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

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

Gibbs Phenomena for LqL^q-Best Approximation in Finite Element Spaces -- Some Examples

Recent developments in the context of minimum residual finite element methods are 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. One of the biggest challenges in designing finite element methods are non-physical oscillations near thin layers and jump discontinuities. In this article we investigate Gibbs phenomena in the context of $L^q$-best approximation of discontinuities in finite element spaces with $1\leq q<\infty$. Using carefully selected examples, we show that on certain meshes the Gibbs phenomenon can be eliminated in the limit as $q$ tends to $1$. The aim here is to show the potential of $L^1$ as a solution space in connection with suitably designed meshes

Eliminating Gibbs Phenomena: A Non-linear Petrov-Galerkin Method for the Convection-Diffusion-Reaction Equation

In this article 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 layers occurring in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in Lq-type Sobolev spaces, with 1 < q < $\infty$. We then apply a non-standard, non-linear PetrovGalerkin 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. 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. For the convection-diffusion-reaction equation, this yields a generalization of a similar approach from the L2-setting to the Lq-setting. 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

Gibbs phenomena for Lq-best approximation in finite element spaces

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretization methods in non-standard function spaces, such as L q-type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of these numerical observations. In particular, we investigate the Gibbs phenomena for L q-best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over-and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon

Eliminating Gibbs phenomena: A non-linear Petrov–Galerkin method for the convection–diffusion–reaction equation

In this article 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 layers occurring in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in Lq-type Sobolev spaces, with 1 < q < $\infty$. We then apply a non-standard, non-linear PetrovGalerkin 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. 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. For the convection-diffusion-reaction equation, this yields a generalization of a similar approach from the L2-setting to the Lq-setting. 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

The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H10(Ω), the Banach Sobolev space W1,q0(Ω), 1 less than ∞ , is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the W1,q0(Ω)-W1,q′0(Ω) functional setting, 1q+1q′=1. The theory is based on directly establishing the inf-sup conditions. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. An elementary consequence of the well-posedness theory is the stability and convergence of Galerkin’s method in this setting, for a diffusion-dominated case and under the assumption of W1,q′-stability of the H10-projector

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

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

Einfluss verschiedener Spüllösungen auf den Haftverbund des Sealers Apexit Plus zum Wurzelkanaldentin

Der Misserfolg einer Wurzelkanalbehandlung hat viele Ursachen, die nicht immer eindeutig sind. Dem Behandler stehen zahlreiche Herangehensweisen und Medikamente zur Verfügung, um eine endodontische Behandlung durchzuführen. Sie umfasst das mechanische Aufbereiten des Kanals, das Spülen mit geeigneter Flüssigkeit sowie einen bakterien- und flüssigkeitsdichten Verschluss. Zum einen sollte dieses Material eine Haftkraft zum umgebenden Wurzeldentin ausbilden, zum anderen sollte zum gewählten Kernmaterial ebenfalls eine Adhäsion entstehen [101]. In dieser In-vitro-Studie sollte der Einfluss verschiedener Spüllösungen auf den Haftverbund zwischen Wurzelkanaldentin und dem calciumsalicylatbasierten Sealers Apexit Plus untersucht werden. In dieser Studie wurden 60 extrahierte, kariesfreie und unbehandelte Zahnwurzeln auf fünf Gruppen randomisiert verteilt (n=12). Sie wurden dekapitiert und mit dem maschinellen Feilensystem BioRaCe bei einer Arbeitslänge von 8 mm bis .02#60 aufbereitet. Die zuvor gebildeten Gruppen wurden einer Spüllösung zugeteilt: A = Chlorhexidindiglukonat (CHX) 2%, B = Ethylendiamintetraacetat (EDTA) 16%, C = Natriumhypochlorit (NaOCl) 3%, D = Zitronensäure 40%, E = Aqua dest. Nachdem die Proben mit ihrer zugeordneten Flüssigkeit gespült wurden, erfolgte die sorgfältige Trocknung mit Papierspitzen und anschließendem Einbringen des vorbereiteten Stahlspreaders mit dem zu untersuchenden Sealer. Die Wurzeln lagerten 14 Tage unter feuchten Bedingungen. Anschließend fanden die Pullout-Tests in einer Universalprüfmaschine statt. Es wurde die maximale Kraft bis zum adhäsiven Versagen bei einer Prüfgeschwindigkeit von 2 mm/min ermittelt. Zusätzlich erfolgte eine Auswertung der Frakturmodi an den experimentellen Spreadern. Apexit Plus zeigte in Kombination mit CHX (0,73 MPa) und NaOCl (0,62 MPa) die höchsten Haftwerte. Nach einer Spülung mit EDTA zeigten sich insgesamt die geringsten Haftwerte (0,21 MPa). Es ergab sich ein signifikanter Einfluss der Spülungen auf die Haftwerte (Kruskal-Wallis Test p= 0,034). Unter den Bedingungen dieser Studie profitierte der Sealer Apexit Plus nicht von der Schmierschichtentfernung durch EDTA. Die Spülung führte zu geringen Haftwerten. Der calciumsalicylatbasierte Sealer verhielt sich somit gegensätzlich zu den meisten anderen Wurzelkanalsealern, die in der Regel zu besseren Haftwerten nach Schmierschichtentfernung gelangen. Basierend auf den Ergebnissen dieser Studie mit Apexit Plus ist eine alleinige Spülung mit EDTA nicht empfehlenswert. Ob Apexit Plus eine Verbesserung der Haftkraft durch eine zusätzliche Spülung mit CHX erlangen kann, war nicht Ziel dieser Arbeit und ist anderweitig zu erforschen

Cognitive decline in Parkinson’s disease: the impact of the motor phenotype on cognition

Objectives Parkinson’s disease (PD) is the second most common neurodegenerative disorder and is further associated with progressive cognitive decline. In respect to motor phenotype, there is some evidence that akinetic-rigid PD is associated with a faster rate of cognitive decline in general and a greater risk of developing dementia.The objective of this study was to examine cognitive profiles among patients with PD by motor phenotypes and its relation to cognitive function.Methods Demographic, clinical and neuropsychological cross-sectional baseline data of the DEMPARK/LANDSCAPE study, a multicentre longitudinal cohort study of 538 patients with PD were analysed, stratified by motor phenotype and cognitive syndrome. Analyses were performed for all patients and for each diagnostic group separately, controlling for age, gender, education and disease duration.Results Compared with the tremor-dominant phenotype, akinetic-rigid patients performed worse in executive functions such as working memory (Wechsler Memory Scale-Revised backward; p=0.012), formal-lexical word fluency (p=0.043), card sorting (p=0.006), attention (Trail Making Test version A; p=0.024) and visuospatial abilities (Leistungsprüfungssystem test 9; p=0.006). Akinetic-rigid neuropsychological test scores for the executive and attentive domain correlated negatively with non-tremor motor scores. Covariate-adjusted binary logistic regression analyses showed significant odds for PD-mild cognitive impairment for not-determined as compared with tremor-dominant (OR=3.198) and akinetic-rigid PD (OR=2.059). The odds for PD-dementia were significant for akinetic-rigid as compared with tremor-dominant phenotype (OR=8.314).Conclusion The three motor phenotypes of PD differ in cognitive performance, showing that cognitive deficits seem to be less severe in tremor-dominant PD. While these data are cross-sectional, longitudinal data are needed to shed more light on these differential findings