DPG Method with Optimal Test Functions for a Fractional Advection Diffusion Equation

Ervin, Vincent J.; Führer, Thomas; Heuer, Norbert; Karkulik, Michael

doi:10.1007/s10915-017-0369-z

Cited by 11 publications

(5 citation statements)

References 46 publications

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“…We finish this section with two propositions which provide us an extension of I α and ∂ α into wider functional spaces. The similar reasoning to the one carried in Proposition 4 may be found in [5,Lemma 5].…”

Section: Introductionsupporting

confidence: 74%

An analytic semigroup generated by a fractional differential operator

Ryszewska

2020

Journal of Mathematical Analysis and Applications

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We study a space-fractional diffusion problem, where the non-local diffusion flux involves the Caputo derivative of the diffusing quantity. We prove the unique existence of regular solutions to this problem by means of the semigroup theory. We show that the operator defined as divergence of product of a positive function with the Caputo derivative is a generator of analytic semigroup. the simplified problem (1). We would like to emphasise that from the modelling point of view, it is important that (1) is a balance low.Our goal is to study the basic solvability problem for equation (1), by the semigroup approach. At this moment we would like to present a broader context. It is worth to mention here, that in the paper [2] the authors studied an array of fractional diffusion equations, for a variety of boundary conditions, but they constructed only the C 0 -semigroup. Besides, we note that the problem (1) was not explicitly studied in [2]. Another paper that presents the probabilistic point of view on space-fractional problems is [4], where the authors consider equations with time-fractional Caputo derivative and non-local space operators. A completely different approach for solving (1) for p ≡ 1 with zero Dirichlet boundary conditions is employed in [12], where the authors obtained the viscosity solutions. Further discussion was made in [3] and [13] where the authors compare the problems with diffusive flux modeled by the Caputo and the Riemann-Liouville derivative and carry a numerical analysis.In the present paper we will present the results concerning solvability of (1) by means of the semigroup theory. At first, we will focus our attention on the case where p ≡ 1. We will describe the domain of ∂ ∂x D α in terms of Sobolev spaces and as a final result we will construct an analytic semigroup. Precisely, we will show that the operator ∂ ∂x D α , considered on the domaingenerates an analytic semigroup. Here, by H α (0, 1) we mean the fractional Sobolev space (see [10, definition 9.1]) and the subspace 0 H α (0, 1) will be introduced in Proposition 3. Subsequently, we will extend our result for the case of any strictly positive lipschitz continuous p. We would like to emphasise that, developing the theory of analytic semigroups for the systems of type (1) seems to be especially important if we notice that the operator ∂ ∂x D α is not self-adjoint, thus we can not expect that its eigenfunctions generate the basis of any natural Hilbert space.The paper is organized as follows. In chapter 2 we give preliminary results concerning fractional operators. Chapter 3 is devoted to the proof of the main result. Namely, we will show that ∂ ∂x D α is a generator of C 0 -semigroup of contractions which can be extended to the analytic semigroup (Theorem 2). In chapter 4, we extend the results of the previous chapter to the case of operator ∂ ∂x (p(x)D α ), where p is a positive lipschitz function (Theorem 3). In chapter 5, we will present the basic applications of this result to solvability of (1). At last, for the sake of comple...

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Section: Introductionsupporting

confidence: 74%

An analytic semigroup generated by a fractional differential operator

Ryszewska

2020

Journal of Mathematical Analysis and Applications

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“…Despite the aforementioned progress, the corresponding results for the variable coefficient fractional diffusion, advection, reaction equations are largely missing. There exists some recent work on Petrov-Galerkin approximations to two-sided fractional diffusion, reaction equations [15,17], onesided fractional diffusion, advection, reaction equations [8,12,20], and two-sided fractional diffusion, advection, reaction equations [36], all with constant diffusivity coefficients. To the best of our knowledge, the only available result for the Petrov-Galerkin method applied to variable coefficient fractional diffusion problems is [29], in which the weak coercivity in the sense of inf-sup condition was proved for the one-sided variable coefficient fractional diffusion operator, i.e., Lα r with r = 1.…”

Section: 6)mentioning

confidence: 99%

Analysis and Petrov-Galerkin numerical approximation for variable coefficient two-sided fractional diffusion, advection, reaction equations

Zheng¹,

Ervin²,

Wang³

2022

Preprint

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In this paper we investigate the variable coefficient two-sided fractional diffusion, advection, reaction equations on a bounded interval. It is known that the fractional diffusion operator may lose coercivity due to the variable coefficient, which makes both the mathematical and numerical analysis challenging. To resolve this issue, we design appropriate test and trial functions to prove the inf-sup condition of the variable coefficient fractional diffusion, advection, reaction operators in suitable function spaces. Based on this property, we prove the well-posedness and regularity of the solutions, as well as analyze the Petrov-Galerkin approximation scheme for the proposed model. Numerical experiments are presented to substantiate the theoretical findings and to compare the behaviors of different models.

