Roggendorf, Sarah
(2019)
Eliminating the Gibbs phenomenon: the nonlinear PetrovGalerkin method for the convectiondiffusionreaction equation.
PhD thesis, University of Nottingham.
Abstract
In this thesis we consider the numerical approximation of the convectiondiffusionreaction equation. One of the main challenges of designing a numerical method for this problem is that boundary and interior layers typically present in the convectiondominated case can lead to nonphysical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The aim of this thesis is to develop a numerical method that eliminates Gibbs phenomena in the numerical approximation.
We consider a weak formulation of the partial differential equation of interest in L^qtype Sobolev spaces, with 1<q<∞. We then apply a nonstandard, nonlinear PetrovGalerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous PetrovGalerkin methods, this method is based on minimizing the residual in a dual norm. By replacing the intractable dual norm by a suitable discrete dual norm gives rise to a nonlinear inexact mixed method. This generalizes the PetrovGalerkin framework developed in the context of discontinuous PetrovGalerkin methods to more general Banach spaces, paving the way for designing finite element methods in nonstandard function spaces. This, in particular, permits the selection of a solution space in which the best approximation of the solution has desirable properties.
For the convectiondiffusionreaction equation, we obtain a generalization of a similar approach from the L^2setting to the L^qsetting and discuss the choices we have made regarding the continuous and discrete test spaces and the corresponding norms. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples.
We furthermore demonstrate that the approximations obtained with our scheme qualitatively behave like the L^qbest approximation of the analytical solution in the same finite element space. We use this observation to study more closely in which cases the oscillations in the numerical approximation vanish as q tends to 1. To this end, we investigate Gibbs phenomena in the context of the L^qbest approximation of discontinuities in finite element spaces with 1≤q∞. Using carefully selected examples, we show that on certain meshes the Gibbs phenomenon can be eliminated in the limit as q tends to 1. We then use these results to design the underlying meshes of the finite element spaces employed for our numerical scheme for the convectiondiffusionreaction equation such that Gibbs phenomena in the numerical approximation are eliminated.
While it is classical in the context of finite element methods to consider the solution of the convection diffusionreaction equation in the Hilbert space H_0^1(Ω)$, the Banach Sobolev space W^{1,q}_0(Ω), 1<q<∞, has received very little attention in this context. However, it is more general allowing for less regular solutions and, moreover, it allows us to consider the nonlinear PetrovGalerkin method that forms the centre of this research. In this thesis, we therefore also present a wellposedness theory for the convectiondiffusionreaction equation in the W^{1,q}_0(Ω)W_0^{1,q'}(Ω) functional setting, 1/q+1/q'=1. The theory is based on directly establishing the infsup conditions which are essential to the analysis of the nonlinear PetrovGalerkin method. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplace operator.
Item Type: 
Thesis (University of Nottingham only)
(PhD)

Supervisors: 
van der Zee, Kristoffer George Houston, Paul 
Keywords: 
ConvectionDiffusion Equation, Gibbs phenomenon, InfSup Condition, Galerkin Methods, PetrovGalerkin Methods, FEM, WellPosedness, Banach Spaces, Elliptic Regularity, Best Approximation, L^q 
Subjects: 
Q Science > QA Mathematics > QA299 Analysis 
Faculties/Schools: 
UK Campuses > Faculty of Science > School of Mathematical Sciences 
Item ID: 
59436 
Depositing User: 
Roggendorf, Sarah

Date Deposited: 
05 Mar 2021 08:10 
Last Modified: 
16 Mar 2021 15:33 
URI: 
https://eprints.nottingham.ac.uk/id/eprint/59436 
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