arXiv subjects

Andrea Cangiani

Publications and source records attributed to Andrea Cangiani.

25 records · Page 2Linked to original sources

arXiv2013

Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes

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.

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arXiv2012

An a posteriori error estimator for discontinuous Galerkin methods for non-stationary convection-diffusion problems

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.

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arXiv2012

On the Stability of Continuous-Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems

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.

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arXiv2012

On low order mimetic finite difference methods

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].

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arXiv2012

Biochemical pathways simulation

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'.

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arXiv2012

On Local Super-Penalization of Interior Penalty Discontinuous Galerkin Methods

We prove in an abstract setting that standard (continuous) Galerkin finite element approximations are the limit of interior penalty discontinuous Galerkin approximations as the penalty parameter tends to infinity. We apply this result to equations of non-negative characteristic form and the non-linear, time dependent system of incompressible miscible displacement. Moreover, we investigate varying the penalty parameter on only a subset of a triangulation and the effects of local super-penalization on the stability of the method, resulting in a partly continuous, partly discontinuous method in the limit. An iterative automatic procedure is also proposed for the determination of the continuous region of the domain without loss of stability of the method.

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arXiv2012

Implementation of the Continuous-Discontinuous Galerkin Finite Element Method

For the stationary advection-diffusion problem the standard continuous Galerkin method is unstable without some additional control on the mesh or method. The interior penalty discontinuous Galerkin method is stable but at the expense of an increased number of degrees of freedom. The hybrid method proposed in [5] combines the computational complexity of the continuous method with the stability of the discontinuous method without a significant increase in degrees of freedom. We discuss the implementation of this method using the finite element library deal.ii and present some numerical experiments.

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