Stable Liftings of Polynomial Traces on Tetrahedra
Charles Parker, Endre Süli
TL;DR
The paper solves the long-standing problem of constructing stable polynomial liftings for trace data on the reference tetrahedron. It develops a four-stage face-by-face lifting scheme, combining fundamental convolution-based liftings with carefully designed corrections to achieve a right inverse of the generalized trace map $(u,\partial_{\mathbf{n}}u,\ldots,\partial_{\mathbf{n}}^{k}u)|_{\partial K}$ that is stable from $\mathrm{Tr}_{k}^{s,p}(\partial K)$ to $W^{s,p}(K)$ for all $p\in(1,\infty)$ and $s>k+1/p$, while preserving polynomial data when the trace originates from a polynomial. A suite of trace theorems, compatibility conditions on edges, and weighted Sobolev estimates underpin the construction and yield a novel range characterization for the trace operator. The results extend known 2D liftings to 3D tetrahedra, enabling stable high-order finite element and spectral-element analyses with minimal regularity requirements. The approach integrates whole-space extensions, Hardy-type estimates, and auxiliary vanishing-trace spaces to deliver robust, polynomial-preserving liftings for a broad range of Sobolev spaces.
Abstract
On the reference tetrahedron $K$, we construct, for each $k \in \mathbb{N}_0$, a right inverse for the trace operator $u \mapsto (u, \partial_{n} u, \ldots, \partial_{n}^k u)|_{\partial K}$. The operator is stable as a mapping from the trace space of $W^{s, p}(K)$ to $W^{s, p}(K)$ for all $p \in (1, \infty)$ and $s \in (k+1/p, \infty)$. Moreover, if the data is the trace of a polynomial of degree $N \in \mathbb{N}_0$, 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.