Low cardinality admissible meshes on quadrangles, triangles and disks

Abstract

Using classical univariate polynomial inequalities (Ehlich and Zeller, 1964), we show that there exist admissible meshes with O(n 2 ) points for total degree bivariate polynomials of degree n on convex quadrangles, triangles and disks. Higher-dimensional extensions are also briefly discussed.2000 AMS subject classification: 41A10, 41A63, 65D10.

Small perturbations of polynomial meshes

“…Indeed, the closed unit disk is a Jackson compact in R 2 with m k = k (as every d-dimensional euclidean ball, cf. [25]), and, as it has been shown in [9], it has an optimal admissible mesh with 4n 2 points (and mesh constant C = 2). Then we can assert that…”

“…In fact, Q = [a, b] is a Jackson compact, and is also polynomially convex (i.e., it coincides with its polynomial convex hull) like all real compacts. Moreover, it has optimal admissible meshes (for real polynomials, but see Remark 2), for example its 2n + 1 Chebyshev-Lobatto points, as it has been shown in [9] using a classical polynomial inequality by Ehlich and Zeller. The φ-image of 2nj(n) + 1 Chebyshev-Lobatto points of [a, b] is then an admissible mesh on the curve (see Corollaries 2 and 3 for the definition of j(n)).…”

“…[8,9]. Admissible and weakly admissible meshes with O(n 2 ) points can then be obtained on any convex or concave simple polygon (by polygon triangulation and finite union).…”

Suboptimal Polynomial Meshes on Planar Lipschitz Domains

“…[6]). In such a way, we can reduce the cardinality N G of the example above from about 148000 to about 43000, and N CL from about 44 millions to about 4.3 millions (with a ratio approximately 100), both meshes having now a mesh constant C = 4.…”

“…cf., e.g., [6,11,13,15]. Moreover, the polynomial inequality (1) can be relaxed, asking that it holds with C = C n , a sequence of constants increasing at most polynomially with n: in such a case, we speak of weakly admissible polynomial meshes.…”