Journal of Scientific Computing

, Volume 50, Issue 2, pp 446–461 | Cite as

Well-conditioned Orthonormal Hierarchical \(\mathcal{L}_{2}\) Bases on ℝn Simplicial Elements

Article

Abstract

We construct well-conditioned orthonormal hierarchical bases for simplicial \(\mathcal{L}_{2}\) finite elements. The construction is made possible via classical orthogonal polynomials of several variables. The basis functions are orthonormal over the reference simplicial elements in two and three dimensions. The mass matrices M are identity while the conditioning of the stiffness matrices S grows as \(\mathcal{O}(p^{3})\) with respect to the order p. The diagonally normalized stiffness matrices are well conditioned. The diagonally normalized composite matrices ζM+S are also well conditioned for a wide range of ζ. For the mass, stiffness and composite matrices, the bases in this study have much better conditioning than existing high-order hierarchical bases.

Keywords

Hierarchical bases Simplicial \(\mathcal{L}_{2}\)-conforming elements Matrix conditioning 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of North Carolina at CharlotteCharlotteUSA

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