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.
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Xin, J., Cai, W. Well-conditioned Orthonormal Hierarchical \(\mathcal{L}_{2}\) Bases on ℝn Simplicial Elements. J Sci Comput 50, 446–461 (2012). https://doi.org/10.1007/s10915-011-9491-5
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DOI: https://doi.org/10.1007/s10915-011-9491-5