Spectral methods on triangles and other domains
Journal of Scientific Computing
Performance of Discontinuous Galerkin Methods for Elliptic PDEs
SIAM Journal on Scientific Computing
Nodal high-order methods on unstructured grids
Journal of Computational Physics
Algorithm 839: FIAT, a new paradigm for computing finite element basis functions
ACM Transactions on Mathematical Software (TOMS)
FEMSTER: An object-oriented class library of high-order discrete differential forms
ACM Transactions on Mathematical Software (TOMS)
New shape functions for triangular p-FEM using integrated Jacobi polynomials
Numerische Mathematik
Optimizing FIAT with level 3 BLAS
ACM Transactions on Mathematical Software (TOMS)
A compiler for variational forms
ACM Transactions on Mathematical Software (TOMS)
Efficient compilation of a class of variational forms
ACM Transactions on Mathematical Software (TOMS)
Algorithm 884: A Simple Matlab Implementation of the Argyris Element
ACM Transactions on Mathematical Software (TOMS)
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The L2-orthogonal polynomials used in finite and spectral element methods on nonrectangular elements may be defined in terms of collapsed coordinates, wherein the shapes are mapped to a square or cube by means of a singular change of variables. The orthogonal basis is a product of specific Jacobi polynomials in these new coordinates. Implementations of these polynomials require special handling of the coordinate singularities. We derive new recurrence relations for these polynomials on triangles and tetrahedra that work directly in the original coordinates. These relations, also applicable to pyramids and prisms, do not require any special treatment of singular points. These recurrences are seen to speed up both symbolic and numerical computation of the orthogonal polynomials.