Hypergeometric summation algorithms for high-order finite elements
type: Item
instance of: scholarly article
title: Hypergeometric summation algorithms for high-order finite elements
zbMATH Open document ID: 1113.65105
author: A. Bećirović
author: Carsten Schneider
author: Peter Paule
author: Veronika Pillwein
author: Axel Riese
author: Joachim Schöberl
published in: Computing
publication date: 2007-01-08T00:00:00Z
review text: The simulation of three-dimensional partial differential equations by finite element methods (FEM) requires the solution of large scale systems of linear algebraic equations. On the one hand high order finite elements are applied to reduce the order of the systems, on the other hand the efficiency of the used iterative solvers with preconditioning depends on the condition number of the system matrix which can be influenced taking appropriate basis functions of the FEM. For this part the authors refer to \textit{C. Pechstein, J. Schöberl, J. Melenk}, and \textit{S. Zaglmayr} [Additive Schwarz preconditioning for \(p\)-version triangular and tetrahedral finite elements. IMA J. Numer. Anal., submitted (2005)]. The aim of the present paper is the design of basis functions for minimizing the condition number and for fast evaluations in block-Jacobi preconditioners. The basis functions have to be designed such that the blocks are nearly orthogonal among each other. The basis functions are derived using hypergeometric summation methods solving the telescoping or the creative telescoping problem. Computer algebra algorithms to compute corresponding recurrence relations and to prove theorems are involved applying the Mathematica package [\textit{C. Schneider}, ``A new Sigma approach to multi-summation.'' Adv. Appl. Math. 34, No.4, 740-767 (2005; Zbl 1078.33021)]. The use of Mathematica is demonstrated for two high-order basis functions. The one is associated with edges applying integrated Legendre polynomials, the other is related to low energy vertex shape functions to reduce the condition number dependence of the polynomial order.
reviewed by: Georg Hebermehl
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33F10
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 5082636
zbMATH Keywords: high order finite elements
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: computer algebra algorithms
zbMATH Keywords: preconditioning
zbMATH Keywords: condition number
zbMATH Keywords: block-Jacobi preconditioners
zbMATH Keywords: Mathematica
describes a project that uses: gfun
describes a project that uses: Mathematica
describes a project that uses: SIGMA
describes a project that uses: GeneratingFunctions
MaRDI profile type: Publication
full work available at URL: s00607-006-0179-x
OpenAlex ID: W1974515217
cites work: Plane partitions. VI: Stembridge's TSPP theorem
cites work: Efficient Preconditioning for thep-Version Finite Element Method in Two Dimensions
cites work: Q4531674
cites work: Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements
cites work: Polynomial Liftings on a Tetrahedron and Applications to the h-p Version of the Finite Element Method in Three Dimensions
cites work: GFUN
cites work: A new Sigma approach to multi-summation
cites work: Basis Functions for Triangular and Quadrilateral High-Order Elements
DOI: 10.1007/S00607-006-0179-X
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description: scientific article; zbMATH DE number 5082636