Stable Liftings of Polynomial Traces on Tetrahedra
Charles Parker, ENDRE E. SÜLI · Foundations of Computational Mathematics · 2024
Abstract On the reference tetrahedron $$K$$ K , we construct, for each $$k \in {\mathbb {N}}_0$$ k ∈ N 0 , a right inverse for the trace operator $$u \mapsto (u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^k u)|_{\partial K}$$ u ↦ ( u , ∂ n u , … , ∂ n k u ) | ∂ K . The operator is stable as a mapping from the trace space of $$W^{s, p}(K)$$ W s , p ( K ) to $$W^{s, p}(K)$$ W s , p ( K ) for all $$p \in (1, \infty )$$ p ∈ ( 1 , ∞ ) and $$s \in (k+1/p, \infty )$$ s ∈ ( k + 1 / p , ∞ ) . Moreover, if the data is the trace of a polynomial of degree $$N \in {\mathbb {N}}_0$$ N ∈ N 0 , then the resulting lifting is a polynomial of degree N. One consequence of the analysis is a novel characterization for the range of the trace operator.