Academic literature on the topic 'Poincaré-Steklov operators'

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Journal articles on the topic "Poincaré-Steklov operators"

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Novikov, R. G., and I. A. Taimanov. "Darboux Moutard Transformations and Poincaré—Steklov Operators." Proceedings of the Steklov Institute of Mathematics 302, no. 1 (August 2018): 315–24. http://dx.doi.org/10.1134/s0081543818060160.

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Deparis, Simone, Marco Discacciati, Gilles Fourestey, and Alfio Quarteroni. "Fluid–structure algorithms based on Steklov–Poincaré operators." Computer Methods in Applied Mechanics and Engineering 195, no. 41-43 (August 2006): 5797–812. http://dx.doi.org/10.1016/j.cma.2005.09.029.

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Demidov, A. S., and A. S. Samokhin. "Explicit Numerically Implementable Formulas for Poincaré–Steklov Operators." Computational Mathematics and Mathematical Physics 64, no. 2 (February 2024): 237–47. http://dx.doi.org/10.1134/s0965542524020040.

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Natarajan, Ramesh. "Domain Decomposition Using Spectral Expansions of Steklov–Poincaré Operators." SIAM Journal on Scientific Computing 16, no. 2 (March 1995): 470–95. http://dx.doi.org/10.1137/0916029.

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Xu, Jinchao, and Shuo Zhang. "Norms of Discrete Trace Functions of (Ω) and (Ω)." Computational Methods in Applied Mathematics 12, no. 4 (2012): 500–512. http://dx.doi.org/10.2478/cmam-2012-0025.

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AbstractThis paper discusses the constructive and computational presentations of several non-local norms of discrete trace functions of H¹(Ω) and H²(Ω) defined on the boundary or interface of an unstructured grid. We transform the nonlocal norms of trace functions to local norms of certain functions defined on the whole domain by constructing isomorphic extension operators. A unified approach is used to explore several typical examples. Additionally, we also discuss exactly invertible Poincaré–Steklov operators and their discretization.
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ACHDOU, YVES, and FREDERIC NATAF. "PRECONDITIONERS FOR THE MORTAR METHOD BASED ON LOCAL APPROXIMATIONS OF THE STEKLOV-POINCARÉ OPERATOR." Mathematical Models and Methods in Applied Sciences 05, no. 07 (November 1995): 967–97. http://dx.doi.org/10.1142/s0218202595000516.

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Many implicit Navier-Stokes solvers involve the discretization of an elliptic partial differential equation of the type −Δu+ηu=f, where η is a large positive parameter. The discretization studied here is the mortar finite element method, a domain decomposition method allowing nonmatching meshes at subdomains interfaces. Two kinds of improvements are proposed here in order to reduce the condition number of the corresponding linear systems: the first one lies on building preconditioners by approximating Steklov-Poincaré operators on subdomains boundaries by second-order partial differential operators; the second one consists in making a nonstandard choice of jump operators at subdomains interfaces. Both ideas are tested numerically.
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Natarajan, Ramesh. "Domain Decomposition using Spectral Expansions of Steklov--Poincaré Operators II: A Matrix Formulation." SIAM Journal on Scientific Computing 18, no. 4 (July 1997): 1187–99. http://dx.doi.org/10.1137/s1064827594274309.

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NICAISE, SERGE, and ANNA-MARGARETE SÄNDIG. "TRANSMISSION PROBLEMS FOR THE LAPLACE AND ELASTICITY OPERATORS: REGULARITY AND BOUNDARY INTEGRAL FORMULATION." Mathematical Models and Methods in Applied Sciences 09, no. 06 (August 1999): 855–98. http://dx.doi.org/10.1142/s0218202599000403.

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This paper is devoted to some transmission problems for the Laplace and linear elasticity operators in two- and three-dimensional nonsmooth domains. We investigate the behaviour of harmonic and linear elastic fields near geometrical singularities, especially near corner points or edges where the interface intersects with the boundaries. We give a short overview about the known results for 2-D problems and add new results for 3-D problems. Numerical results for the calculation of the singular exponents in the asymptotic expansion are presented for both two- and three-dimensional problems. Some spectral properties of the corresponding parameter depending operator bundles are also given. Furthermore, we derive boundary integral equations for the solution of the transmission problems, which lead finally to "local" pseudo-differential operator equations with corresponding Steklov–Poincaré operators on the interface. We discuss their solvability and uniqueness. The above regularity results are used in order to characterize the regularity of the solutions of these integral equations.
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Hao, Sijia, and Per-Gunnar Martinsson. "A direct solver for elliptic PDEs in three dimensions based on hierarchical merging of Poincaré–Steklov operators." Journal of Computational and Applied Mathematics 308 (December 2016): 419–34. http://dx.doi.org/10.1016/j.cam.2016.05.013.