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“…A number of papers study the convection-diffusion-reaction equations with or without singular pertubation [DG10; DGN12; DH13; CEQ14; CHBD14; BS14; BS15; FH17; HK17a; MZ17; BDS18; Füh18]. Other applications include Maxwell's equations [DL13;CDG16], wave propagation [DGMZ12; GMO14; PD17], the Schrödinger equation [DGNS17], the fractional Laplacian [ABH18] and fractional advection-diffusion [EFHK17], the heat equation [FHS17a], transmission problems [HK15; FHK17; FHKR17] and a hypersingular integral equation [HP14;HK17b]. The dPG methodology has also been applied to nonlinear model problems [CBHW18], to a contact problem [FHS17b], in nonlinear fluid mechanics [CDM14;RDM15] and viscoelasticity [KKRE+17;FDW17].…”

Section: Motivationmentioning

confidence: 99%

Optimal Convergence Rates for Adaptive Lowest-Order Discontinuous Petrov--Galerkin Schemes

Carstensen¹,

Hellwig²

2018

SIAM J. Numer. Anal.

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The thesis »Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods« proves optimal convergence rates for four lowest-order discontinuous Petrov-Galerkin methods for the Poisson model problem for a sufficiently small initial mesh-size in two different ways by equivalences to two other non-standard classes of finite element methods, the reduced mixed and the weighted Least-Squares method. The first is a mixed system of equations with first-order conforming Courant and nonconforming Crouzeix-Raviart functions. The second is a generalized Least-Squares formulation with a midpoint quadrature rule and weight functions. The thesis generalizes a result on the primal discontinuous Petrov-Galerkin method from [Carstensen, Bringmann, Hellwig, Wriggers 2018] and characterizes all four discontinuous Petrov-Galerkin methods simultaneously as particular instances of these methods. It establishes alternative reliable and efficient error estimators for both methods. A main accomplishment of this thesis is the proof of optimal convergence rates of the adaptive schemes in the axiomatic framework [Carstensen, Feischl, Page, Praetorius 2014]. The optimal convergence rates of the four discontinuous Petrov-Galerkin methods then follow as special cases from this rate-optimality. Numerical experiments verify the optimal convergence rates of both types of methods for different choices of parameters. Moreover, they complement the theory by a thorough comparison of both methods among each other and with their equivalent discontinuous Petrov-Galerkin schemes. ZusammenfassungDie vorliegende Arbeit »Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods« beweist optimale Konvergenzraten für vier diskontinuierliche Petrov-Galerkin (dPG) Finite-Elemente-Methoden für das Poisson-Modell-Problem für genügend feine Anfangstriangulierung. Sie zeigt dazu die Äquivalenz dieser vier Methoden zu zwei anderen Klassen von Methoden, den reduzierten gemischten Methoden und den verallgemeinerten Least-Squares-Methoden. Die erste Klasse benutzt ein gemischtes System aus konformen Courantund nichtkonformen Crouzeix-Raviart-Finite-Elemente-Funktionen. Die zweite Klasse verallgemeinert die Standard-Least-Squares-Methoden durch eine Mittelpunktsquadratur und Gewichtsfunktionen. Diese Arbeit verallgemeinert ein Resultat aus [Carstensen, Bringmann, Hellwig, Wriggers 2018], indem die vier dPG-Methoden simultan als Spezialfälle dieser zwei Klassen charakterisiert werden. Sie entwickelt alternative Fehlerschätzer für beide Methoden und beweist deren Zuverlässigkeit und Effizienz. Ein Hauptresultat der Arbeit ist der Beweis optimaler Konvergenzraten der adaptiven Methoden durch Beweis der Axiome aus [Carstensen, Feischl, Page, Praetorius 2014]. Daraus folgen dann insbesondere die optimalen Konvergenzraten der vier dPG-Methoden. Numerische Experimente bestätigen diese optimalen Konvergenzraten für beide Klassen von Methoden. Außerdem ergänzen sie die Theorie durch ausführliche Vergleiche beider Methoden untereinander und mit den äquivalenten dPG-Methoden. C...

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DPG Method with Optimal Test Functions for a Fractional Advection Diffusion Equation

Cited by 11 publications

References 46 publications

An analytic semigroup generated by a fractional differential operator

An analytic semigroup generated by a fractional differential operator

Analysis and Petrov-Galerkin numerical approximation for variable coefficient two-sided fractional diffusion, advection, reaction equations

Optimal Convergence Rates for Adaptive Lowest-Order Discontinuous Petrov--Galerkin Schemes

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