University of Leicester
2015metcalfesaphd.pdf (2.08 MB)
Download file

Adaptive discontinuous Galerkin methods for nonlinear parabolic problems

Download (2.08 MB)
posted on 2015-04-22, 14:56 authored by Stephen Arthur Metcalfe
This work is devoted to the study of a posteriori error estimation and adaptivity in parabolic problems with a particular focus on spatial discontinuous Galerkin (dG) discretisations. We begin by deriving an a posteriori error estimator for a linear non-stationary convection-diffusion problem that is discretised with a backward Euler dG method. An adaptive algorithm is then proposed to utilise the error estimator. The effectiveness of both the error estimator and the proposed algorithm is shown through a series of numerical experiments. Moving on to nonlinear problems, we investigate the numerical approximation of blow-up. To begin this study, we first look at the numerical approximation of blow-up in nonlinear ODEs through standard time stepping schemes. We then derive an a posteriori error estimator for an implicit-explicit (IMEX) dG discretisation of a semilinear parabolic PDE with quadratic nonlinearity. An adaptive algorithm is proposed that uses the error estimator to approach the blow-up time. The adaptive algorithm is then applied in a series of test cases to gauge the effectiveness of the error estimator. Finally, we consider the adaptive numerical approximation of a nonlinear interface problem that is used to model the mass transfer of solutes through semi-permiable membranes. An a posteriori error estimator is proposed for the IMEX dG discretisation of the model and its effectiveness tested through a series of numerical experiments.



Georgoulis, Emmanuil; Cangiani, Andrea

Date of award


Author affiliation

Department of Mathematics

Awarding institution

University of Leicester

Qualification level

  • Doctoral

Qualification name

  • PhD



Usage metrics

    University of Leicester Theses