Core Concepts
Development of H(div)-conforming finite element tensors based on geometric decomposition.
Abstract
This work presents a unified construction of H(div)-conforming finite element tensors, including vector div element, symmetric div matrix element, and traceless div matrix element. The approach involves decomposing the tensor at each sub-simplex into tangential and normal components. Boundary degrees of freedom are explored to discover various finite elements that are H(div)-conforming and satisfy the discrete inf-sup condition. An explicit basis for the constraint tensor space is established. The content delves into Hilbert complexes' role in numerical methods for partial differential equations and introduces a systematic approach to derive new complexes from well-understood differential complexes like the de Rham complex. The article discusses the algebraic operator sk,n−1 and its role in deriving new complexes. It also explores the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex.
Stats
n = 2, r = 1 (Lemma 3.9)
r ≥ 2 (Lemma 3.9)
m = -1, m = 0, m = n - 2 (Lemma 3.8)
r ≥ 1 (Proposition 3.10)
Quotes
"The developed finite element spaces are H(div)-conforming and satisfy the discrete inf-sup condition."
"A significant aspect of our contribution is the expansion of geometric decomposition techniques to effectively manage tensors subjected to specific constraints."
"An explicit basis for the constraint tensor space is also established."