Eliminating the Gibbs phenomenon: the non-linear Petrov-Galerkin method for the convection-diffusion-reaction equation

Abstract

In this thesis we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary and interior layers typically present in the convection-dominated case can lead to non-physical 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^q-type Sobolev spaces, with 1<q<∞. We then apply a non-standard, non-linear Petrov-Galerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov-Galerkin 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 non-linear inexact mixed method. This generalizes the Petrov-Galerkin framework developed in the context of discontinuous Petrov-Galerkin methods to more general Banach spaces, paving the way for designing finite element methods in non-standard 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 convection-diffusion-reaction equation, we obtain a generalization of a similar approach from the L^2-setting to the L^q-setting 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^q-best 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^q-best 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 convection-diffusion-reaction 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 diffusion-reaction 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 non-linear Petrov-Galerkin method that forms the centre of this research. In this thesis, we therefore also present a well-posedness theory for the convection-diffusion-reaction 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 inf-sup conditions which are essential to the analysis of the non-linear Petrov-Galerkin 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