Properties of the multidimensional finite elements

Abstract

The paper describes properties of some multidimensional simplicial finite elements with respect to their applications for solving multidimensional boundary and eigenvalue problems. New classes of similarity are obtained and compared with Freudenthal’s simplicial class. The degeneracy measure of each simplicial class grows up to infinity when the dimensions of the Euclidean spaces increase unboundedly. Naturally, the question arises: how fast degenerate the elements from each simplicial class? To give an answer to this question we calculate the exact rate of divergence for all considered sequences of simplicial classes. Analytic relations between the investigated simplicial classes are proved. Examples supporting the theoretical results are presented. The cosine of the abstract angle between multidimensional finite element spaces is calculated in the isotropic and anisotropic cases. The dependence of the Cauchy–Bunyakovsky–Schwarz (CBS) constant on the shape of the used elements is illustrated. We confirm the Brandts et al. [10] conjecture concerning the contraction number for the Laplace operator and red refined Freudenthal’s simplicial elements. Additionally, we formulate new conjectures concerning the invariance of simplicial elements. We found multidimensional simplices invariant regarding the red and blue refinement strategies.