Sparsity optimized high order finite element functions for H(curl) on tetrahedra

作者：Beuchler Sven; Pillwein Veronika^*; Zaglmayr Sabine

来源：Advances in Applied Mathematics, 2013, 50(5): 749-769.

DOI：10.1016/j.aam.2012.11.004

• High order finite elements
• Orthogonal polynomials
• Symbolic computation
• Solution of discretized equations

H(curl) conforming finite element discretizations are a powerful tool for the numerical solution of the system of Maxwell%26apos;s equations in electrodynamics. In this paper we construct a basis for conforming high-order finite element discretizations of the function space H(curl) in 3 dimensions. We introduce a set of hierarchic basis functions on tetrahedra with the property that both the L-2-inner product and the H(curl)-inner product are sparse with respect to the polynomial degree. The construction relies on a tensor-product based structure with properly weighted Jacobi polynomials as well as an explicit splitting of the basis functions into gradient and non-gradient functions. The basis functions yield a sparse system matrix with O(1) nonzero entries per row. %26lt;br%26gt;The proof of the sparsity result on general tetrahedra defined in terms of their barycentric coordinates is carried out by an algorithm that we implemented in Mathematica. A rewriting procedure is used to explicitly evaluate the inner products. The precomputed matrix entries in this general form for the cell-based basis functions are available online.

• 出版日期2013-5