Hypergeometric summation algorithms for high-order finite elements
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.
scientific article; zbMATH DE number 5082636
scientific article