Sparse shape functions for tetrahedral p-FEM using integrated Jacobi polynomials

Beuchler, Sven; Pillwein, Veronika

doi:10.1007/s00607-007-0236-0

Cited by 22 publications

(33 citation statements)

References 24 publications

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“…These relations have been proved in [10], [8] and [9]. In the present paper the main relations required for proving the orthogonality of our basis function in H(div) are…”

Section: Properties Of Jacobi Polynomials With Weight (1 − X) αmentioning

confidence: 68%

“…This sparsity result has been obtained by evaluating the entries of the mass matrix on the reference triangle symbolically using the algorithm developed in [8]. Carrying out such computations manually, as done e.g.…”

Section: Testing With the Constant Low-order Contributionsmentioning

confidence: 99%

“…It can be shown that the number of nonzero entries per row in the block M II is O(1) and hence independent of the polynomial degree p. Since the number of combinations as well as the integrals to be computed in the sparsity proof of the mass matrix become very involved, we do not evaluate them by hand, but utilize the algorithm and implementation presented in [8,9]. An upper bound for the non-zero entries is summarized in the following lemma.…”

Section: Let Us Define the Element Stiffness Matrix With Respect To Tmentioning

confidence: 99%

“…The set of divergence-free basis functions [Ψ (1) ] are chosen as the curl of the hierarchic H(curl)-conforming completion functions as presented in [35]. But in this work, we rely on products of mixed-weighted Jacobi-polynomials (instead of Legendre-type polynomials), where the weights are chosen in analogy to the H 1 -conforming basis suggested in [10,8].…”

Section: Introductionmentioning

confidence: 99%

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Sparsity Optimized High Order Finite Element Functions on Simplices

Beuchler

Pillwein

Schöberl

et al. 2011

Texts &Amp; Monographs in Symbolic Computation

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This paper deals with conforming high-order finite element discretizations of the vectorvalued function space H(div) in 2 and 3 dimensions. A new set of hierarchic basis functions on simplices with the following two main properties is introduced. Provided an affine simplicial triangulation, first, the divergence of the basis functions is L2-orthogonal, and secondly, the L2-inner product of the interior basis functions is sparse with respect to the polynomial order p. The construction relies on a tensor-product based construction with properly weighted Jacobi polynomials as well as an explicit splitting of the higher-order basis functions into solenoidal and non-solenoidal ones. The basis is suited for fast assembling strategies. The general proof of the sparsity result is done by the assistance of computer algebra software tools. Several numerical experiments document the proved sparsity patterns and practically achieved condition numbers for the parameter-dependent div-div problem. Even on curved elements a tremendous improvement in condition numbers is observed. The precomputed mass and stiffness matrix entries in general form are available online.

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“…These relations have been proved in [10], [8] and [9]. In the present paper the main relations required for proving the orthogonality of our basis function in H(div) are…”

Section: Properties Of Jacobi Polynomials With Weight (1 − X) αmentioning

confidence: 68%

Section: Testing With the Constant Low-order Contributionsmentioning

confidence: 99%

Section: Let Us Define the Element Stiffness Matrix With Respect To Tmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sparsity Optimized High Order Finite Element Functions on Simplices

Beuchler

Pillwein

Schöberl

et al. 2011

Texts &Amp; Monographs in Symbolic Computation

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show abstract

“…For the proof of the sparsity of the mass matrix we will need additional relations which are proven in [10,8,9] and summarized in the appendix A. Jacobi-and integrated Jacobi-type polynomials can be evaluated efficiently by three term recurrences as summarized in the appendix.…”

Section: Properties Of Jacobi Polynomials With Weight (1 − X) αmentioning

confidence: 99%

Sparsity optimized high order finite element functions for H(div) on simplices

Beuchler

Pillwein²,

Zaglmayr³

2012

Numer. Math.

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Improved p‐version finite element matrix conditioning using semi‐orthogonal modal approximation functions

Winterscheidt

2022

Numerical Meth Engineering

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Recursive equations for the efficient computation of nearly orthogonal, C 0 continuous p-version approximation functions ("max-orthogonal") are developed in one dimension and used to construct hexagonal ("brick") element approximation functions. A modification to the functions is presented to simplify adaptive p-refinement, resulting in "semi-orthogonal" approximation functions. These functions are found to produce Laplace equation stiffness matrices, as well as mass matrices, with substantially lower condition numbers compared to standard Legendre-based modal functions. Sample problems are presented using a conjugate gradient matrix solver to compare the matrix solution computation times using max-orthogonal, semi-orthogonal, and standard modal functions.Automatic adaptive polynomial refinement is also discussed and applied to these sample problems. The semi-orthogonal functions are found to be an efficient and practical means of achieving very well-conditioned p-version finite element matrices.

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Sparse shape functions for tetrahedral p-FEM using integrated Jacobi polynomials

Cited by 22 publications

References 24 publications

Sparsity Optimized High Order Finite Element Functions on Simplices

Sparsity Optimized High Order Finite Element Functions on Simplices

Sparsity optimized high order finite element functions for H(div) on simplices

Improved p‐version finite element matrix conditioning using semi‐orthogonal modal approximation functions

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