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Virtual Element Method for Quasilinear Elliptic Problems (2019)
Journal Article
Cangiani, A., Chatzipantelidis, P., Diwan, G., & Georgoulis, E. (2020). Virtual Element Method for Quasilinear Elliptic Problems. IMA Journal of Numerical Analysis, 40(4), 2450–2472. https://doi.org/10.1093/imanum/drz035

A Virtual Element Method (VEM) for the quasilinear equation ?div(? ? ?(u)gradu) = f using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtua... Read More about Virtual Element Method for Quasilinear Elliptic Problems.

Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems (2019)
Journal Article
Cangiani, A., Georgoulis, E. H., & Sabawi, Y. (2020). Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems. Journal of Computational and Applied Mathematics, 367, Article 112397. https://doi.org/10.1016/j.cam.2019.112397

We prove a basic error contraction result of an adaptive discontinuous Galerkin method for an elliptic interface problem. The interface conditions considered model mass transfer of solutes through semi-permeable membranes and other filtering processe... Read More about Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems.

Revealing new dynamical patterns in a reaction–diffusion model with cyclic competition via a novel computational framework (2018)
Journal Article
Cangiani, A., Georgoulis, E. H., Morozov, A. Y., & Sutton, O. J. (2018). Revealing new dynamical patterns in a reaction–diffusion model with cyclic competition via a novel computational framework. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 474(2213), Article 20170608. https://doi.org/10.1098/rspa.2017.0608

Adaptive discontinuous Galerkin methods for elliptic interface problems (2018)
Journal Article
Cangiani, A., Georgoulis, E. H., & Sabawi, Y. A. (2018). Adaptive discontinuous Galerkin methods for elliptic interface problems. Mathematics of Computation, 87(314), 2675-2707. https://doi.org/10.1090/mcom/3322

© 2017 American Mathematical Society. An interior-penalty discontinuous Galerkin (dG) method for an elliptic interface problem involving, possibly, curved interfaces, with fluxbalancing interface conditions, e.g., modelling mass transfer of solutes t... Read More about Adaptive discontinuous Galerkin methods for elliptic interface problems.

A posteriori error estimates for the virtual element method (2017)
Journal Article
Cangiani, A., Georgoulis, E. H., Pryer, T., & Sutton, O. J. (2017). A posteriori error estimates for the virtual element method. Numerische Mathematik, 137(4), 857-893. https://doi.org/10.1007/s00211-017-0891-9

An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully com... Read More about A posteriori error estimates for the virtual element method.

$hp$-Version Space-Time Discontinuous Galerkin Methods for Parabolic Problems on Prismatic Meshes (2017)
Journal Article
Cangiani, A., Dong, Z., & Georgoulis, E. H. (2017). $hp$-Version Space-Time Discontinuous Galerkin Methods for Parabolic Problems on Prismatic Meshes. SIAM Journal on Scientific Computing, 39(4), A1251-A1279. https://doi.org/10.1137/16m1073285

We present a new $hp$-version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic evolution equations on general spatial meshes consisting of polygonal/polyhedral (polytopic) elements, giving rise... Read More about $hp$-Version Space-Time Discontinuous Galerkin Methods for Parabolic Problems on Prismatic Meshes.

Adaptivity and Blow-Up Detection for Nonlinear Evolution Problems (2016)
Journal Article
Cangiani, A., Georgoulis, E. H., Kyza, I., & Metcalfe, S. (2016). Adaptivity and Blow-Up Detection for Nonlinear Evolution Problems. SIAM Journal on Scientific Computing, 38(6), A3833-A3856. https://doi.org/10.1137/16m106073x

This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for a semilinear convection-diffusion problem which may exhibit blow-up in finite time. More specifically, a posterio... Read More about Adaptivity and Blow-Up Detection for Nonlinear Evolution Problems.

The nonconforming virtual element method for the stokes equations (2016)
Journal Article
Cangiani, A., Gyrya, V., & Manzini, G. (2016). The nonconforming virtual element method for the stokes equations. SIAM Journal on Numerical Analysis, 54(6), 3411-3435. https://doi.org/10.1137/15M1049531

© 2016 Society for Industrial and Applied Mathematics. We present the nonconforming virtual element method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous p... Read More about The nonconforming virtual element method for the stokes equations.

Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains (2016)
Book Chapter
Antonietti, P. F., Cangiani, A., Collis, J., Dong, Z., Georgoulis, E. H., Giani, S., & Houston, P. (2016). Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains. In G. R. Barrenechea, F. Brezzi, A. Cangiani, & E. H. Georgoulis (Eds.), Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations (281-310). Cham: Springer Publishing Company. https://doi.org/10.1007/978-3-319-41640-3_9

The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of st... Read More about Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains.

Conforming and nonconforming virtual element methods for elliptic problems (2016)
Journal Article
Cangiani, A., Manzini, G., & Sutton, O. J. (2017). Conforming and nonconforming virtual element methods for elliptic problems. IMA Journal of Numerical Analysis, 37(3), 1317-1354. https://doi.org/10.1093/imanum/drw036

We present in a unified framework new conforming and nonconforming Virtual Element Methods (VEM) for general second order elliptic problems in two and three dimensions. The differential operator is split into its symmetric and non-symmetric parts and... Read More about Conforming and nonconforming virtual element methods for elliptic problems.

hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes (2016)
Journal Article
Cangiani, A., Dong, Z., Georgoulis, E. H., & Houston, P. (2016). hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes. ESAIM: Mathematical Modelling and Numerical Analysis, 50(3), 699-725. https://doi.org/10.1051/m2an/2015059

We consider the hp-version interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the advection-diffusion-reaction equation on general computational meshes consisting of polygonal/polyhedral (polytopi... Read More about hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes.

Discontinuous Galerkin methods for fast reactive mass transfer through semi-permeable membranes (2014)
Journal Article
Cangiani, A., Georgoulis, E. H., & Jensen, M. (2016). Discontinuous Galerkin methods for fast reactive mass transfer through semi-permeable membranes. Applied Numerical Mathematics, 104, 3-14. https://doi.org/10.1016/j.apnum.2014.06.007

A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is analysed. The study of interface problems is motivate... Read More about Discontinuous Galerkin methods for fast reactive mass transfer through semi-permeable membranes.

hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes (2014)
Journal Article
Cangiani, A., Georgoulis, E. H., & Houston, P. (2014). hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes. Mathematical Models and Methods in Applied Sciences, 24(10), 2009-2041. https://doi.org/10.1142/S0218202514500146

An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analy... Read More about hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes.

Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems (2013)
Journal Article
Cangiani, A., Georgoulis, E. H., & Metcalfe, S. (2014). Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems. IMA Journal of Numerical Analysis, 34(4), 1578-1597. https://doi.org/10.1093/imanum/drt052

© 2013 The authors. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This work is concerned with the derivation of a robust a posteriori error estimator for a discontinuous Gale... Read More about Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems.