Variable-degree conforming hp spaces on matching affine meshes

Let $d\ge 2$ and let $\mathcal{T}$ be a finite matching affine simplicial mesh of a bounded polyhedral open set $\mathrm{\Omega }\subset {\mathbb{R}}^d$ in the sense of Matching affine simplicial meshes and shape regularity. A degree map is a function $p:\mathcal{T}\to {\mathbb{N}}_0$ with finite values. The broken simplex space associated with $(\mathcal{T},p)$ is

$$V^{\mathrm{b}\mathrm{r}}(\mathcal{T},p):=\{v\in L^2(\mathrm{\Omega }):v{\mathrm{\mid }}_K\in {\mathbb{P}}_{p_K}(K)\text{ for every }K\in \mathcal{T}\},$$

where $p_K:=p(K)$ and ${\mathbb{P}}_m(K)$ denotes restrictions of real polynomials of total degree at most $m$. Its conforming hp space is the subspace of functions whose two polynomial traces agree on every interior facet. The zero-boundary hp space further requires the polynomial trace to vanish on every boundary facet. For an interior facet $F=K\cap K^{\mathrm{\prime }}$, its trace degree is $p_F:=\min (p_K,p_{K^{\mathrm{\prime }}})$; equality of the two traces means that their common polynomial belongs to ${\mathbb{P}}_{p_F}(F)$.

A matching affine tensor-box mesh is a finite face-to-face partition of a bounded polytope into affine images of $[-1,1{]}^d$, with pairwise disjoint interiors. For a scalar degree $p_K$ on each box, the local tensor space is ${\mathbb{Q}}_{p_K}(K)$, the pullback of polynomials of degree at most $p_K$ in each reference coordinate. The conforming box hp space is defined by equality of polynomial traces across every shared face, and its zero-boundary subspace by zero traces on the exterior faces. For a shared face $F=K\cap K^{\mathrm{\prime }}$, set $p_F=\min (p_K,p_{K^{\mathrm{\prime }}})$; its common trace lies in ${\mathbb{Q}}_{p_F}(F)$ in tangential reference coordinates. This definition of tensor boxes does not include curved or non-affine cells.

On either mesh, h-refinement replaces one cell by a finite matching partition of that cell and assigns degrees to the children; p-refinement raises at least one cell degree while leaving the mesh fixed; hp-refinement performs both operations. A local polynomial basis is hierarchical if its ordered degree layers span the nested spaces ${\mathbb{P}}_0\subset \cdots \subset {\mathbb{P}}_p$ (or ${\mathbb{Q}}_0\subset \cdots \subset {\mathbb{Q}}_p$), with each new layer extending the preceding span. For example, pullbacks of reference monomials ${\widehat{x}}^{\nu }$ ordered by $\mathrm{\mid }\nu \mathrm{\mid }$ give such a basis for ${\mathbb{P}}_p$, and tensor monomials ${\xi }^{\nu }$ ordered by ${\max }_i{\nu }_i$ give one for ${\mathbb{Q}}_p$. This basis convention does not assert stability of a particular basis or projector.