scholarly journals Positivity of continuous piecewise polynomials

2012 ◽  
Vol 44 (4) ◽  
pp. 749-757
Author(s):  
Daniel Plaumann
Keyword(s):  
Piecewise Polynomials ◽  
Continuous Piecewise Polynomials

On a Right-Inverse for the Divergence Operator in Spaces of Continuous Piecewise Polynomials

1997 ◽  
Vol 07 (04) ◽  
pp. 487-505 ◽  
Author(s):  
E. Boillat
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Stokes Problem ◽  
Higher Order ◽  
Operator Norm ◽  
Boundary Vertex ◽  
Piecewise Polynomials ◽  
Divergence Operator ◽  
Triangular Elements ◽  
Continuous Piecewise Polynomials

This paper is concerned with the norm of a right-inverse for the divergence operator between spaces of piecewise polynomials on triangular elements. More specifically, one tries to construct a right-inverse acting from the space W of continuous piecewise polynomials of degree p into the space V of R2-valued piecewise polynomials of degree p+1. Our results are as follows. Assume that we are dealing with a quasiuniform family of triangulations {Mh} satisfying two additional hypotheses: Mh can be transformed into a quadriangular mesh by grouping its element two by two and Mh has no boundary vertex shared by only one or exactly three elements. In that context, one proves that the divergence has a right-inverse with an operator norm growing at most like [Formula: see text] when both W and V are equipped with the H1-norm and at most like [Formula: see text] if W is equipped with the L2-norm only. An application of the first of these two results is the approximation of the thermoelasticity equations by the p-version of the finite element methods. One also shows how the second result can be used in the context of higher-order Hood–Taylor method for Stokes problem.

scholarly journals On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

2020 ◽  
Author(s):  
Markus Faustmann ◽  
Jens Markus Melenk ◽  
Maryam Parvizi
Keyword(s):  
Fractional Laplacian ◽  
Stability Result ◽  
Fractional Diffusion ◽  
Eigenvalue Bounds ◽  
Piecewise Polynomials ◽  
Refined Meshes ◽  
Continuous Piecewise Polynomials ◽  
The Stability ◽  
Locally Refined Meshes ◽  
Multilevel Preconditioning

We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from $H^{3/2}$ into $B^{3/2}_{2,\infty}$; for elementwise polynomials these are bounded from $H^{1/2}$ into $B^{1/2}_{2,\infty}$. As an application, we obtain a multilevel decomposition based on Scott-Zhang operators on a hierarchy of meshes generated by newest vertex bisection with equivalent norms up to (but excluding) the endpoint case. A local multilevel diagonal preconditioner for the fractional Laplacian on locally refined meshes with optimal eigenvalue bounds is presented.

scholarly journals A C0-Collocation-like method for elliptic equations on rectangular regions

1997 ◽  
Vol 38 (3) ◽  
pp. 368-387 ◽  
Author(s):  
Zbigniew Leyk
Keyword(s):  
Quadrature Formula ◽  
Elliptic Equations ◽  
Tensor Products ◽  
Two Dimensional ◽  
Piecewise Polynomial ◽  
Dirichlet Problems ◽  
Piecewise Polynomials ◽  
Galerkin Procedure ◽  
Continuous Piecewise Polynomials ◽  
Collocation Solution

AbstractWe describe a C0-collocation-like method for solving two-dimensional elliptic Dirichlet problems on rectangular regions, using tensor products of continuous piecewise polynomials. Nodes of the Lobatto quadrature formula are taken as the points of collocation. We show that the method is stable and convergent with order hr(r ≥ 1) in the H1–norm and hr+1(r ≥ 2) in the L2–norm, if the collocation solution js a piecewise polynomial of degree not greater than r with respect to each variable. The method has an advantage over the Galerkin procedure for the same space in that no integrals need be evaluated or approximated.

Multivariate piecewise polynomials

Acta Numerica ◽  
1993 ◽  
Vol 2 ◽  
pp. 65-109 ◽  
Author(s):  
C. de Boor
Keyword(s):  
Approximation Theory ◽  
Variational Approach ◽  
Polynomial Function ◽  
Piecewise Polynomial ◽  
Multivariate Splines ◽  
Piecewise Polynomial Function ◽  
Multivariate Spline ◽  
Piecewise Polynomials ◽  
The Subject ◽  
Function Of Several Arguments

This article was supposed to be on ‘multivariate splines». An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a ‘multivariate spline’, resulted in the answer that a multivariate spline is a possibly smooth piecewise polynomial function of several arguments. In particular the potentially very useful thin-plate spline was thought to belong more to the subject of radial basis funtions than in the present article. This is all the more surprising to me since I am convinced that the variational approach to splines will play a much greater role in multivariate spline theory than it did or should have in the univariate theory. Still, as there is more than enough material for a survey of multivariate piecewise polynomials, this article is restricted to this topic, as is indicated by the (changed) title.

Estimation of Distributed Unbalance of Rotors

2001 ◽  
Author(s):  
Tachung Yang ◽  
Chunyi Lin
Keyword(s):  
Finite Element ◽  
Singular Value Decomposition ◽  
Element Model ◽  
Singular Value ◽  
Lumped Mass ◽  
Piecewise Polynomials ◽  
Mass Unbalance ◽  
Thick Disks ◽  
Value Decomposition ◽  
Thin Disks

Mass unbalance commonly causes vibration of rotor-bearing systems. Lumped mass modeling of unbalance was adapted in most previous research. The lumped unbalance assumption is adequate for thin disks or impellers, but not for thick disks or shafts. Lee et al. (1993) proposed that the unbalance of shafts should be continuously distributed. Balancing methods based on discrete unbalance models may not be very appropriate for rotors with distributed unbalance. A better alternative is to identify the distributed unbalance of shafts before balancing. In this study, the eccentricity distribution of the shaft is assumed in piecewise polynomials. A finite element model for the distributed unbalance is provided. Singular value decomposition is used to identify the eccentricity curves of the rotor. Numerical validation of this method is presented and examples are given to show the effectiveness of the identification method.

scholarly journals Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

2010 ◽  
Vol 2 (2) ◽  
pp. 264-272 ◽  
Author(s):  
A. Shirin ◽  
M. S. Islam
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Fredholm Integral Equations ◽  
Matrix Formulation ◽  
Piecewise Polynomials ◽  
The Galerkin Method

In this paper, Bernstein piecewise polynomials are used to solve the integral equations numerically. A matrix formulation is given for a non-singular linear Fredholm Integral Equation by the technique of Galerkin method. In the Galerkin method, the Bernstein polynomials are used as the approximation of basis functions. Examples are considered to verify the effectiveness of the proposed derivations, and the numerical solutions guarantee the desired accuracy.  Keywords: Fredholm integral equation; Galerkin method; Bernstein polynomials. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i2.4483               J. Sci. Res. 2 (2), 264-272 (2010) 

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