Andrea Cangiani

Andrea Cangiani, Emmanuil H. Georgoulis, Tristan Pryer et al.

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 computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing.

Andrea Cangiani, Panagiotis Chatzipantelidis, Ganesh Diwan et al.

We present a Virtual Element Method (VEM) for the solution of Dirichlet problems for the quasilinear equation $-\text{div} (k(u)\text{grad} u)=f$ with essential boundary conditions. Within the VEM 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 $L^2$ norms are proven. In addition, the convergence of fixed point iterations for the solution of the resulting nonlinear system is established. Numerical examples confirm the convergence analysis.

Andrea Cangiani, Emmanuil H. Georgoulis, Irene Kyza et al.

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 posteriori error bounds are derived in the $L^{\infty}(L^2)+L^2(H^1)$-type norm for a first order in time implicit-explicit (IMEX) interior penalty discontinuous Galerkin (dG) in space discretization of the problem, although the theory presented is directly applicable to the case of conforming finite element approximations in space. The choice of the discretization in time is made based on a careful analysis of adaptive time stepping methods for ODEs that exhibit finite time blow-up. The new adaptive algorithm is shown to accurately estimate the blow-up time of a number of problems, including one which exhibits regional blow-up.

Andrea Cangiani, Emmanuil H. Georgoulis, Andrew Yu. Morozov et al.

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 work, 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 characterised 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 proven 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 both in two and three spatial dimensions.

Andrea Cangiani, John Chapman, Emmanuil Georgoulis et al.

We consider a finite element method which couples the continuous Galerkin method away from internal and boundary layers with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. The stability properties of the coupled method are illustrated by numerical experiments.

Andrea Cangiani, Emmanuil H. Georgoulis, Max Jensen

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 presented and analysed. The study of interface problems is motivated by models of mass transfer of solutes through semi-permeable membranes. More specifically, a model problem consisting of a system of semilinear parabolic advection-diffusion-reaction partial differential equations in each compartment, equipped with respective initial and boundary conditions, is considered. Nonlinear interface conditions modelling selective permeability, congestion and partial reflection are applied to the compartment interfaces. An interior penalty dG method is presented for this problem and it is analysed in the space-discrete setting. The a priori analysis shows that the method yields optimal a priori bounds, provided the exact solution is sufficiently smooth. Numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection-dominated regime.

Andrea Cangiani, Emmanuil H. Georgoulis, Stephen Metcalfe

This work is concerned with the derivation of a robust a posteriori error estimator for a discontinuous Galerkin method discretisation of linear non-stationary convection-diffusion initial/boundary value problems and with the implementation of a corresponding adaptive algorithm. More specifically, we derive a posteriori bounds for the error in the $L^2(H^1)$-type norm for an interior penalty discontinuous Galerkin (dG) discretisation in space and a backward Euler discretisation in time. An important feature of the estimator is robustness with respect to the Péclet number of the problem which is verified in practice by a series of numerical experiments. Finally, an adaptive algorithm is proposed utilising the error estimator. Optimal rate of convergence of the adaptive algorithm is observed in a number of test problems.

Andrea Cangiani

In this short note we review deterministic simulation of biochemical pathways, i.e. networks of biochemical reactions obeying the law of mass action. It is meant as a basis for the MATLAB code, written by the author, which permits easy input and simulation of general biochemical networks. This work was carried out for the European Project `CardioWorkBench'.

Andrea Cangiani

These pages review two families of mimetic finite difference methods: the mixed-type methods presented in [Brezzi, Lipnikov, and Simoncini, M3AS, 2005] and the nodal methods of [Brezzi, Buffa, and Lipnikov, M2AN, 2009]. The purpose of this exercise it to highlight the similitudes underlying the construction of the two families. The comparison prompts the definition of a piecewise linear postprocessing of the nodal mimetic finite difference solution, as it was done for the mixed-type method in [Cangiani and Manzini, CMAME, 2008].

Andrea Cangiani, Mauricio Munar

We present an a posteriori error analysis for the mixed virtual element method (mixed VEM) applied to second order elliptic equations in divergence form with mixed boundary conditions. The resulting error estimator is of residual-type. It only depends on quantities directly available from the VEM solution and applies on very general polygonal meshes. The proof of the upper bound relies on a global inf-sup condition, a suitable Helmholtz decomposition, and the local approximation properties of a Clément-type interpolant. In turn, standard inverse inequalities and localization techniques based on bubble functions are the main tools yielding the lower bound. Via the inclusion of a fully local postprocessing of the mixed VEM solution, we also show that the estimator provides a reliable and efficient control on the broken $\mathrm{H}(\mathrm{div})$-norm error between the exact and the postprocessed flux. Numerical examples confirm the theoretical properties of our estimator, and show that it can be effectively used to drive an adaptive mesh refinement algorithm.

Andrea Cangiani, Vitaliy Gyrya, Gianmarco Manzini

We present the non-conforming Virtual Element Method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable non-polynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two-and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the non-conforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.

Andrea Cangiani, Gianmarco Manzini, Oliver J. Sutton

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 conditions for stability and accuracy on their discrete counterparts are established. These conditions are shown to lead to optimal $H^1$- and $L^2$-error estimates, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the two methods is shown to be comparable.