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Zhang, Yi, Varun Jain, Artur Palha, and Marc Gerritsma. "The Discrete Steklov–Poincaré Operator Using Algebraic Dual Polynomials." Computational Methods in Applied Mathematics 19, no. 3 (July 1, 2019): 645–61. http://dx.doi.org/10.1515/cmam-2018-0208.

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AbstractIn this paper, we will use algebraic dual polynomials to set up a discrete Steklov–Poincaré operator for the mixed formulation of the Poisson problem. The method will be applied in curvilinear coordinates and to a test problem which contains a singularity. Exponential convergence of the trace variable in {H^{1/2}}-norm will be shown.
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Dissertations / Theses on the topic "Poincaré-Steklov operators"

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Zreik, Mahdi. "Spectral properties of Dirac operators on certain domains." Electronic Thesis or Diss., Bordeaux, 2024. http://www.theses.fr/2024BORD0085.

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Cette thèse se focalise sur l'étude spectrale des modèles de perturbations de l'opérateur de Dirac libre en dimensions 2 et 3.Le premier chapitre de cette thèse étudie la perturbation de l'opérateur de Dirac par une grande masse M, supportée sur un domaine. Notre objectif principal est d'établir, sous la condition d'une masse M suffisamment grande, la convergence de l'opérateur perturbé vers l'opérateur de Dirac avec la condition au bord MIT bag, au sens de la norme de la résolvante. Pour se faire, nous introduisons ce que nous appelons les opérateurs Poincaré-Steklov (PS) (comme un analogue des opérateurs Dirichlet-to-Neumann pour l'opérateur de Laplace) et les analysons d'un point de vue microlocal, afin de comprendre précisément le taux de convergence de la résolvante. D'une part, nous montrons que les opérateurs PS s'intègrent dans le cadre des opérateurs pseudodifférentiels et nous déterminons leurs symboles principaux. D'autre part, comme nous nous intéressons principalement aux grandes masses, nous traitons notre problème du point de vue semiclassique, où le paramètre semiclassique est h = M^{-1}. Enfin, en établissant une formule de Krein reliant la résolvante de l'opérateur perturbé à celle de l'opérateur MIT bag, et en utilisant les propriétés pseudodifférentielles des opérateurs PS combinées aux structures matricielles des symboles principaux, nous établissons la convergence requise avec un taux de convergence de O(M^{-1}.Dans le chapitre 2, nous définissons un voisinage tubulaire de la frontière d'un domaine régulier donné. Nous considérons la perturbation de l'opérateur de Dirac libre par une grande masse M, supportée dans ce voisinage d'épaisseur varepsilon:=M^{-1}. Notre objectif principal est d'étudier la convergence de l'opérateur de Dirac perturbé lorsque M tend vers l'infini. En comparaison avec la première partie, nous obtenons ici deux opérateurs limites MIT bag, qui agissent en dehors de la frontière. Il est intéressant de noter que le découplage de ces deux opérateurs MIT bag peut être considéré comme la version confinée de delta-interaction scalaire de Lorentz de l'opérateur de Dirac, supportée sur une surface fermée. La méthodologie suivie, comme au problème précédent, porte sur l'étude des propriétés pseudodifférentielles des opérateurs PS. Cependant, la nouveauté de ce problème réside dans le contrôle de ces opérateurs en suivant la dépendance du paramètre varepsilon, et par conséquent, dans la convergence lorsque varepsilon tend vers 0 et M tend vers l'infini. Avec ces ingrédients, nous prouvons que l'opérateur perturbé converge au sens de la norme de la résolvante vers l'opérateur de Dirac couplé à une delta-interaction scalaire de Lorentz.Dans le chapitre 3, nous généralisation une approximation de l'opérateur de Dirac tridimensionnel couplé à une combinaison singulière de delta-interactions électrostatiques et scalaires de Lorentz supportée sur une surface fermée, par un opérateur de Dirac avec un potentiel régulier localisé dans une couche mince contenant la surface. Dans les cas non-critiques et non-confinants, nous montrons que l'opérateur de Dirac perturbé régulier converge au sens de la résolvante forte vers la delta-interaction singulière de l'opérateur de Dirac.Dans le dernier chapitre, notre étude porte sur l'opérateur de Dirac bidimensionnel couplé à une delta-interaction électrostatique et scalaire de Lorentz. Nous traitons dans des espaces de Sobolev d'ordre un-demi l'auto-adjonction de certaines réalisations de ces opérateurs dans divers contextes de courbes. Le cas le plus important se présente lorsque les courbes considérées sont des polygones curvilignes. Sous certaines conditions sur les constantes de couplage, en utilisant la propriété de Fredholm de certains opérateurs intégraux de frontière, et en exploitant la forme explicite de la transformée de Cauchy sur des courbes non lisses, nous établissons l'auto-adjonction de l'opérateur perturbé
