Longest-edgen-section algorithms: Properties and open problems

On interpolation error on degenerating prismatic elements

“…Taking the partial derivative with respect tox at both sides of (24), we get From estimates (15), (18), and the fact that ∂z/∂x = 0, we come to…”

“…It influences the interpolation properties of finite elements and through Cea's lemma also the convergence of the finite element method [9]; various regularity mesh properties are required in derivation of a posteriori error estimates for various finite element-type approximations, in the discrete maximum principles (see e.g. [13]), for acceleration of convergence of finite element approximations, also in computer graphics (see [18] and references therein), etc.…”

“…Many algorithms for constructing families of triangulations satisfying (1) can be found e.g. in the review paper [18]. The weaker limitation on angles of triangles, called the maximum angle condition (see [28]), reads as follows: there exists a constant γ 0 < π such that for any triangulation T h ∈ F and any triangle T ∈ T h one has (2) γ…”

Similarity Classes of the Longest-Edge Trisection of Triangles

“…For the longest-edge bisection of triangles, the non-degeneracy was proved by Rosenberg and Stenger [7] and similar results have been presented recently for the longest-edge trisection or in general for the longest n-section of triangles. See [8] and references therein.…”