How to implement hierarchical multigrid using hypre?

[olhaiy]

Hello,

we are trying to solve a scalar PDE via the hierarchical finite elements method with the hp-multigrid approach, based on the paper
Hierarchical multigrid approaches for the finite cell method on uniform and multi-level hp-refined grids

The hierarchical multigrid method stems from the choice of basis functions in FEM, specifically using high-order integrated Legendre shape functions.

The highest level of the multigrid contains all shape functions on the base level and all refinement levels. Then, the following reduction steps are performed:

• a reduction in the $p$-level
• a reduction in the levels of refinement until the base level is reached, (p.9, Figure 4).
  

The set of basis functions that belong to the multigrid level $\ell$ contains the set of basis functions that belong to the multigrid level $\ell-1$, (see p.8, eq. (8)).
Therefore, the vector $u_\ell$ of DOFs on level $\ell$ is made up of

• entries $w_\ell$ solely on level $\ell$
• coefficients $u_{\ell-1}$ from level $\ell-1$

$u_\ell=[w_\ell, u_{\ell-1} ]^T$

Hence,

$u_{\ell-1} =[ 0 , I]u_\ell$,

that is, the restriction operator is given by $R_\ell =[ 0 , I]$, and the prolongation $R_\ell = R_\ell^T$.
All restriction and prolongation operators reduce to binary matrices.

The hierarchical nature of the FE-basis is reflected in the system matrix of level $\ell$ (see p.9, eq. (11)):

where

• entries $\tilde{A}_{\ell}$ belong to basis functions contained solely in the highest level,
• a term ${A}_{\ell-1}$ contains entries of all lower levels up to $\ell-1$,
• a term $\tilde{A}_{\ell, \ell-1}$ couples degrees of freedom (DOFs) on level $\ell$ with all other DOFs.

Could you please advise on, how the matrix ${A}_{\ell}$ can be efficiently inverted using Hypre?

cc @karthichockalingam, @amg56