This thesis mainly focused on the spectral analysis of perturbation models of the free Dirac operator, in 2-D and 3-D space.The first chapter of this thesis examines perturbation of the Dirac operator by a large mass M, supported on a domain. Our main objective is to establish, under the condition of sufficiently large mass M, the convergence of the perturbed operator, towards the Dirac operator with the MIT bag condition, in the norm resolvent sense. To this end, we introduce what we refer to the Poincaré-Steklov (PS) operators (as an analogue of the Dirichlet-to-Neumann operators for the Laplace operator) and analyze them from the microlocal point of view, in order to understand precisely the convergence rate of the resolvent. On one hand, we show that the PS operators fit into the framework of pseudodifferential operators and we determine their principal symbols. On the other hand, since we are mainly concerned with large masses, we treat our problem from the semiclassical point of view, where the semiclassical parameter is h = M^{-1}. Finally, by establishing a Krein formula relating the resolvent of the perturbed operator to that of the MIT bag operator, and using the pseudodifferential properties of the PS operators combined with the matrix structures of the principal symbols, we establish the required convergence with a convergence rate of mathcal{O}(M^{-1}).In the second chapter, we define a tubular neighborhood of the boundary of a given regular domain. We consider perturbation of the free Dirac operator by a large mass M, within this neighborhood of thickness varepsilon:=M^{-1}. Our primary objective is to study the convergence of the perturbed Dirac operator when M tends to +infty. Comparing with the first part, we get here two MIT bag limit operators, which act outside the boundary. It's worth noting that the decoupling of these two MIT bag operators can be considered as the confining version of the Lorentz scalar delta interaction of Dirac operator, supported on a closed surface. The methodology followed, as in the previous problem study the pseudodifferential properties of Poincaré-Steklov operators. However, the novelty in this problem lies in the control of these operators by tracking the dependence on the parameter varepsilon, and consequently, in the convergence as varepsilon goes to 0 and M goes to +infty. With these ingredients, we prove that the perturbed operator converges in the norm resolvent sense to the Dirac operator coupled with Lorentz scalar delta-shell interaction.In the third chapter, we investigate the generalization of an approximation of the three-dimensional Dirac operator coupled with a singular combination of electrostatic and Lorentz scalar delta-interactions supported on a closed surface, by a Dirac operator with a regular potential localized in a thin layer containing the surface. In the non-critical and non-confining cases, we show that the regular perturbed Dirac operator converges in the strong resolvent sense to the singular delta-interaction of the Dirac operator. Moreover, we deduce that the coupling constants of the limit operator depend nonlinearly on those of the potential under consideration.In the last chapter, our study focuses on the two-dimensional Dirac operator coupled with the electrostatic and Lorentz scalar delta-interactions. We treat in low regularity Sobolev spaces (H^{1/2}) the self-adjointness of certain realizations of these operators in various curve settings. The most important case in this chapter arises when the curves under consideration are curvilinear polygons, with smooth, differentiable edges and without cusps. Under certain conditions on the coupling constants, using the Fredholm property of certain boundary integral operators, and exploiting the explicit form of the Cauchy transform on non-smooth curves, we achieve the self-adjointness of the perturbed operator
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Hilal, Mohammed Azeez. "Domain decomposition like methods for solving an electrocardiography inverse problem." Thesis, Nantes, 2016. http://www.theses.fr/2016NANT4060.

