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Search Results: 1 - 10 of 2591 matches for " Ralf Hiptmair "
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Discrete Compactness for p-Version of Tetrahedral Edge Elements
Ralf Hiptmair
Mathematics , 2009,
Abstract: We consider the first family of $\Hcurl$-conforming Ned\'el\'ec finite elements on tetrahedral meshes. Spectral approximation ($p$-version) is achieved by keeping the mesh fixed and raising the polynomial degree $p$ uniformly in all mesh cells. We prove that the associated subspaces of discretely weakly divergence free piecewise polynomial vector fields enjoy a long conjectured discrete compactness property as $p\to\infty$. This permits us to conclude asymptotic spectral correctness of spectral Galerkin finite element approximations of Maxwell eigenvalue problems.
Conservative discretization of the Einstein-Dirac equations in spherically symmetric spacetime
Benedikt Zeller,Ralf Hiptmair
Physics , 2006, DOI: 10.1088/0264-9381/23/16/S17
Abstract: In computational relativity, critical behaviour near the black hole threshold has been studied numerically for several models in the last decade. In this paper we present a spatial Galerkin method, suitable for finding numerical solutions of the Einstein-Dirac equations in spherically symmetric spacetime (in polar/areal coordinates). The method features exact conservation of the total electric charge and allows for a spatial mesh adaption based on physical arclength. Numerical experiments confirm excellent robustness and convergence properties of our approach. Hence, this new algorithm is well suited for studying critical behaviour.
Local Multigrid in H(curl)
Ralf Hiptmair,Weiying Zheng
Mathematics , 2009,
Abstract: We consider H(curl)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H1-context along with local discrete Helmholtz-type decompositions of the edge element space.
Eulerian and Semi-Lagrangian Methods for Convection-Diffusion for Differential Forms
Holger Heumann,Ralf Hiptmair
Mathematics , 2010,
Abstract: We consider generalized linear transient convection-diffusion problems for differential forms on bounded domains in $\mathbb{R}^{n}$. These involve Lie derivatives with respect to a prescribed smooth vector field. We construct both new Eulerian and semi-Lagrangian approaches to the discretization of the Lie derivatives in the context of a Galerkin approximation based on discrete differential forms. Details of implementation are discussed as well as an application to the discretization of eddy current equations in moving media.
Convergence of the Natural hp-BEM for the Electric Field Integral Equation on Polyhedral Surfaces
Alexei Bespalov,Norbert Heuer,Ralf Hiptmair
Mathematics , 2009,
Abstract: We consider the variational formulation of the electric field integral equation (EFIE) on bounded polyhedral open or closed surfaces. We employ a conforming Galerkin discretization based on div-conforming Raviart-Thomas boundary elements (BEM) of locally variable polynomial degree on shape-regular surface meshes. We establish asymptotic quasi-optimality of Galerkin solutions on sufficiently fine meshes or for sufficiently high polynomial degree.
A Survey of Trefftz Methods for the Helmholtz Equation
Ralf Hiptmair,Andrea Moiola,Ilaria Perugia
Mathematics , 2015,
Abstract: Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces. We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties. One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.
Discrete compactness for the p-version of discrete differential forms
Daniele Boffi,Martin Costabel,Monique Dauge,Leszek Demkowicz,Ralf Hiptmair
Mathematics , 2009,
Abstract: In this paper we prove the discrete compactness property for a wide class of p-version finite element approximations of non-elliptic variational eigenvalue problems in two and three space dimensions. In a very general framework, we find sufficient conditions for the p-version of a generalized discrete compactness property, which is formulated in the setting of discrete differential forms of any order on a d-dimensional polyhedral domain. One of the main tools for the analysis is a recently introduced smoothed Poincar\'e lifting operator [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincar\'e integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2010)]. For forms of order 1 our analysis shows that several widely used families of edge finite elements satisfy the discrete compactness property in p-version and hence provide convergent solutions to the Maxwell eigenvalue problem. In particular, N\'ed\'elec elements on triangles and tetrahedra (first and second kind) and on parallelograms and parallelepipeds (first kind) are covered by our theory.
Self-adjoint curl operators
R. Hiptmair,P. R. Kotiuga,S. Tordeux
Mathematics , 2008,
Abstract: We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain $D$. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl) equipped with a symplectic pairing arising from the $\wedge$-product of 1-forms on $\partial D$. Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extension. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed.
Trends of Abundance of Salton Sea Fish: A Reversible Collapse or a Permanent Condition?  [PDF]
Ralf Riedel
Natural Resources (NR) , 2016, DOI: 10.4236/nr.2016.710045
Abstract: The Salton Sea is a closed-basin, 980 km2 salt lake in the Sonoran Desert of southern California. Three marine species, bairdiella (Bairdiella icistia), orangemouth corvina (Cynoscion xanthulus), and sargo (Anisotremus davidsoni), established from introductions of over 34 species beginning in 1929. During the late 1960s and early 1970s, a hybrid tilapia (Oreochromis mossambicus × O. urolepis hornorum) invaded the Salton Sea and became dominant by number and weight. Recent surveys show a precipitous decline of all four species above starting sometime between 2001 and 2002. Declines were more evident in nearshore than in estuarine habitats. Corvina has probably declined the soonest, followed by Gulf croaker. Tilapia declines were followed by more recent increases in population numbers. The tilapia rebound observed are probably only sustainable if a curb in Salton Sea salinity levels is realized. The marine species will likely need restocking to reach historic levels, if the salinity of the lake is managed at 40 gL1 or below. Restoration alternatives for the Salton Sea must take into consideration estuarine areas as essential fish habitats and fish refuge against high salinities.
Combination of Novel Virtual and Real Prototyping Methods in a Rapid Product Development Methodology / Kombinacija novih prividnih (virtualnih) i stvarnih postupaka proizvodnje prototipova i metodologija brzog razvoja proizvoda
Polimeri , 2012,
Abstract: The applicability of novel rapid prototyping methods and techniques for improving various stages of rapid product development is described in this study. In addition to the conventional prototyping, functional prototypes with various properties may be generated. Finite element simulations and novel experimental techniques are successfully used for improve the prototyping process both on a macroscopic and on a microscopic scale. The 2 component prototyping offers new options for 2k industrial component design and for biomodelling. / Ovaj rad opisuje primjenjivost novih postupaka brze proizvodnje prototipova i metoda za pobolj anje raznih stupnjeva brzog razvoja proizvoda. Osim konvencionalne proizvodnje prototipova mogu se na initi i funkcionalni prototipovi razli itih svojstava. Metode simuliranja s kona nim elementima i nove eksperimentalne metode uspje no se upotrebljavaju za pobolj anje proizvodnje prototipova na makroskopskoj i mikroskopskoj razini. Dvokomponentna proizvodnja prototipova nudi nove mogu nosti za 2k konstruiranje industrijskih dijelova i za biomodeliranje.
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