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64
Spectral geometry processing with manifold harmonics
 Computer Graphics Forum
, 2008
"... the geometry into frequency space by computing the Manifold Harmonic Transform (MHT). C: Apply the frequency space filter on the transformed geometry. D: Transform back into geometric space by computing the inverse MHT. We present a new method to convert the geometry of a mesh into frequency space. ..."
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Cited by 51 (1 self)
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the geometry into frequency space by computing the Manifold Harmonic Transform (MHT). C: Apply the frequency space filter on the transformed geometry. D: Transform back into geometric space by computing the inverse MHT. We present a new method to convert the geometry of a mesh into frequency space. The eigenfunctions of the LaplaceBeltrami operator are used to define Fourierlike function basis and transform. Since this generalizes the classical Spherical Harmonics to arbitrary manifolds, the basis functions will be called Manifold Harmonics. It is well known that the eigenvectors of the discrete Laplacian define such a function basis. However, important theoretical and practical problems hinder us from using this idea directly. From the theoretical point of view, the combinatorial graph Laplacian does not take the geometry into account. The discrete Laplacian (cotan weights) does not have this limitation, but its eigenvectors are not orthogonal. From the practical point of view, computing even just a few eigenvectors is currently impossible for meshes with more than a few thousand vertices. In this paper, we address both issues. On the theoretical side, we show how the FEM (Finite Element Modeling) formulation defines a function basis which is both geometryaware and orthogonal. On the practical side, we propose a bandbyband spectrum computation algorithm and an outofcore implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering and interactive shading design.
Mixed finite element methods for linear elasticity with weakly imposed symmetry
, 2005
"... In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger–Reissner variational principl ..."
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Cited by 10 (5 self)
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In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger–Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.
EQUILIBRATED RESIDUAL ERROR ESTIMATOR FOR EDGE ELEMENTS
"... Abstract. Reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem. A cheap computation of such functions via equilibration is well known for scalar equations of second order. We simpl ..."
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Cited by 9 (1 self)
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Abstract. Reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem. A cheap computation of such functions via equilibration is well known for scalar equations of second order. We simplify and modify the equilibration such that it can be applied to the curlcurl equation and edge elements. The construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions. For this reason, Raviart–Thomas elements are extended in the spirit of distributions. 1.
SMOOTHED PROJECTIONS IN FINITE ELEMENT EXTERIOR CALCULUS
"... Abstract. The development of smoothed projections, constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator, have proved to be an effective tool in finite element exterior calculus. The advantage of these operators is that they are ..."
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Cited by 8 (3 self)
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Abstract. The development of smoothed projections, constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator, have proved to be an effective tool in finite element exterior calculus. The advantage of these operators is that they are L 2 bounded projections, and still they commute with the exterior derivative. In the present paper we generalize the construction of these smoothed projections, such that also non quasi–uniform meshes and essential boundary conditions are covered. The new tool introduced here is a space dependent smoothing operator which commutes with the exterior derivative. 1.
Convergence and optimality of adaptive mixed finite element methods
, 2009
"... The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved ..."
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Cited by 8 (3 self)
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The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation.
On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains
 Math. Z
"... We study integral operators related to a regularized version of the classical Poincaré path integral and the adjoint class generalizing Bogovskiĭ’s integral operator, acting on differential forms in R n. We prove that these operators are pseudodifferential operators of order −1. The Poincarétype op ..."
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Cited by 8 (2 self)
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We study integral operators related to a regularized version of the classical Poincaré path integral and the adjoint class generalizing Bogovskiĭ’s integral operator, acting on differential forms in R n. We prove that these operators are pseudodifferential operators of order −1. The Poincarétype operators map polynomials to polynomials and can have applications in finite element analysis. For a domain starlike with respect to a ball, the special support properties of the operators imply regularity for the de Rham complex without boundary conditions (using Poincarétype operators) and with full Dirichlet boundary conditions (using Bogovskiĭtype operators). For bounded Lipschitz domains, the same regularity results hold, and in addition we show that the cohomology spaces can always be represented by C ∞ functions. 2000 Mathematics Subject Classification. Primary 35B65, 35C15; Secondary 58J10, 47G30 Key words and phrases. Exterior derivative, differential forms, Lipschitz domain, Sobolev
Isogeometric analysis in electromagnetics: Bsplines approximation
 Comput. Methods Appl. Mech. Engrg
, 1143
"... We introduce a new discretization scheme for Maxwell equations in two space dimension. Inspired by the new paradigm of Isogeometric analysis introduced in [16], we propose an algorithm based on the use of bivariate Bsplines spaces suitably adapted to electromagnetics. We construct Bsplines spaces ..."
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Cited by 7 (0 self)
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We introduce a new discretization scheme for Maxwell equations in two space dimension. Inspired by the new paradigm of Isogeometric analysis introduced in [16], we propose an algorithm based on the use of bivariate Bsplines spaces suitably adapted to electromagnetics. We construct Bsplines spaces of variable interelement regularity on the parametric domain. These spaces (and their pushforwards on the physical domain) form a De Rham diagram and we use them to solve the Maxwell source and eigen problem. Our scheme has the following features: (i) is adapted to treat complex geometries, (ii) is spectral correct, (iii) provides regular (e.g., globally C 0) discrete solutions of Maxwell equations. 1
Computational electromagnetism with variational integrators and discrete differential forms,” preprint at arXiv: 0707.4470 [math.NA
"... In this paper, we introduce a general family of variational, multisymplectic numerical methods for solving Maxwell’s equations, using discrete differential forms in spacetime. In doing so, we demonstrate several new results, which apply both to some wellestablished numerical methods and to new meth ..."
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Cited by 6 (1 self)
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In this paper, we introduce a general family of variational, multisymplectic numerical methods for solving Maxwell’s equations, using discrete differential forms in spacetime. In doing so, we demonstrate several new results, which apply both to some wellestablished numerical methods and to new methods introduced here. First, we show that Yee’s finitedifference timedomain (FDTD) scheme, along with a number of related methods, are multisymplectic and derive from a discrete Lagrangian variational principle. Second, we generalize the
2010), Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces
"... Soc. 47 (2010), 281–354] showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkinlikeapproach, one solvesavariational problemonafinitedimensional subcomplex. In a seemingly unrelated research direction, Dziuk [Lecture Notes ..."
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Cited by 4 (3 self)
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Soc. 47 (2010), 281–354] showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkinlikeapproach, one solvesavariational problemonafinitedimensional subcomplex. In a seemingly unrelated research direction, Dziuk [Lecture Notes in Math., vol. 1357 (1988), 142–155] analyzed a class of nodal finite elements for the Laplace–Beltrami equation on smooth 2surfaces approximated by a piecewiselinear triangulation; Demlow later extended this analysis [SIAM J. Numer. Anal., 47 (2009), 805–827] to 3surfaces, as well as to higherorder surface approximation. In this article, we bring these lines of research together, Hilbert complexes, and then applying this abstract framework to the setting of finite element exterior calculus on hypersurfaces. Our framework extends the work of Arnold, Falk, and Winther to problems that violate their subcomplex assumption, allowing for the extension of finite element exterior calculus to approximate domains, most notably the Hodge–de Rham complex on approximate