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L’objectif de cette thèse est d’étudier un problème électrocardiographique (ECG), modélisant l’activité électrique cardiaque en utilisant un modèle bidomaine stationnaire. Deux types de modélisation sont considérées : la modélisation basée sur un modèle mathématique directe et la modélisation basée sur un problème inverse de Cauchy. Dans le premier cas, le problème directe est résolu en utilisant la méthode de décomposition de domaine et l’approximation par la méthode des éléments finis. Dans le deuxième cas le problème inverse de Cauchy de l’ECG a été reformulé en un problème de point fixe. Puis, un résultat d’existence et l’unicité du point fixe basé sur les degrés topologique de Leray-Schauder a été démontré. Ensuite, quelques algorithmes itératifs régularisant et stables basés sur les techniques de décomposition de domaine ont été développés. Enfin, l’efficacité et la précision des résultats obtenus a été discutés
The aim of the this thesis is to study an electrocardiography (ECG) problem, modeling the cardiac electrical activity by using the stationary bidomain model. Tow types of modeling are considered :The modeling based on direct mathematical model and the modeling based on an inverse Cauchy problem. In the first case, the direct problem is solved by using domain decomposition methods and the approximation by finite elements method. For the inverse Cauchy problem of ECG, it was reformulated into a fixed point problem. In the second case, the existence and uniqueness of fixed point based on the topological degree of Leray-Schauder is showed. Then, some regularizing and stable iterative algorithms based on the techniques of domain decomposition method was developed. Finally, the efficiency and the accurate of the obtained results was discussed
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Perlich, Lars. "Holomorphic Semiflows and Poincaré-Steklov Semigroups." Doctoral thesis, 2019. https://tud.qucosa.de/id/qucosa%3A36097.

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Die Arbeit untersucht einen überraschenden Zusammenhang zwischen Halbflüssen von holomorphen Selbstabbildungen auf einfach zusammenhängenden Gebieten und Halbgruppen, die von Poincaré-Steklov Operatoren erzeugt werden. Mithilfe von Erzeuger von Kompositionshalbgruppen auf Banachräumen von analytischen Funktionen werden insbesondere Dirichlet-zu-Neumann und Dirichlet-zu-Robin Operatoren konstruiert. Dieser Zugang eröffnet einen neuen Ansatz für das Studium partiellen Differentialgleichungen, die mit solchen Operatoren assoziiert sind.
We study a surprising connection between semiflows of holomorphic selfmaps of a simply connected domain and semigroups generated by Poincaré-Steklov operators. In particular, by means of generators of semigroups of composition operators on Banach spaces of analytic functions, we construct Dirichlet-to-Neumann and Dirichlet-to-Robin operators. This approach gives new insights to the theory of partial differential equations associated with such operators.
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Held, Joachim. "Ein Gebietszerlegungsverfahren für parabolische Probleme im Zusammenhang mit Finite-Volumen-Diskretisierung." Doctoral thesis, 2006. http://hdl.handle.net/11858/00-1735-0000-0006-B39E-E.

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Book chapters on the topic "Poincaré-Steklov operators"

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Khoromskij, Boris N., and Gabriel Wittum. "Elliptic Poincaré-Steklov Operators." In Lecture Notes in Computational Science and Engineering, 37–62. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-642-18777-3_2.

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Quarteroni, A., and A. Valli. "Theory and Application of Steklov-Poincaré Operators for Boundary-Value Problems." In Applied and Industrial Mathematics, 179–203. Dordrecht: Springer Netherlands, 1991. http://dx.doi.org/10.1007/978-94-009-1908-2_14.

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Hu, Qiya. "A New Kind of Multilevel Solver for Second Order Steklov-Poincaré Operators." In Lecture Notes in Computational Science and Engineering, 391–98. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-75199-1_49.

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Novotny, Antonio André, Jan Sokołowski, and Antoni Żochowski. "Steklov–Poincaré Operator for Helmholtz Equation." In Applications of the Topological Derivative Method, 41–50. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-05432-8_3.

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Gosse, Laurent. "Viscous Equations Treated with $$\mathcal{L}$$ -Splines and Steklov-Poincaré Operator in Two Dimensions." In Innovative Algorithms and Analysis, 167–95. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-49262-9_6.

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