36 research outputs found

    Instance optimal Crouzeix-Raviart adaptive finite element methods for the Poisson and Stokes problems

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    We extend the ideas of Diening, Kreuzer, and Stevenson [Instance optimality of the adaptive maximum strategy, Found. Comput. Math. (2015)], from conforming approximations of the Poisson problem to nonconforming Crouzeix-Raviart approximations of the Poisson and the Stokes problem in 2D. As a consequence, we obtain instance optimality of an AFEM with a modified maximum marking strategy

    Comparison results for the Stokes equations

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    This paper enfolds a medius analysis for the Stokes equations and compares different finite element methods (FEMs). A first result is a best approximation result for a P1 non-conforming FEM. The main comparison result is that the error of the P2-P0-FEM is a lower bound to the error of the Bernardi-Raugel (or reduced P2-P0) FEM, which is a lower bound to the error of the P1 non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The paper discusses the converse direction, as well as other methods such as the discontinuous Galerkin and pseudostress FEMs. Furthermore this paper provides counterexamples for equivalent convergence when different pressure approximations are considered. The mathematical arguments are various conforming companions as well as the discrete inf-sup condition

    Thermo-optical interactions in a dye-microcavity photon Bose-Einstein condensate

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    Superfluidity and Bose-Einstein condensation are usually considered as two closely related phenomena. Indeed, in most macroscopic quantum systems, like liquid helium, ultracold atomic Bose gases, and exciton-polaritons, condensation and superfluidity occur in parallel. In photon Bose-Einstein condensates realized in the dye microcavity system, thermalization does not occur by direct interaction of the condensate particles as in the above described systems, i.e. photon-photon interactions, but by absorption and re-emission processes on the dye molecules, which act as a heat reservoir. Currently, there is no experimental evidence for superfluidity in the dye microcavity system, though effective photon interactions have been observed from thermo-optic effects in the dye medium. In this work, we theoretically investigate the implications of effective thermo-optic photon interactions, a temporally delayed and spatially non-local effect, on the photon condensate, and derive the resulting Bogoliubov excitation spectrum. The calculations suggest a linear photon dispersion at low momenta, fulfilling the Landau's criterion of superfluidity . We envision that the temporally delayed and long-range nature of the thermo-optic photon interaction offer perspectives for novel quantum fluid phenomena.Comment: 21 pages, 5 figure

    Rot-free mixed finite elements for gradient elasticity at finite strains

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    Through enrichment of the elastic potential by the second-order gradient of deformation, gradient elasticity formulations are capable of taking nonlocal effects into account. Moreover, geometry-induced singularities, which may appear when using classical elasticity formulations, disappear due to the higher regularity of the solution. In this contribution, a mixed finite element discretization for finite strain gradient elasticity is investigated, in which instead of the displacements, the first-order gradient of the displacements is the solution variable. Thus, the C1 continuity condition of displacement-based finite elements for gradient elasticity is relaxed to C0. Contrary to existing mixed approaches, the proposed approach incorporates a rot-free constraint, through which the displacements are decoupled from the problem. This has the advantage of a reduction of the number of solution variables. Furthermore, the fulfillment of mathematical stability conditions is shown for the corresponding small strain setting. Numerical examples verify convergence in two and three dimensions and reveal a reduced computing cost compared to competitive formulations. Additionally, the gradient elasticity features of avoiding singularities and modeling size effects are demonstrated

    A class of mixed finite element methods based on the Helmholtz decomposition in computational mechanics

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    ´╗┐Diese Dissertation verallgemeinert die nichtkonformen Finite-Elemente-Methoden (FEMn) nach Morley und Crouzeix und Raviart durch neue gemischte Formulierungen f├╝r das Poisson-Problem, die Stokes-Gleichungen, die Navier-Lam├ę-Gleichungen der linearen Elastizit├Ąt und m-Laplace-Gleichungen der Form (Ôłĺ1)m╬ömu=f(-1)^m\Delta^m u=f f├╝r beliebiges m=1,2,3,... Diese Formulierungen beruhen auf Helmholtz-Zerlegungen. Die neuen Formulierungen gestatten die Verwendung von Ansatzr├Ąumen beliebigen Polynomgrades und ihre Diskretisierungen stimmen f├╝r den niedrigsten Polynomgrad mit den genannten nicht-konformen FEMn ├╝berein. Auch f├╝r h├Âhere Polynomgrade ergeben sich robuste Diskretisierungen f├╝r fast-inkompressible Materialien und Approximationen f├╝r die L├Âsungen der Stokes-Gleichungen, die punktweise die Masse erhalten. Dieser Ansatz erlaubt au├čerdem eine Verallgemeinerung der nichtkonformen FEMn von der Poisson- und der biharmonischen Gleichung auf m-Laplace-Gleichungen f├╝r beliebiges m>2. Erm├Âglicht wird dies durch eine neue Helmholtz-Zerlegung f├╝r tensorwertige Funktionen. Die neuen Diskretisierungen lassen sich nicht nur f├╝r beliebiges m einheitlich implementieren, sondern sie erlauben auch Ansatzr├Ąume niedrigster Ordnung, z.B. st├╝ckweise affine Polynome f├╝r beliebiges m. Hat eine L├Âsung der betrachteten Probleme Singularit├Ąten, so beeintr├Ąchtigt dies in der Regel die Konvergenz so stark, dass h├Âhere Polynomgrade in den Ansatzr├Ąumen auf uniformen Gittern dieselbe Konvergenzrate zeigen wie niedrigere Polynomgrade. Deshalb sind gerade f├╝r h├Âhere Polynomgrade in den Ansatzr├Ąumen adaptiv generierte Gitter unabdingbar. Neben der A-priori- und der A-posteriori-Analysis werden in dieser Dissertation optimale Konvergenzraten f├╝r adaptive Algorithmen f├╝r die neuen Diskretisierungen des Poisson-Problems, der Stokes-Gleichungen und der m-Laplace-Gleichung bewiesen. Diese werden auch in den numerischen Beispielen dieser Dissertation empirisch nachgewiesen.´╗┐This thesis generalizes the non-conforming finite element methods (FEMs) of Morley and Crouzeix and Raviart by novel mixed formulations for the Poisson problem, the Stokes equations, the Navier-Lam├ę equations of linear elasticity, and mth-Laplace equations of the form (Ôłĺ1)m╬ömu=f(-1)^m\Delta^m u=f for arbitrary m=1,2,3,... These formulations are based on Helmholtz decompositions. The new formulations allow for ansatz spaces of arbitrary polynomial degree and its discretizations coincide with the mentioned non-conforming FEMs for the lowest polynomial degree. Also for higher polynomial degrees, this results in robust discretizations for almost incompressible materials and approximations of the solution of the Stokes equations with pointwise mass conservation. Furthermore this approach also allows for a generalization of the non-conforming FEMs for the Poisson problem and the biharmonic equation to mth-Laplace equations for arbitrary m>2. A new Helmholtz decomposition for tensor-valued functions enables this. The new discretizations allow not only for a uniform implementation for arbitrary m, but they also allow for lowest-order ansatz spaces, e.g., piecewise affine polynomials for arbitrary m. The presence of singularities usually affects the convergence such that higher polynomial degrees in the ansatz spaces show the same convergence rate on uniform meshes as lower polynomial degrees. Therefore adaptive mesh-generation is indispensable especially for ansatz spaces of higher polynomial degree. Besides the a priori and a posteriori analysis, this thesis proves optimal convergence rates for adaptive algorithms for the new discretizations of the Poisson problem, the Stokes equations, and mth-Laplace equations. This is also demonstrated in the numerical experiments of this thesis
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