p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

Table 1 Notation for the p-multilevel solver. We only consider the equal-order version of Scheme I, where both element and face velocity unknowns have the same polynomial degree

Scheme index	Scheme label	Fine discrete space	Coarse discrete spaces	Coarsest level
I	HHO-dp	\({\underline{\varvec{V}}}_h^{k_0,k_0}\), \({\mathcal {P}}^{k_0}({\mathcal {T}}_h)\)	\({\underline{\varvec{V}}}_h^{k_\ell ,k_\ell }\), \({\mathcal {P}}^{k_\ell }({\mathcal {T}}_h)\)	\(k_L \geqslant 0\)
II	HHO-hp	\({\underline{\varvec{V}}}_h^{k_0+1,k_0}\), \({\mathcal {P}}^{k_0}({\mathcal {T}}_h)\)	\({\underline{\varvec{V}}}_h^{k_\ell +1,k_\ell }\), \({\mathcal {P}}^{k_\ell }({\mathcal {T}}_h)\)	\(k_L \geqslant 0\)
III	DG	\({\mathcal {P}}^{k_0}({\mathcal {T}}_h)^d\), \({\mathcal {P}}^{k_0}({\mathcal {T}}_h)\)	\({\mathcal {P}}^{k_\ell }({\mathcal {T}}_h)^d\), \({\mathcal {P}}^{k_\ell }({\mathcal {T}}_h)\)	\(k_L \geqslant 1\)