaonan A1 - E.H. Georgoulis A1 - Houston, Paul VL - 50 UR - https://doi.org/10.1051/m2an/2015059 ER - TY - JOUR T1 - The nonconforming virtual element method for the Stokes equations JF - SIAM J. Numer. Anal. Y1 - 2016 A1 - Andrea Cangiani A1 - Gyrya, Vitaliy A1 - Manzini, Gianmarco VL - 54 UR - https://doi.org/10.1137/15M1049531 ER - TY - CHAP T1 - Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains T2 - Building bridges: connections and challenges in modern approaches to numerical partial differential equations Y1 - 2016 A1 - Antonietti, Paola F. A1 - Andrea Cangiani A1 - Collis, Joe A1 - Dong, Zhaonan A1 - E.H. Georgoulis A1 - Stefano Giani A1 - Houston, Paul JF - Building bridges: connections and challenges in modern approaches to numerical partial differential equations T3 - Lect. Notes Comput. Sci. Eng. PB - Springer, [Cham] VL - 114 ER - TY - JOUR T1 - Hourglass stabilization and the virtual element method JF - International Journal for Numerical Methods in Engineering Y1 - 2015 A1 - Andrea Cangiani A1 - Manzini, G. A1 - Russo, A. A1 - Sukumar, N. KW - consistency matrix KW - hourglass control KW - polygonal and polyhedral finite elements KW - stabilization matrix KW - underintegration KW - virtual element method AB - 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. VL - 102 UR - https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.4854 ER - TY - JOUR T1 - Hourglass stabilization and the virtual element method JF - Internat. J. Numer. Methods Engrg. Y1 - 2015 A1 - Andrea Cangiani A1 - Manzini, G. A1 - Russo, A. A1 - Sukumar, N. VL - 102 UR - https://doi.org/10.1002/nme.4854 ER - TY - JOUR T1 - Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems JF - IMA J. Numer. Anal. Y1 - 2014 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Metcalfe, Stephen VL - 34 UR - https://doi.org/10.1093/imanum/drt052 ER - TY - JOUR T1 - $hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes JF - Math. Models Methods Appl. Sci. Y1 - 2014 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Houston, Paul VL - 24 UR - https://doi.org/10.1142/S0218202514500146 ER - TY - JOUR T1 - On local super-penalization of interior penalty discontinuous Galerkin methods JF - Int. J. Numer. Anal. Model. Y1 - 2014 A1 - Andrea Cangiani A1 - Chapman, John A1 - E.H. Georgoulis A1 - Jensen, Max VL - 11 ER - TY - JOUR T1 - Basic principles of virtual element methods JF - Math. Models Methods Appl. Sci. Y1 - 2013 A1 - Beirão da Veiga, L. A1 - Brezzi, F. A1 - Andrea Cangiani A1 - Manzini, G. A1 - Marini, L. D. A1 - Russo, A. VL - 23 UR - https://doi.org/10.1142/S0218202512500492 ER - TY - JOUR T1 - Discontinuous Galerkin methods for mass transfer through semipermeable membranes JF - SIAM J. Numer. Anal. Y1 - 2013 A1 - Andrea Cangiani A1 - E.H. Georgoulis A1 - Jensen, Max VL - 51 UR - https://doi.org/10.1137/120890429 ER - TY - CHAP T1 - Implementation of the continuous-discontinuous Galerkin finite element method T2 - Numerical mathematics and advanced applications 2011 Y1 - 2013 A1 - Andrea Cangiani A1 - Chapman, J. A1 - E.H. Georgoulis A1 - Jensen, M. JF - Numerical mathematics and advanced applications 2011 PB - Springer, Heidelberg ER - TY - JOUR T1 - On the stability of continuous-discontinuous Galerkin methods for advection-diffusion-reaction problems JF - J. Sci. Comput. Y1 - 2013 A1 - Andrea Cangiani A1 - Chapman, John A1 - E.H. Georgoulis A1 - Jensen, Max VL - 57 UR - https://doi.org/10.1007/s10915-013-9707-y ER - TY - JOUR T1 - Convergence of the mimetic finite difference method for eigenvalue problems in mixed form JF - Comput. Methods Appl. Mech. Engrg. Y1 - 2011 A1 - Andrea Cangiani A1 - Gardini, Francesca A1 - Manzini, Gianmarco VL - 200 UR - https://doi.org/10.1016/j.cma.2010.06.011 ER - TY - JOUR T1 - A spatial model of cellular molecular trafficking including active transport along microtubules JF - J. Theoret. Biol. Y1 - 2010 A1 - Andrea Cangiani A1 - Natalini, R. VL - 267 UR - https://doi.org/10.1016/j.jtbi.2010.08.017 ER - TY - JOUR T1 - Convergence analysis of the mimetic finite difference method for elliptic problems JF - SIAM J. Numer. Anal. Y1 - 2009 A1 - Andrea Cangiani A1 - Manzini, Gianmarco A1 - Russo, Alessandro VL - 47 UR - https://doi.org/10.1137/080717560 ER - TY - JOUR T1 - Flux reconstruction and solution post-processing in mimetic finite difference methods JF - Comput. Methods Appl. Mech. Engrg. Y1 - 2008 A1 - Andrea Cangiani A1 - Manzini, Gianmarco VL - 197 UR - https://doi.org/10.1016/j.cma.2007.09.019 ER - TY - JOUR T1 - The residual-free-bubble finite element method on anisotropic partitions JF - SIAM J. Numer. Anal. Y1 - 2007 A1 - Andrea Cangiani A1 - Süli, Endre VL - 45 UR - https://doi.org/10.1137/060658011 ER - TY - CHAP T1 - Enhanced residual-free bubble method for convection-diffusion problems T2 - Internat. J. Numer. Methods Fluids Y1 - 2005 A1 - Andrea Cangiani A1 - Süli, E. JF - Internat. J. Numer. Methods Fluids VL - 47 UR - https://doi.org/10.1002/fld.859 N1 - 8th ICFD Conference on Numerical Methods for Fluid Dynamics. Part 2 ER - TY - JOUR T1 - Enhanced RFB method JF - Numer. Math. Y1 - 2005 A1 - Andrea Cangiani A1 - Süli, Endre VL - 101 UR - https://doi.org/10.1007/s00211-005-0620-7 ER -