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 Laplace-Beltrami operator are used to define Fourier-like 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 geometry-aware and orthogonal. On the practical side, we propose a band-by-band spectrum computation algorithm and an out-of-core 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.

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.

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 quasi-orthogonality 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.

Abstract. 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 Maxwell’s equations 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.

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 curl-curl 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.

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 well-established numerical methods and to new methods introduced here. First, we show that Yee’s finite-difference time-domain (FDTD) scheme, along with a number of related methods, are multisymplectic and derive from a discrete Lagrangian variational principle. Second, we generalize the

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.

Abstract — In this paper we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineering, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topology-preserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multi-linear transformations of the Hasse matrix representing the complex. Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. This paper is a further contribution towards bridging the subject of computer representations for solid and physical modeling—which flourished border-line between computer graphics, engineering mechanics and computer science with its own methods and data structures—under the general cover of linear algebra and algebraic topology. The main advantage of such an approach is that topology, geometry and physics may coexist in one and the same formalized framework, concurring together to define, represent and simulate the behavior of the model. Our approach is based on first principles and is general in that it applies to most representational domains that can be characterized as cell complexes, without any restrictions on their type, dimension, codimension, orientability, manifoldness, connectedness. Contrary to what might appear at first sight, the theoretical complexity of the present approach is not greater than that of current methods, provided that sparse-matrix techniques with double element access (by rows and by columns) are exploited. Last but not least, our tensor-based approach is a significant step forward in achieving close integration of geometrical representations and physics-based simulations, i.e., in the concurrent modeling of shape and behavior.

Abstract We give an overview of multilevel methods, such as V-cycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasi-uniform meshes and extend them to graded meshes and completely unstructured grids. We first discuss the classical multigrid theory on the basis of the method of subspace correction of Xu and a key identity of Xu and Zikatanov. We next extend the classical multilevel methods in H(grad) to graded bisection grids upon employing the decomposition of bisection grids of Chen, Nochetto, and Xu. We finally discuss a class of multilevel preconditioners developed by Hiptmair and Xu for problems discretized on unstructured grids and extend them to H(curl) and H(div) systems over graded bisection grids. 1

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 Galerkin-likeapproach, one solvesavariational problemonafinite-dimensional 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 2-surfaces approximated by a piecewise-linear triangulation; Demlow later extended this analysis [SIAM J. Numer. Anal., 47 (2009), 805–827] to 3-surfaces, as well as to higher-order 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