Quadratures on a sphere

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

“…This allows for functions to be approximated through L 2 -orthogonal projections using exact quadratures up to a desired order [2,30]. To achieve discretizations with favorable symmetry on the sphere, we use the nodes of Lebedev quadrature [17,18]. This is in contrast to the more common approach of using samping points based on lattitude and longitude coordinates.…”

“…For a comperable number of points, the Lebedev sampling provides a more uniform resolution of functions. The Lebedev quadrature points also have the feature of being invariant under rotations corresponding to octohedral symmetry [17,18]. We show Lebedev nodes on example radial manifolds in Figure 3.…”

“…The Lebedev quadrature points are more regularly distributed on the surface and for a comperable number of points provide over the surface a more uniform resolution of functions. Furthermore, the Lebedev quadrature points are invariant under rotations corresponding to octohedral symmetry [17,18]. We use an L 2 -projection to spherical harmonics to approximate the exterior derivative operator d and Hodge star operator on the surface.…”

“…This allows for the treatment of approximation instead using approaches such as L 2 -orthogonal projection based on exact quadratures up to a desired order [2,30]. To achieve discretizations on spherical toplogies with favorable symmetry, we use the nodes of Lebedev quadrature [17,18]. While the more common approach of using samping points based on lattitude and longitude does provide some computational advantage through fast transforms, the sampling points have poor symmetry and exhibit a significant inhomogeneous distribution over the surface with many sample points clustering near the poles [9,12,27].…”

Orthogonal polynomials and cubature formulae on spheres and on simplices

“…Since this result helps to reduce the number of polynomials that need to be evaluated by a great deal, it has been used to construct a number of cubature formulae on the unit sphere 5 2 (cf. [6,11,15,17] and the references there). In the case we are considering, the results in the previous section shows that the invariant polynomials on S d can be identified with the polynomials on the simplex S d under a proper change of variables.…”