1,721,013 research outputs found
Unisolvent and minimal physical degrees of freedom for the second family of polynomial differential forms
Full text linkThe principal aim of this work is to provide a family of unisolvent and minimal physical degrees of freedom, called weights, for Nédélec second family of finite elements. Such elements are thought of as differential forms whose coefficients are polynomials of degree r. In this paper we confine ourselves in the two dimensional case ℝ2, as in this framework the Five Lemma offers a neat and elegant treatment avoiding computations on the middle space. The majority of definitions and constructions are meaningful for n > 2 as well and, when possible, they are thus given in such a generality, although more complicated techniques shall be invoked to replace the graceful role of the Five Lemma. In particular, we use techniques of homological algebra to obtain degrees of freedom for the whole diagram
An algorithm for the estimation of the segmental Lebesgue constant
No full textThe main goal of this work is to provide an explicit algorithm for the estimation of the segmental Lebesgue constant, an extension of the nodal Lebesgue constant that arise, for instance, in histopolation problems. With the help of two simple but efficacious lemmas, we reverse the already known technology and sensibly speed up the numerical estimation of such quantities. Results are comparable with the known literature, although cpu time of the presented method is sensibly smaller. It is worth pointing out that the numerical approach is the only known for analyzing the majority of families of supports
The Fekete problem in segmental polynomial interpolation
No full textIn this article, we study the Fekete problem in segmental and combined nodal-segmental univariate polynomial interpolation by investigating sets of segments, or segments combined with nodes, such that the Vandermonde determinant for the respective polynomial interpolation problem is maximized. For particular families of segments, we will be able to find explicit solutions of the corresponding maximization problem. The quality of the Fekete segments depends hereby strongly on the utilized normalization of the segmental information in the Vandermonde matrix. To measure the quality of the Fekete segments in interpolation, we analyse the asymptotic behaviour of the generalized Lebesgue constant linked to the interpolation problem. For particular sets of Fekete segments we will get, similar to the nodal case, a favourable logarithmic growth of this constant
Polynomial Interpolation of Function Averages on Interval Segments
Full text linkMotivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise to new theoretical and practical aspects that distinguish this problem considerably from classical nodal interpolation. We will analyze fundamental mathematical properties of this problem as existence, uniqueness, and numerical conditioning of its solution. In a few selected scenarios, we will provide concrete conditions for unisolvence and explicit Lagrange-type basis systems for its representation. To study the numerical conditioning, we will provide respective concrete bounds for the Lebesgue constant
A pluripotential theoretic framework for polynomial interpolation of vector-valued functions and differential forms
Full text linkWe consider the problem of uniform interpolation of functions with values in a complex inner product space of finite dimension. This problem can be cast within a modified weighted pluripotential theoretic framework. Indeed, in the proposed modification a vector valued weight is considered, allowing to partially extend the main asymptotic results holding for interpolation of scalar valued functions to the case of vector valued ones. As motivating example and main application we specialize our results to interpolation of differential forms by differential forms with polynomial coefficients
The numerical linear algebra of weights. Part I : from the spectral analysis to conditioning and preconditioning in the one-dimensional Laplacian case
Full text linkWeights are geometrical degrees of freedom that allow to generalize Lagrangian finite elements. They are defined through integrals over specific supports, well understood in terms of differential forms and integration, and lie within the framework of finite element exterior calculus. We adopt this formalism with the target of identifying supports that are appealing for a finite element approximation, describing weights in terms of a single parameter in the third- and fourth-order methods. To do so, we study the related parametric matrix-sequences, with the matrix order tending to infinity as the mesh size tends to zero. Using the generalized locally Toeplitz theory, we analyze the performance of weights-based finite elements on an elliptic operator. In particular, for degrees 3 and 4, we identify an optimal value for the weights location, which sits in a rather large interval where weights give rise to better conditioned stiffness matrices. With this at hand, we propose and test ad hoc preconditioners, in dependence of the discretization parameters and in connection with conjugate gradient method. The model problem we consider is a one-dimensional Laplacian, both with constant and non-constant coefficients. Numerical visualizations and experimental tests are reported and critically discussed, showing a confidence interval for the choice of the parameter
Minimal Sets of Unisolvent Weights for High Order Whitney Forms on Simplices
No full textWhitney forms—degree one trimmed polynomials—are a crucial tool for finite element analysis of electromagnetic problem. They not only induce several finite element methods, but they also bear interesting geometrical features. If, on the one hand, features of degree one elements are well understood, when it comes to higher degree elements one is forced to choose between an analytical approach and a geometric one, that is, the duality that holds for the lower degree gets lost. Using tools of finite element exterior calculus, we show a correspondence between the usual basis of a high order Whitney forms space and a subset of the weights, that is, degrees of freedom obtained by integration over subsimplices of the mesh
Computing weights for high order Whitney edge elements
Full text linkThe interpolation of differential forms is a challenging problem that is getting increasing attention. The issue of finding unisolvent degrees of freedom to describe a differential form in terms of high-order Whitney forms is an active area of research nowadays. In this paper we deal with a family of such degrees of freedom, called weights, that fits with the physical and geometrical nature of the field to interpolate. These weights play the role of interpolation coefficients when reconstructing scalar/vector fields in terms of a set of selected multivariate polynomial forms. Weights are a generalization of the evaluations of a scalar function at a set of nodes in view of its reconstruction on multivariate polynomial bases. As in the nodal case, different sets of such weights are compared in terms of a Lebesgue constant. In this contribution, we briefly recall their definition and provide examples of algorithms in low dimension to compute their associated Lebesgue constant value. Insights to greater dimensions are offered as well
Towards nonuniform distributions of unisolvent weights for high-order Whitney edge elements
Full text linkWe propose to extend results on the interpolation theory for scalar functions to the case of differential k-forms. More precisely, we consider the interpolation of fields in Pr-Λk(T), the finite element spaces of trimmed polynomial k-forms of arbitrary degree r≥ 1 , from their weights, namely their integrals on k-chains. These integrals have a clear physical interpretation, such as circulations along curves, fluxes across surfaces, densities in volumes, depending on the value of k. In this work, for k= 1 , we rely on the flexibility of the weights with respect to their geometrical support, to study different sets of 1-chains in T for a high order interpolation of differential 1-forms, constructed starting from “good” sets of nodes for a high order multi-variate polynomial representation of scalar fields, namely 0-forms. We analyse the growth of the generalized Lebesgue constant with the degree r and preliminary numerical results for edge elements support the nonuniform choice, in agreement with the well-known nodal case
Polynomial Histopolation On Mock-Chebyshev Segments
No full textIn computational practice, we often encounter situations where only measurements at equally spaced points are available. Using standard polynomial interpolation in such cases can lead to highly inaccurate results due to numerical ill-conditioning of the problem. Several techniques have been developed to mitigate this issue, such as the mock-Chebyshev subset interpolation and the constrained mock-Chebyshev least squares approximation. The high accuracy and the numerical stability achieved by these techniques motivate us to extend these methods to histopolation, a polynomial interpolation method based on segmental function averages. While classical polynomial interpolation relies on function evaluations at specific nodes, histopolation leverages averages of the function over subintervals. In this work, we introduce three types of mock-Chebyshev approaches for segmental interpolation and theoretically analyse the stability of their Lebesgue constants, which measure the numerical conditioning of the histopolation problem under small perturbations of the segments. We demonstrate that these segmental mock-Chebyshev approaches yield a quasi-optimal logarithmic growth of the Lebesgue constant in relevant scenarios. Additionally, we compare the performance of these new approximation techniques through various numerical experiments
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