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Hypergeometric summation algorithms for high-order finite elements. (English) Zbl 1113.65105

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 C. Pechstein, J. Schöberl, J. Melenk, and 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 [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.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
68W30 Symbolic computation and algebraic computation
65F35 Numerical computation of matrix norms, conditioning, scaling
35J25 Boundary value problems for second-order elliptic equations
65F10 Iterative numerical methods for linear systems

Citations:

Zbl 1078.33021
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References:

[1] Andrews, G. E., Askey, R. Roy, R.: Special functions. Cambridge UP, 2000.
[2] Andrews, G. E., Paule, P., Schneider, C.: Plane partitions VI: Stembridge’ s TSPP theorem. Adv. Appl. Math. Special Issue dedicated to Dr. David P. Robbins (Bressoud, D., ed.), 34(4), 709–739 (2005). · Zbl 1066.05019
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[7] Pechstein, C., Schöberl, J., Melenk, J., Zaglmayr, S.: Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements. IMA J. Numer. Anal. (2005) (submitted). · Zbl 1153.65113
[8] Mallinger, C.: Algorithmic manipulations and transformations of univariate holonomic functions and sequences. Master’s thesis, RISC, J. Kepler University, Linz, Austria, August 1996.
[12] Schneider, C.: A new Sigma approach to multi-summation. Adv. Appl. Math. Special Issue dedicated to Dr. David P. Robbins. (Bressoud, D., ed.), 34(4), 740–767 (2005). · Zbl 1078.33021
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